Browse Differentiation

$\text{f}'(x) = 4\sqrt{x} - \dfrac{k}{x^2}$ where $k$ is a constant
the equation of the tangent to the curve at $P$ has equation $y = 11x + c$ where $c$ is a constant

(a) (i) show that

k = 81

(ii) find the value of

c

(4 marks)
(b) Hence find the value of

\text{f}''(x)

P

.

(3 marks)
(c) Find

\text{f}(x)

.

(4 marks)

●●●●●Level 511 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = \text{f}(x)

x > 0

Given that

the point $P(9, 420)$ lies on $C$
$\text{f}'(x) = \dfrac{5x^2 + ax + b}{\sqrt{x}}$ , where $a$ and $b$ are constants
the gradient of the tangent to $C$ at $P$ is $120$

(a) show that

9a + b = -45

(2 marks)

Given also that

a + b = 3

(b) find, in simplest form,

\text{f}(x)

(7 marks)

Curve

C

is transformed to the curve with equation

y = \text{f}(x + 5)

Given that point

P

is transformed to the point

Q

,
(c) state the coordinates of

Q

.

(1 mark)

●●●●●Level 510 marksStart

An open rectangular tank is to be made with a square base of side

x

metres and vertical sides. It must hold

45

^{3}

of water.

The material for the base costs

10

dollars per square metre and the material for the four sides costs

3

dollars per square metre.

Find the least possible cost of the material for the tank, justifying that your value is a minimum. (9 marks)

●●●●●Level 59 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

Figure 1 shows a sketch of the curve

C

with equation

y = \mathrm{f}(x)

, where

\mathrm{f}(x) = (2x + 3)(x - 4)^2

(a) Deduce the range of values of

x

for which

\mathrm{f}(x) \geqslant 0

(1 mark)
(b) Find

\mathrm{f}'(x)

, giving your answer in simplest form.

(3 marks)

The point

R(-1,\ 25)

lies on

C

.

Given that the tangent to

C

at the point

P

is parallel to the tangent to

C

at the point

R

(c) (i) show that the

x

coordinate of

P

satisfies the equation

3x^2 - 13x - 16 = 0

(ii) Hence find the

x

coordinate of

P

.

(4 marks)
(d) Find the point to which

R

is transformed when the curve with equation

y = \mathrm{f}(x)

is transformed to the curve with equation

(i)

y = \mathrm{f}(x + 2)

(ii)

y = \mathrm{f}\left(\tfrac12 x\right)

(2 marks)

●●●●●Level 510 marksStart

Figure 1In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve with equation

y = 10x - 2x^{\frac{3}{2}} \qquad x \geqslant 0

The curve has a turning point at the point

M

, as shown in Figure 1.
(a) Using calculus, find the coordinates of

M

.

(5 marks)

The curve crosses the

x

-axis at the point

P

, as shown in Figure 1.
(b) Use algebra to find the

x

coordinate of

P

.

(2 marks)

The finite region

R

, shown shaded in Figure 1, is bounded by the curve, the line through

M

parallel to the

x

-axis and the line through

P

parallel to the

y

-axis.
(c) Use algebraic integration to find the area of

R

, giving your answer to one decimal place.

(5 marks)

●●●●●Level 512 marksStart

Figure 1Figure 1 shows a sketch of an open container in the shape of a prism.

The faces

ABFE

and

CDEF

are rectangles.

The ends

ADE

and

BCF

are congruent right-angled triangles, with the right angles at

A

and

B

.

The top

ABCD

is open, and the container is made from sheet steel of negligible thickness.

The diagram is not drawn to scale.

Given that

$AE = BF = 3x$ metres
$AD = BC = 4x$ metres
$AB = DC = EF = L$ metres

and the capacity of the container is

6

^3

(a) show that the area of steel used to make the container,

S

^2

, is given by

S = Px^2 + \dfrac{Q}{x}

where

P

and

Q

are positive integers to be found.

(4 marks)

Given that

x

can vary,
(b) use algebraic calculus to find the minimum value of

S

, giving your answer to one decimal place.

(5 marks)
(c) Justify that the value of

S

found in part (b) is a minimum.

(2 marks)

●●●●●Level 511 marksStart

Tangents and normals

55 questions

LessonNot started

The gradient of the normal to the curve

y = 3x^2 - 2

at the point where

x = 1

is

Circle your answer.

A $6$
B $-6$
C $\dfrac16$
D $-\dfrac16$

●●●●●Level 11 markStart

Find an equation of the tangent to the curve

y = (x - 3)^5

at the point where

x = 0

(3 marks)

●●●●●Level 23 marksStart

In this question you must show detailed reasoning.

The curve

C

has equation

y = \sqrt{2x+1}

\quad x \ge -\dfrac{1}{2}

,

and meets the

x

-axis at the point

A

. The tangent to

C

at the point

P

has gradient

\dfrac{1}{3}

.
(a) Find the coordinates of

P

. (3 marks)

The normal to

C

P

meets the

x

-axis at the point

Q

. The region

R

is bounded by the arc

AP

of the curve, the line segment

PQ

and the

x

-axis.
(b) Determine the exact area of

R

. (5 marks)

●●●●●Level 38 marksStart

The curve

C

has equation

y = 3\sin 2x + \cos 2x

.
(a) Find

\dfrac{dy}{dx}

. (2 marks)
(b) Find an equation of the tangent to

C

at the point where

x = \dfrac{\pi}{4}

. (3 marks)
(c) This tangent crosses the

x

-axis at the point

P

. Find the exact

x

-coordinate of

P

. (2 marks)

●●●●●Level 37 marksStart

The curve

C

has equation

y = x^2 - 5x + 6

.
(a) Find an equation of the tangent to

C

at the point where

x = 1

. (4 marks)
(b) This tangent meets the

x

-axis at

A

and the

y

-axis at

B

. Find the area of triangle

OAB

, where

O

is the origin. (4 marks)

●●●●●Level 38 marksStart

The curve

C

has equation

y = \sqrt{x}

.
(a) Find an equation of the tangent to

C

at the point

P(4, 2)

. (3 marks)
(b) Find an equation of the normal to

C

P

. (3 marks)

●●●●●Level 36 marksStart

The curve

C

has equation

y = 4e^{3x} - 2x

The point

P(0, 4)

lies on

C

.

Find an equation of the normal to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 34 marksStart

The tangent to the curve

y = x^2

at the point

(2, 4)

meets the

x

-axis at

A

and the

y

-axis at

B

. Find the area of triangle

OAB

, where

O

is the origin.

●●●●●Level 36 marksStart

(a) Show that the line

y = 4x - 4

is a tangent to the curve

y = x^2

. (4 marks)
(b) State the coordinates of the point of contact. (2 marks)

●●●●●Level 36 marksStart

The curve

C

has equation

x^2 + xy + y^2 = 12

Find the coordinates of the two points on

C

at which the tangent is parallel to the

x

-axis. (7 marks)

●●●●●Level 47 marksStart

A curve has equation

y = 4\sqrt{x} - \dfrac{2}{x} \quad \text{for } x > 0

(a) Find an expression for

\dfrac{dy}{dx}

(3 marks)
(b) The point

P

with coordinates

(4, 7.5)

lies on the curve.

Find an equation of the tangent to the curve at the point

P

(2 marks)
(c) Show that the curve has no stationary points.

(2 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

2x^2 + xy + y^2 = 16

.
(a) Show that

\dfrac{dy}{dx} = -\dfrac{4x + y}{x + 2y}

. (4 marks)
(b) The point

P(2, 2)

lies on

C

. Find an equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers. (2 marks)
(c) Find a relationship between

x

and

y

that holds at any point of

C

where the tangent is parallel to the

x

-axis. (1 mark)

●●●●●Level 47 marksStart

The curve

C

has parametric equations

x = 4\cos t, \qquad y = 3\sin t, \qquad 0 \leqslant t < 2\pi

(a) Find

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

. (3 marks)
(b) The point

P

C

has parameter

t = \dfrac{\pi}{3}

.

Find an equation of the normal to

C

P

, giving your answer in the form

y = mx + c

, where

m

and

c

are exact constants. (6 marks)

●●●●●Level 49 marksStart

The curve

C

has equation

x^2 - xy + y^2 = 7

. The point

A(3, 2)

lies on

C

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{y - 2x}{2y - x}

. (3 marks)
(b) Find the equation of the tangent to

C

A

. (2 marks)
(c) The normal to

C

A

meets the curve again at the point

B

. Determine the coordinates of

B

. (3 marks)

●●●●●Level 48 marksStart

The curve

C

has equation

x^2 + 3xy - y^2 = 3

The point

P(1, 2)

lies on

C

.

Find an equation of the tangent to

C

P

. (7 marks)

●●●●●Level 47 marksStart

A curve has equation

y = \dfrac{x^2}{2} + 6\sqrt{x} \quad \text{for } x > 0

(a) Find an expression for

\dfrac{dy}{dx}

(3 marks)
(b) The point

P

with coordinates

(4, 20)

lies on the curve.

Find an equation of the tangent to the curve at the point

P

(2 marks)
(c) Show that the curve has no stationary points.

(2 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

2x^2 - 3xy + 2y^2 = 14

The points

P

and

Q

lie on

C

. The tangent to

C

at each of the points

P

and

Q

is parallel to the

x

-axis.

Find the coordinates of

P

and

Q

, showing all your working. (7 marks)

●●●●●Level 47 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A curve

C

has equation

2x^2 + xy + y^2 = 44

(a) Find

\dfrac{dy}{dx}

in terms of

x

and

y

.

(4 marks)

The point

P(4, 2)

lies on

C

.
(b) Find an equation of the normal to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers to be found.

(3 marks)

●●●●●Level 47 marksStart

The curve

C

has parametric equations

x = t^2 - 4t - 12 \qquad\qquad y = 8\ln(t - 2) \qquad\qquad t > 2

(a) Show that a Cartesian equation for

C

y = A\ln(x + B) \qquad\qquad x > -B

where

A

and

B

are integers to be found.

(3 marks)

The curve

C

cuts the

y

-axis at the point

P

.
(b) Show that the equation of the normal to

C

P

can be written in the form

ax + by = c\ln 2

where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 47 marksStart

The curve

C

has parametric equations

x = t^2 - 14t + 40 \qquad\qquad y = 12\ln(t - 7) \qquad\qquad t > 7

(a) Show that a Cartesian equation for

C

y = A\ln(x + B) \qquad\qquad x > -B

where

A

and

B

are integers to be found.

(3 marks)

The curve

C

cuts the

y

-axis at the point

P

.
(b) Show that the equation of the tangent to

C

P

can be written in the form

ax + by = c\ln 3

where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 47 marksStart

The curve

C

has parametric equations

x = 2t^2 + 3 \qquad\qquad y = 3t^2 - t^3 \qquad\qquad 0 \leqslant t \leqslant 3

The point

P

with coordinates

(5, 2)

lies on

C

.
(a) Using parametric differentiation, show that the tangent to

C

P

has equation

3x - 4y = 7

(5 marks)

The curve

C

is used to model the profile of a humpback bridge. Units are in metres, with

y

being the height of the bridge above the level ground on either side of it.
(b) Find, according to the model, the greatest height of the bridge above the ground.

(1 mark)

●●●●●Level 46 marksStart

The curve

C

has parametric equations

x = t^3 + 6t \qquad\qquad y = 18 - 2t^2 \qquad\qquad -2 \leqslant t \leqslant 2

The point

P

with coordinates

(7, 16)

lies on

C

.
(a) Using parametric differentiation, show that the tangent to

C

P

has equation

4x + 9y = 172

(5 marks)

The curve

C

is used to model the profile of a section of a roller coaster track. Units are in metres, with

y

being the height of the track above the ground.
(b) Find, according to the model, the greatest height of the track above the ground.

(1 mark)

●●●●●Level 46 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = \text{f}(x)

, where

\text{f}'(x) = 12x - 3\cos x \qquad x \in \mathbb{R}

C

has a stationary point at

x = \alpha

, where

\alpha

is small and positive.
(a) Use the small angle approximation for

\cos x

to find an estimate for

\alpha

, giving your answer to 3 decimal places.

(3 marks)

C

passes through the point

P(0, 5)

.
(b) Find an equation of the tangent to

C

P

, giving your answer in the form

y = mx + c

where

m

and

c

are integers.

(2 marks)

●●●●●Level 45 marksStart

In this question you must show all stages of your working.

The curve

C

has equation

y = \text{f}(x)

x > 0

Given that

the point $P(3, -120)$ lies on $C$
$\text{f}'(x) = 3x^2 - k\sqrt{x} + 20$ where $k$ is a constant
$\text{f}''(x) = 0$ at $P$

(a) find the exact value of

k

,

(4 marks)
(b) find

\text{f}(x)

, giving your answer in simplest form.

(4 marks)

●●●●●Level 48 marksStart

Figure 1Figure 1 shows a sketch of the curve

C

with parametric equations

x = 5 - 2\cos 2\theta \qquad y = 6\sin\theta \qquad 0 \leqslant \theta \leqslant \dfrac{\pi}{2}

(a) Show that

\dfrac{dy}{dx} = k\operatorname{cosec}\theta \qquad \theta \neq 0

where

k

is a constant to be found.

(3 marks)

The point

P

lies on

C

where

\theta = \dfrac{\pi}{6}

(b) Find the equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

where

a

b

and

c

are integers.

(3 marks)
(c) Show that

C

has Cartesian equation

y^2 = 9(x - q) \qquad 3 \leqslant x \leqslant 7

where

q

is a constant to be found.

(3 marks)

●●●●●Level 49 marksStart

The diagram shows part of the curve

y = 4x - x^2

. The normal to the curve at the point

A(3, 3)

meets the curve again at the point

B

. The angle

AOB

\alpha

.(a) Determine the coordinates of

B

.

(6 marks)
(b) Hence determine the exact value of

\tan\alpha

.

(3 marks)

●●●●●Level 49 marksStart

The line

x + 2y = 24

is the normal to the curve

y = a\sqrt{x} + \dfrac{b}{x}

at the point where

x = 4

.

Determine the values of the constants

a

and

b

.

(8 marks)

●●●●●Level 48 marksStart

A curve has equation

x^3 - 6xy + 3y^2 = -7

.

Show that there are no points on this curve where the tangent is parallel to

y = x

.

(8 marks)

●●●●●Level 48 marksStart

The diagram shows the curve with equation

x^2 - 4xy + 5y^2 + 2y - k = 0

, where

k

is a positive integer.

At the points

P

and

Q

on the curve, the tangents to the curve are parallel to the

y

-axis.

Given that the difference in the

y

-coordinates of

P

and

Q

4

, determine the

x

-coordinates of

P

and

Q

.

(7 marks)

●●●●●Level 47 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of part of the curve with equation

y = \sqrt{3x - 11} \qquad x \geqslant \dfrac{11}{3}

The line

l

, shown in Figure 1, is the normal to the curve at the point

P(9, 4)

(a) Use calculus to show that an equation of

l

8x + 3y - 84 = 0

(5 marks)

The region

R

, shown shaded in Figure 1, is bounded by the curve, the

x

-axis and

l

.
(b) Use algebraic integration to find the exact area of

R

.

(4 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve

C

has equation

y = 3x^2 - 4x\sqrt{x} + 7x \qquad x > 0

(a) Find

\dfrac{\mathrm{d}y}{\mathrm{d}x}

, giving your answer in simplest form.

(2 marks)

Given that

the point $P$ lies on $C$
the line with equation $y = 19x + k$ , where $k$ is a constant, is the tangent to $C$ at $P$

(b) show that the

x

coordinate of

P

satisfies the equation

x - \sqrt{x} - 2 = 0

(2 marks)
(c) Hence find the value of

k

.

(4 marks)

●●●●●Level 48 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with equation

2^x + xy^2 - 5y = 1

The point

P

has coordinates

(2, 1)

.
(a) Verify that

P

lies on

C

.

(1 mark)
(b) Find

\dfrac{dy}{dx}

in terms of

x

and

y

.

(5 marks)

The normal to

C

P

crosses the

x

-axis at a point

Q

.
(c) Find the

x

coordinate of

Q

, giving your answer in the form

a + b\ln 2

where

a

and

b

are integers.

(3 marks)

●●●●●Level 49 marksStart

A curve

C

has equation

y = \text{f}(x)

x > 0

The point

P

with

x

coordinate

9

lies on

C

.

Given that

$\text{f}'(x) = 2x - \dfrac{k}{\sqrt{x}} + 1$ where $k$ is a constant
the normal to $C$ at $P$ has equation $x + 17y = 60$

(a) show that

k = 6

(3 marks)
(b) Hence find

\text{f}(x)

.

(4 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

x^2 + 2xy + 3y^2 = 6

(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{x + y}{x + 3y}

(3 marks)
(b) Hence find the exact coordinates of the two points on

C

at which the tangent is parallel to the

x

-axis. (4 marks)
(c) Show that there are exactly two points on

C

at which

\dfrac{\mathrm{d}y}{\mathrm{d}x}

is not defined, and find their coordinates. (3 marks)

●●●●●Level 510 marksStart

The curve

C

has parametric equations

x = t + \dfrac{1}{t}

y = t - \dfrac{1}{t}

t > 0

(a) Show that a Cartesian equation of

C

x^2 - y^2 = 4

. (2 marks)
(b) Find an equation of the normal to

C

at the point where

t = 2

, giving your answer in the form

ax + by = c

, where

a

b

and

c

are integers. (4 marks)
(c) The finite region

R

is bounded by

C

, the

x

-axis and the normal found in part (b).

Find the exact area of

R

. (5 marks)

●●●●●Level 511 marksStart

The curve

C

has equation

x^2 - 4xy + 5y^2 = 4

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2y - x}{5y - 2x}

. (3 marks)
(b) Find the exact coordinates of the two points on

C

at which the tangent is parallel to the

y

-axis. (3 marks)
(c) The point

P(4, 2)

lies on

C

. The normal to

C

P

meets

C

again at the point

Q

. Find the coordinates of

Q

. (3 marks)

●●●●●Level 59 marksStart

In this question you must show detailed reasoning.

The curve

C

has equation

x^2 - 2xy + 2y^2 = 8

The tangent to

C

at the point

P

is parallel to the

x

-axis, and the tangent to

C

at the point

Q

is parallel to the

y

-axis. Both

P

and

Q

lie in the first quadrant.

Determine the exact area of triangle

OPQ

, where

O

is the origin. (7 marks)

●●●●●Level 57 marksStart

The curve

C

has equation

x^2 - xy + y^2 = 12

(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2x - y}{x - 2y}

. (3 marks)
(b) Find the coordinates of the two points on

C

at which the tangent is parallel to the

y

-axis. (4 marks)
(c) The line with equation

y = x + k

, where

k

is a constant, is a tangent to

C

.

Find the possible values of

k

. (3 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y^3 + x^2y = 10

(a) Find

\dfrac{dy}{dx}

in terms of

x

and

y

.

(5 marks)

The point

P(3, 1)

lies on

C

.
(b) Show that the normal to

C

P

has equation

y = 2x - 5

(2 marks)
(c) Prove that the normal to

C

P

does not meet

C

again.

You should use algebra for your proof and make your reasoning clear.

(5 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = \frac{18}{2x + 3} \qquad x \geqslant 0

The point

P

C

has

x

coordinate

3

(a) Verify that the

y

coordinate of

P

2

(1 mark)

The line

l

is the tangent to

C

P

.
(b) Use calculus to show that an equation of

l

4x + 9y = 30

(3 marks)

The finite region

R

is bounded by the curve

C

, the line

l

and the

y

-axis.
(c) Use algebraic integration to find the exact area of

R

, giving your answer in the form

a\ln 3 + b

, where

a

and

b

are integers to be found.

(5 marks)

●●●●●Level 59 marksStart

The curve

C

has parametric equations

x = \tan t + 5 \qquad\qquad y = \sec^2 t + 2 \qquad\qquad -\frac{\pi}{2} < t < \frac{\pi}{4}

The line

l

is the normal to

C

at the point

P

where

t = -\dfrac{\pi}{4}

(a) Using parametric differentiation, show that an equation for

l

y = \tfrac{1}{2}x + 2

(5 marks)
(b) Show that all points on

C

satisfy the equation

y = (x - 5)^2 + 3

(2 marks)

The straight line with equation

y = \tfrac{1}{2}x + k

, where

k

is a constant, intersects

C

at two distinct points.
(c) Find the range of possible values for

k

.

(5 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = \text{f}(x)

where

x > 0

Given that

$\text{f}'(x) = (\sqrt{x} + 3)(2\sqrt{x} - 5)$
the point $P(9, -30)$ lies on $C$

(a) (i) show that the gradient of

C

P

6

(ii) Hence find the equation of the normal to

C

P

, giving your answer in the form

ax + by + c = 0

where

a

b

and

c

are integers to be found.

(4 marks)
(b) Find

\text{f}(x)

.

(6 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of the curve

C

with parametric equations

x = \cos 2t \qquad y = \sin t\cos 2t \qquad -\dfrac{\pi}{2} < t < \dfrac{\pi}{2}

where

t

is a parameter.

The point

P

lies on

C

where

t = \dfrac{\pi}{6}

The line

l

, shown in Figure 1, is the tangent to

C

P

.
(a) Use parametric differentiation to show that

(i)

\dfrac{dy}{dx} = p\sin t + q\operatorname{cosec} t

where

p

and

q

are constants to be found

(ii) an equation for

l

2x - 8y + 1 = 0

(6 marks)

The line

l

intersects the curve

C

again at the point

Q

.
(b) Using algebra and showing detailed reasoning, find the exact coordinates of

Q

.

(6 marks)

●●●●●Level 512 marksStart

A curve has parametric equations

x = 2t^2

y = t^3

. The point

P

on the curve has parameter

p

, where

p > 0

.
(a) Show that the equation of the tangent at

P

can be written as

4y = 3px - 2p^3

.

(4 marks)

The tangent to this curve at

P

crosses the

x

-axis at the point

A

and the normal to this curve at

P

crosses the

x

-axis at the point

B

.
(b) Show that the ratio

PA : PB

4 : 3p

.

(8 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.The diagram shows the curve with equation

(3x - y)^2 = 12x + 4y

.
(a) Show that

\dfrac{dy}{dx} = \dfrac{9x - 3y - 6}{3x - y + 2}

.

(3 marks)

At the point

P

on the curve the tangent to the curve is parallel to the

y

-axis and at the point

Q

on the curve the tangent to the curve is parallel to the

x

-axis.
(b) Show that the distance

PQ

k\sqrt{10}

, where

k

is a rational number to be determined.

(8 marks)

●●●●●Level 511 marksStart

(a) Use the substitution

u = 4 - x

to show that

\int x\sqrt{4 - x}\,dx = -\tfrac{2}{15}(3x + 8)(4 - x)^{\frac32} + c.

(5 marks)
(b) In this question you must show detailed reasoning.The diagram shows part of the curve

y = x\sqrt{4 - x}

, for

0 \le x \le 4

, and the normal to the curve at the point

P(3, 3)

. The shaded region is enclosed by the curve, the normal to the curve at

P

and the

x

-axis.

Find the exact area of the shaded region.

(7 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.The diagram shows the curve

y = \tan x

, for

0 \le x < \tfrac12\pi

, and the normal to the curve at the point

P\left(\tfrac13\pi, \sqrt3\right)

.

Determine the exact area of the shaded region enclosed by the curve, the normal to the curve and the

y

-axis, giving your answer in the form

p\pi\sqrt3 + q\pi^2 + r\ln 2

, where

p

q

and

r

are rational numbers to be determined.

(10 marks)

●●●●●Level 510 marksStart

Figure 1In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve

C

with equation

y = 2x^{\frac{3}{2}} - 9x + 30 \qquad x \geqslant 0

(a) Find, using calculus, the range of values of

x

for which

y

is increasing.

(4 marks)

The point

P

lies on

C

and has coordinates

(16, 14)

.

The line

l

is the tangent to

C

at the point

P

.

The finite region

R

, shown shaded in Figure 1, is bounded by the curve

C

, the line

l

, the

x

-axis and the

y

-axis.
(b) Find, using calculus, the exact area of

R

.

(8 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A curve has equation

y = \text{f}(x)

x > 0

The point

P(9, 40)

lies on the curve.

Given that

$\text{f}'(x) = 4\sqrt{x} - \dfrac{k}{x^2}$ where $k$ is a constant
the equation of the tangent to the curve at $P$ has equation $y = 11x + c$ where $c$ is a constant

(a) (i) show that

k = 81

(ii) find the value of

c

(4 marks)
(b) Hence find the value of

\text{f}''(x)

P

.

(3 marks)
(c) Find

\text{f}(x)

.

(4 marks)

●●●●●Level 511 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = \text{f}(x)

x > 0

Given that

the point $P(9, 420)$ lies on $C$
$\text{f}'(x) = \dfrac{5x^2 + ax + b}{\sqrt{x}}$ , where $a$ and $b$ are constants
the gradient of the tangent to $C$ at $P$ is $120$

(a) show that

9a + b = -45

(2 marks)

Given also that

a + b = 3

(b) find, in simplest form,

\text{f}(x)

(7 marks)

Curve

C

is transformed to the curve with equation

y = \text{f}(x + 5)

Given that point

P

is transformed to the point

Q

,
(c) state the coordinates of

Q

.

(1 mark)

●●●●●Level 510 marksStart

Figure 1

Figure 1 shows a sketch of part of the curve

C

with equation

y = \mathrm{f}(x)

, where

\mathrm{f}(x) = (x + 3)(2x - 1)(x - 4)

(a) Use the given information to state the values of

x

for which

\mathrm{f}(x) > 0

(2 marks)
(b) Expand

(x + 3)(2x - 1)(x - 4)

, writing your answer as a polynomial in simplest form.

(3 marks)

The straight line

l

is the tangent to

C

at the point where

C

cuts the

y

-axis.

Given that

l

cuts

C

at the point

P

, as shown in Figure 1,
(c) find, using algebra, the

x

coordinate of

P

(Solutions relying on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

Figure 1 shows a sketch of the curve

C

with equation

y = \mathrm{f}(x)

, where

\mathrm{f}(x) = (2x + 3)(x - 4)^2

(a) Deduce the range of values of

x

for which

\mathrm{f}(x) \geqslant 0

(1 mark)
(b) Find

\mathrm{f}'(x)

, giving your answer in simplest form.

(3 marks)

The point

R(-1,\ 25)

lies on

C

.

Given that the tangent to

C

at the point

P

is parallel to the tangent to

C

at the point

R

(c) (i) show that the

x

coordinate of

P

satisfies the equation

3x^2 - 13x - 16 = 0

(ii) Hence find the

x

coordinate of

P

.

(4 marks)
(d) Find the point to which

R

is transformed when the curve with equation

y = \mathrm{f}(x)

is transformed to the curve with equation

(i)

y = \mathrm{f}(x + 2)

(ii)

y = \mathrm{f}\left(\tfrac12 x\right)

(2 marks)

●●●●●Level 510 marksStart

Figure 1

Figure 1 shows a sketch of the curve

C

with equation

y = 2x^2 - 12x + 22

(a) Write

2x^2 - 12x + 22

in the form

a(x + b)^2 + c

where

a

b

and

c

are constants to be found.

(3 marks)

The point

M

is the minimum turning point of

C

, as shown in Figure 1.
(b) Deduce the coordinates of

M

(2 marks)

The line

l

is the normal to

C

at the point

P

, as shown in Figure 1.

Given that

l

has equation

x - 4y + k = 0

, where

k

is a constant,
(c) (i) find the coordinates of

P

(ii) show that

k = 22

(6 marks)

The finite region

R

, shown shaded in Figure 1, is bounded by

C

l

and the

y

-axis.
(d) Use inequalities to define the region

R

.

(3 marks)

●●●●●Level 514 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with equation

y = \text{f}(x)

where

\text{f}(x) = \dfrac{x^2 + 4x - 5}{x + 1} \qquad x > -1

(a) Find

\text{f}'(x)

, giving the answer in simplest form.

(3 marks)

The line

l

is the normal to

C

at the point

P(3, 4)

(b) Show that an equation for

l

2x + 3y = 18

(3 marks)
(c) Write

\text{f}(x)

in the form

Ax + B + \dfrac{D}{x + 1}

where

A

B

and

D

are constants.

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by

C

l

and the

x

-axis.
(d) Use algebraic integration to find the exact area of

R

, giving your answer in the form

p + q\ln 2

, where

p

and

q

are rational constants.

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 514 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with parametric equations

x = t^2 + 2t \qquad y = 2t - \dfrac{2}{t} \qquad t > 0

The curve

C

intersects the

x

-axis at the point

Q

.
(a) Find the

x

coordinate of

Q

.

(1 mark)

The line

l

is the normal to

C

at the point

P

as shown in Figure 1.

Given that

t = 2

P

(b) write down the coordinates of

P

(1 mark)
(c) Using calculus, show that an equation of

l

12x + 5y = 111

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by the curve

C

, the line

l

and the

x

-axis.
(d) Using algebraic integration, find the exact volume of the solid of revolution formed when the region

R

is rotated through

2\pi

radians about the

x

-axis. Give your answer in the form

\pi(p + q\ln 2)

where

p

and

q

are rational constants.

(7 marks)

●●●●●Level 512 marksStart

Stationary points (maxima and minima)

66 questions

LessonNot started

The curve

C

has equation

y = 2x^3 - 9x^2 + 12x

.
(a) Find the coordinates of the two stationary points of

C

. (4 marks)
(b) Determine the nature of each stationary point. (3 marks)

●●●●●Level 37 marksStart

In this question you must show detailed reasoning.

A curve has equation

y = x^3 - 6x^2 + 15x - 2

.
(a) Show that the curve has no stationary points.

(2 marks)
(b) Show that the curve has exactly one point of inflection.

(2 marks)

●●●●●Level 34 marksStart

The curve

C

has equation

y = \frac{x^2 + 5}{x + 2} \qquad x \in \mathbb{R},\ x \neq -2

(a) Find

\dfrac{dy}{dx}

, giving your answer as a single fraction in its simplest form.

(3 marks)
(b) Hence find the

x

coordinates of the points on

C

at which the tangent to

C

is parallel to the

x

-axis.

(2 marks)

●●●●●Level 35 marksStart

The curve

C

has equation

y = \text{f}(x)

The curve

passes through the point $P(-2, 7)$
has a turning point at $P$

Given that

\frac{dy}{dx} = 6x^2 + 2x + k

where

k

is a constant,
(a) show that

k = -20

(2 marks)
(b) Hence find the coordinates of the point where

C

crosses the

y

-axis.

(3 marks)

●●●●●Level 35 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A curve has equation

y = \dfrac{3x - 5}{(x - 1)^2} \qquad x \neq 1,\ x \in \mathbb{R}

Use calculus to find the range of values of

x

for which

y

is increasing.

(6 marks)

●●●●●Level 36 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = 2x^{\frac{3}{2}} + \dfrac{96}{\sqrt{x}} - 15 \qquad x > 0

(a) Find

\dfrac{dy}{dx}

, giving each term in simplest form.

(2 marks)
(b) Hence find the

x

coordinate of the stationary point of

C

.

(2 marks)
(c) (i) Find

\dfrac{d^2y}{dx^2}

, giving each term in simplest form.

(ii) Hence determine the nature of the stationary point of

C

, giving a reason for your answer.

(2 marks)
(d) State the range of values of

x

for which

y

is decreasing.

(1 mark)

●●●●●Level 37 marksStart

Figure 1The curve

C

, shown in Figure 1, has equation

2y^2 - x^2y + 12x = k \qquad y > 0

where

k

is a constant.
(a) Find

\dfrac{dy}{dx}

in terms of

x

and

y

(5 marks)

The point

P(p, 3)

, where

p

is a constant, lies on

C

.

Given that

P

is the minimum turning point on

C

,
(b) find

(i) the value of

p

(ii) the value of

k

(4 marks)

●●●●●Level 49 marksStart

The curve

C

has equation

y = x^3 - 6x^2 + 9x

.(a) Show that

y = x(x - 3)^2

, and hence write down the coordinates of the points where

C

meets the

x

-axis. (2 marks)
(b) Find the coordinates of the stationary points of

C

, and determine the nature of each. (4 marks)
(c) Find the area of the finite region enclosed by

C

and the

x

-axis. (3 marks)

●●●●●Level 49 marksStart

Figure 1 shows the curve

C

with equation

y = x^2\sqrt{3 - 2x}, \qquad x < \dfrac32

Figure 1
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{x(6 - 5x)}{\sqrt{3 - 2x}}

. (4 marks)
(b) Find the exact coordinates of the stationary points of

C

. (3 marks)

●●●●●Level 47 marksStart

A curve has equation

y = x + \dfrac{4}{x-2}

, for

x > 2

.

Find the coordinates of the stationary point of the curve and determine its nature. (6 marks)

●●●●●Level 46 marksStart

A curve has equation

y = 4\sqrt{x} - \dfrac{2}{x} \quad \text{for } x > 0

(a) Find an expression for

\dfrac{dy}{dx}

(3 marks)
(b) The point

P

with coordinates

(4, 7.5)

lies on the curve.

Find an equation of the tangent to the curve at the point

P

(2 marks)
(c) Show that the curve has no stationary points.

(2 marks)

●●●●●Level 47 marksStart

Relative to a fixed origin

O

, the points

A

and

B

have position vectors

\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}

and

3\mathbf{i} + 2\mathbf{j} - \mathbf{k}

respectively.

The point

P

lies on the straight line through

A

and

B

, and is the point on this line that is closest to

O

.
(a) Find the position vector of

P

. (4 marks)
(b) Hence find the exact area of triangle

OAB

. (3 marks)

●●●●●Level 47 marksStart

Figure 5 shows an open-topped storage tank in the shape of a cuboid. The base is a square of side

x

metres and the height of the tank is

h

metres.Figure 5 (not accurately drawn)

The tank must hold exactly

32

m³ of liquid. The base is made from a reinforced material costing £

15

per m², and the four vertical sides are made from a lighter material costing £

6

per m².
(a) Show that the total cost, £

C

, of the material used to make the tank is given by

C = 15x^2 + \dfrac{768}{x}

(4 marks)
(b) Use calculus to find the value of

x

for which

C

is a minimum, giving your answer to

3

significant figures, and prove that your value gives a minimum. (5 marks)
(c) Hence find the minimum cost of the material, to the nearest pound. (1 mark)

●●●●●Level 410 marksStart

The curve

C

has equation

y = x^3 e^{-x}

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = x^2 e^{-x}(3 - x)

. (2 marks)
(b) Find the coordinates of the stationary points of

C

. (2 marks)
(c) Determine the nature of each stationary point. (3 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = \dfrac{2x + 1}{\sqrt{x}}, \qquad x > 0

Find the exact coordinates of the stationary point of

C

. (6 marks)

●●●●●Level 46 marksStart

A curve has equation

y = \dfrac{\ln x}{x^2}

\quad x > 0

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}

. (3 marks)
(b) Determine the exact coordinates of the stationary point of the curve, and show that it is a maximum. (4 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = \dfrac{x}{x^2 + 1}

(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - x^2}{\left(x^2 + 1\right)^2}

. (3 marks)
(b) Find the coordinates of the two stationary points of

C

, and determine the nature of each. (4 marks)

●●●●●Level 47 marksStart

\text{f}(x) = 5\sin 2x - 3x + 2 \qquad 0 \le x \le \pi

where

x

is measured in radians.

The curve with equation

y = \text{f}(x)

has a single maximum turning point in the interval

0 \le x \le \pi

(a) Using calculus, find the

x

coordinate of this maximum turning point, giving your answer to 3 decimal places.

(4 marks)

The equation

\text{f}(x) = 0

has a single root

\alpha

in the interval

0 \le x \le \pi

Given that

\text{f}(1.3) = 0.678

and

\text{f}(1.4) = -0.525

, to 3 decimal places,
(b) explain why

\alpha

must lie in the interval

[1.3,\ 1.4]

(1 mark)
(c) Using

x_0 = 1.4

as a first approximation for

\alpha

, apply the Newton–Raphson method once to obtain a second approximation for

\alpha

.

Show your method and give your answer to 3 significant figures.

(2 marks)

●●●●●Level 47 marksStart

A curve has equation

y = \dfrac{x^2}{2} + 6\sqrt{x} \quad \text{for } x > 0

(a) Find an expression for

\dfrac{dy}{dx}

(3 marks)
(b) The point

P

with coordinates

(4, 20)

lies on the curve.

Find an equation of the tangent to the curve at the point

P

(2 marks)
(c) Show that the curve has no stationary points.

(2 marks)

●●●●●Level 47 marksStart

The curve

y = 18 - x^2

and the isosceles triangle

OPQ

are shown on the diagram below.Vertices

P

and

Q

lie on the curve such that

Q

lies vertically above some point

(q, 0)

The line

PQ

is parallel to the

x

-axis.
(a) Show that the area,

A

, of the triangle

OPQ

is given by

A = 18q - q^3 \quad \text{for } 0 < q < k

where

k

is a constant to be found.

(3 marks)
(b) Find the exact maximum area of triangle

OPQ

Fully justify your answer.

(6 marks)

●●●●●Level 49 marksStart

The curve

y = 27 - 3x^2

and the rectangle

ABCD

are shown on the diagram below.Vertices

A

and

B

lie on the

x

-axis and vertices

C

and

D

lie on the curve.

B

has coordinates

(q, 0)

and

C

lies vertically above

B

(a) Show that the area,

A

, of the rectangle

ABCD

is given by

A = 54q - 6q^3 \quad \text{for } 0 < q < k

where

k

is a constant to be found.

(3 marks)
(b) Find the exact maximum area of rectangle

ABCD

Fully justify your answer.

(6 marks)

●●●●●Level 49 marksStart

A curve has equation

xe^{2y} + y^2 - 3x^2 = k

where

k

is a constant.

The curve has a stationary point at which

y = \ln 3

Determine the exact value of

k

(7 marks)

●●●●●Level 47 marksStart

A curve has equation

2y^3e^{3x} + 5y + 48x = k

where

k

is a constant.

The curve has a stationary point on the

y

-axis.

Determine the value of

k

(7 marks)

●●●●●Level 47 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with equation

y = 3\sqrt{x}\,(8 - x) \qquad x \geqslant 0

The point

P

is the stationary point of

C

.
(a) Find, using calculus, the

x

coordinate of

P

.

(4 marks)

The region

R_1

, shown shaded in Figure 1, is bounded by

C

and the

x

-axis.

The region

R_2

, also shown shaded in Figure 1, is bounded by

C

, the

x

-axis and the line with equation

x = k

, where

k

is a constant.

Given that the area of

R_1

is equal to the area of

R_2

(b) find, using calculus, the exact value of

k

.

(4 marks)

●●●●●Level 48 marksStart

A curve

C

has equation

x^2 + xy + y^2 = 12

(a) Show that

\frac{dy}{dx} = -\frac{2x + y}{x + 2y}

(4 marks)
(b) Hence find the coordinates of the two points on

C

at which the tangent to

C

is parallel to the

x

-axis.

(4 marks)

●●●●●Level 48 marksStart

The curve

C

has equation

y = (2x - 1)e^{3x}

(a) Show that

\frac{dy}{dx} = (6x - 1)e^{3x}

(2 marks)
(b) Hence find the exact coordinates of the stationary point of

C

.

(3 marks)
(c) Use further differentiation to determine the nature of this stationary point.

(2 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = x^2 \ln x

\quad x > 0

.
(a) Show that

\dfrac{dy}{dx} = x(2\ln x + 1)

. (3 marks)
(b) Hence find the exact coordinates of the stationary point of

C

. (3 marks)
(c) Determine the nature of this stationary point. (2 marks)
(d) Find the range of values of

x

for which

C

is convex. (1 mark)

●●●●●Level 49 marksStart

[For a cone with base radius

r

, height

h

and slant height

l

, the following formulae are given.

Curved surface area,

S = \pi rl

Volume,

V = \tfrac13\pi r^2h

]

A sweet shop makes paper cones, open at the top, to hold sweets. Each cone has base radius

r

cm and holds a volume of

V

^3

when full. Each cone uses

25\pi

^2

of paper.
(a) Show that

V = \tfrac13\pi r\sqrt{625 - r^4}

.

(4 marks)
(b) In this question you must show detailed reasoning.

It is given that

V

has a maximum value for a certain value of

r

.

Find the maximum value of

V

, giving your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 49 marksStart

(a) Differentiate

(x^2 - 2x + 4)e^{3x}

with respect to

x

.

(3 marks)
(b) Hence show that

(x^2 - 2x + 4)e^{3x}

is increasing for all values of

x

.

(4 marks)

●●●●●Level 47 marksStart

A curve has equation

y = 2^{x^2 - 6x}

.
(a) Determine the coordinates of any stationary points on the curve.

(4 marks)
(b) Show that the curve is convex for all values of

x

.

(5 marks)

●●●●●Level 49 marksStart

A curve has parametric equations

x = \dfrac{2t^2 + 1}{t}

and

y = \dfrac{t^2 - 3}{t}

, for

t \ne 0

.
(a) Find

\dfrac{dy}{dx}

in terms of

t

, giving your answer in its simplest form.

(4 marks)
(b) Explain why the curve has no stationary points.

(2 marks)
(c) By considering

x - 2y

, or otherwise, find a cartesian equation of the curve, giving your answer in a form not involving fractions or brackets.

(4 marks)

●●●●●Level 410 marksStart

Figure 1Figure 1 shows an open-topped tray used for growing seedlings.

The tray is in the shape of a cuboid and is made of thin sheet plastic.

The base of the tray is a rectangle

5x

cm by

2x

cm.

The height of the tray is

h

cm, as shown in Figure 1.

The diagram is not drawn to scale.

Given that the capacity of the tray is

1500

^3

(a) show that the area

S

^2

of the plastic used to make the tray is given by

S = Px^2 + \dfrac{Q}{x}

where

P

and

Q

are positive constants to be found.

(4 marks)
(b) Use calculus to find the value of

x

for which

S

has a stationary value, giving your answer to 3 significant figures.

(3 marks)
(c) Find

\dfrac{d^2S}{dx^2}

and hence show that the value of

x

found in part (b) gives the minimum value of

S

.

(2 marks)

●●●●●Level 49 marksStart

Figure 1Figure 1 shows a sketch of the curve

C

with equation

y = \text{f}(x)

, where

\text{f}(x) = \dfrac{2x + 12}{\sqrt{2x + 3}} \qquad x > -\dfrac{3}{2}

(a) Show that

\text{f}'(x) = \dfrac{k(x - 3)}{(2x + 3)^{\frac{3}{2}}}

where

k

is a constant to be found.

(4 marks)

The point

P

, shown in Figure 1, is the minimum turning point on

C

.
(b) Find the coordinates of

P

.

(2 marks)
(c) Hence find the range of the function g defined by

\text{g}(x) = -3\text{f}(x) \qquad x > -\dfrac{3}{2}

(2 marks)

The point

Q

with coordinates

(-1, 10)

lies on

C

.
(d) Find the coordinates of the point to which

Q

is mapped when

C

is transformed to the curve with equation

y = \dfrac{1}{2}\text{f}(2x - 5) + 4

(2 marks)

●●●●●Level 410 marksStart

The time taken,

X

hours, for a mechanic to service a car is modelled by the continuous random variable

X

with cumulative distribution function given by

\mathrm{F}(x) = \begin{cases} 0 & x < 0 \\ k(8x^2 - x^4) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}

where

k

is a positive constant.
(a) Show that

k = \dfrac{1}{16}

(1 mark)
(b) Find

\mathrm{P}(X > 1.5)

(2 marks)
(c) (i) Show that the median,

m

, of

X

satisfies

m^4 + pm^2 + q = 0

where

p

and

q

are integers to be found.

(ii) Hence find the value of

m

(3 marks)
(d) Find, using calculus, the mode of

X

(3 marks)

Given that

\mathrm{E}(X) = \dfrac{16}{15}

(e) state, giving a reason, the skewness of the distribution.

(2 marks)

●●●●●Level 411 marksStart

The curve

C

has equation

y = \dfrac{x^4 - 6x^3 + 13x^2 - 14x}{\sqrt{x}} \qquad x > 0

(a) Write

y

in the form

x^{\frac{7}{2}} + ax^{\frac{5}{2}} + bx^{\frac{3}{2}} + cx^{\frac{1}{2}}

, where

a

b

and

c

are constants to be found.

(2 marks)
(b) Using calculus, show that the

x

coordinates of the turning points of

C

satisfy

7x^3 - 30x^2 + 39x - 14 = 0

(4 marks)

Given that one of the turning points of

C

has

x

coordinate 2
(c) find, using algebra, the exact

x

coordinates of the other two turning points of

C

.

(3 marks)

●●●●●Level 49 marksStart

(a) The graph of the function

y = f(x)

passes through the point

P

with coordinates

(-3, 8)

, and is a one-one function. State the coordinates of the point corresponding to

P

on each of the following curves.

(i)

y = f(x - 4)

(ii)

y = 3f(x) + 2

(iii)

y = f^{-1}(x)

(4 marks)
(b) The diagram shows part of the graph of

y = g'(x)

. This is the graph of the gradient function of

y = g(x)

. The graph of

y = g'(x)

meets the

x

-axis at

x = -3

and

x = 3

only, and has a maximum point where

x = -1

.

(i) State the

x

-coordinate of any stationary points on the graph of

y = g(x)

.

(ii) State the set of values of

x

for which

y = g(x)

is a decreasing function.

(iii) State the

x

-coordinate of any points of inflection on the graph of

y = g(x)

.

(3 marks)

●●●●●Level 47 marksStart

The curve

y = (x + 3)\text{e}^{-x} + x^2

has one stationary point for

x > 0

, which is close to

x = 0.7

.
(a) Show that the

x

-coordinate of this stationary point satisfies the equation

2x\text{e}^{x} - x - 2 = 0

.

(2 marks)
(b) Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a) can be written in the form

x_{n+1} = \frac{2x_n^2\text{e}^{x_n} + 2}{2(x_n + 1)\text{e}^{x_n} - 1}.

(4 marks)
(c) Apply the Newton-Raphson formula with initial value

x_1 = 0.7

to find

x_2

and

x_3

.

(1 mark)
(d) Find the coordinates of this stationary point, giving each coordinate correct to 3 decimal places.

(2 marks)

●●●●●Level 49 marksStart

An open-topped box has a square base of side

x

cm and height

h

cm. The total surface area (the base and four sides) is

48\ \text{cm}^2

.
(a) Show that the volume is

V = 12x - \tfrac{1}{4}x^3

.
(b) Find the value of

x

that maximises

V

, and the maximum volume, justifying that it is a maximum.

●●●●●Level 57 marksStart

A closed cylinder has volume

1000\ \text{cm}^3

. Its radius is

r

cm.
(a) Show that the total surface area is

S = 2\pi r^2 + \dfrac{2000}{r}

. (3 marks)
(b) Find the value of

r

that minimises

S

, to 3 s.f. (4 marks)
(c) Justify that this value gives a minimum. (2 marks)

●●●●●Level 59 marksStart

Figure 2 shows the curve

C

with equation

y = \dfrac{8}{x} + 2x, \qquad x > 0

together with the line

l

with equation

y = 10

. The region

R

, shown shaded, is bounded by

C

and

l

.Figure 2
(a) Find the coordinates of the stationary point of

C

, and determine its nature. (5 marks)
(b) Find the

x

-coordinates of the two points where

l

meets

C

. (3 marks)
(c) Find the exact area of

R

. (5 marks)
(d) Write down an equation of each of the two asymptotes of

C

. (2 marks)

●●●●●Level 515 marksStart

\mathrm{f}(x) = (2x - 1)\mathrm{e}^{-x}

(a) Show that

\mathrm{f}'(x) = (3 - 2x)\mathrm{e}^{-x}

, and hence find the exact coordinates of the stationary point of the curve

y = \mathrm{f}(x)

. (5 marks)
(b) Determine the nature of the stationary point. (3 marks)
(c) Find the exact value of

\displaystyle\int_0^1 (2x - 1)\mathrm{e}^{-x}\;\mathrm{d}x

. (3 marks)

●●●●●Level 511 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x^2}, \qquad x > 0

(a) Find the exact coordinates of the stationary point of

C

, and determine its nature. (6 marks)
(b) The finite region

R

is bounded by

C

, the

x

-axis and the line

x = \mathrm{e}

.

Find the exact area of

R

. (4 marks)

●●●●●Level 510 marksStart

A closed cylinder of radius

r

and height

h

is inscribed in a sphere of radius

6

cm, so that both circular ends of the cylinder touch the sphere.(a) Show that the volume of the cylinder,

V

cm³, is given by

V = 36\pi h - \dfrac{\pi h^3}{4}

. (4 marks)
(b) Use calculus to find the value of

h

that maximises

V

, justifying that your value gives a maximum. Give the maximum volume in the form

k\sqrt{3}\,\pi

cm³. (6 marks)
(c) Show that the ratio of the maximum cylinder volume to the volume of the sphere is

\sqrt{3} : 3

. (4 marks)

●●●●●Level 514 marksStart

The curve

C

has equation

2x^3 + y^2 - 6xy = 0

C

passes through the origin

O

and has a stationary point at

P

, where

P

is not

O

, as shown in the diagram below.(a) Show that the

x

-coordinate of any stationary point of

C

satisfies the equation

x^3(x - 4) = 0

(7 marks)
(b) Hence, find the coordinates of

P

(2 marks)

●●●●●Level 59 marksStart

A flat metal weather vane is to be made in the shape shown in the diagram below.

The points

P

O

and

B

lie on a straight line with

PO = 2OB

.

The shape consists of the sector

OBD

of a circle with centre

O

and radius

OB

, together with the triangle

OPD

.

The angle

POD

\theta

radians, where

0 < \theta < \pi

The weather vane is to be made with

PB = 1.5

metres.
(a) Show that the area of the weather vane,

A

square metres, is given by

A = \frac{1}{8}\left(\pi - \theta + 2\sin\theta\right)

(4 marks)
(b) (i) Show that the maximum value of

A

occurs when

\theta = \dfrac{\pi}{3}

Fully justify your answer.

(ii) Find the exact maximum value of

A

(8 marks)
(c) Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if

PB

were increased to 3 metres.

(2 marks)

●●●●●Level 514 marksStart

The curve

C

has parametric equations

x = 3\sin\theta

\qquad y = \sin 2\theta

\qquad 0 \le \theta \le \dfrac{\pi}{2}

.
(a) Find the exact coordinates of the maximum point

M

C

. (4 marks)

The finite region

R

is bounded by

C

and the

x

-axis.
(b) Find the exact area of

R

. (3 marks)
(c) Show that the line through

M

parallel to the

y

-axis divides

R

into two regions whose areas are in the ratio

\left(2\sqrt{2} - 1\right) : 1

. (3 marks)

●●●●●Level 510 marksStart

In this question you must show detailed reasoning.

Figure 1 shows the curve

C

with equation

y = x\sqrt{4 - x}, \qquad 0 \leqslant x \leqslant 4

The region

R

, shown shaded, is bounded by

C

and the

x

-axis.Figure 1
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{8 - 3x}{2\sqrt{4 - x}}

. (4 marks)
(b) Find the exact coordinates of the maximum point of

C

. (3 marks)
(c) Using the substitution

u = 4 - x

, show that the area of

R

\dfrac{128}{15}

. (5 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.

Figure 1 shows a closed box in the shape of a cuboid with a square base of side

x

cm and height

h

cm. The total surface area of the box is

600

cm².Figure 1 (not accurately drawn)
(a) Show that the volume

V

cm³ of the box is given by

V = 150x - \dfrac{x^3}{2}

. (3 marks)
(b) Find the value of

x

for which

V

is a maximum, and find the maximum volume. You must show that your value gives a maximum. (5 marks)
(c) Find the corresponding value of

h

, and describe the shape of the box in this case. (2 marks)

●●●●●Level 510 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x}, \qquad x > 0.

(a) Determine the exact coordinates of the stationary point of

C

, and show that it is a maximum point. (5 marks)
(b) The equation

\ln x = kx,

where

k

is a constant, has exactly two distinct real roots. Determine the set of possible values of

k

, justifying your answer fully. (4 marks)

●●●●●Level 59 marksStart

A rectangular enclosure of area

200\ \text{m}^2

is built against a straight wall, so only three sides need fencing. The side parallel to the wall has length

x

m.
(a) Show that the total length of fencing is

L = x + \dfrac{400}{x}

. (3 marks)
(b) Find the value of

x

that minimises the fencing used, and the minimum length. (4 marks)

●●●●●Level 57 marksStart

A rectangle has its base on the

x

-axis and its two upper vertices on the curve

y = 12 - x^2

(with

y \ge 0

), symmetric about the

y

-axis. The upper-right vertex is at

(x, 12 - x^2)

.
(a) Show that the area of the rectangle is

A = 24x - 2x^3

. (2 marks)
(b) Find the value of

x

that maximises the area, and the maximum area. (5 marks)

●●●●●Level 57 marksStart

The diagram below shows the approximate shape of the vertical cross section of a natural rock arch above a horizontal footpath.

The inside of the arch meets the footpath at the points

O

and

P

Rhiannon models the shape of the inside of the arch using the equation

x^2 + 2y^2 = a\sqrt{x}

where

a

is a constant, and

x

and

y

are the horizontal and vertical distances respectively, in metres, measured from

O

(a) The distance

OP

5

metres.

Find the exact value of

a

that Rhiannon should use in the model.

(2 marks)
(b) Show that the maximum height of the inside of the arch above the footpath is approximately

2.43

metres.

(6 marks)
(c) Suggest one limitation of the model Rhiannon has used.

(1 mark)

●●●●●Level 59 marksStart

The function

\text{f}

is defined by

\text{f}(x) = \frac{x^2 + 7}{2x + 2}

where

\text{f}

has its maximum possible domain.

The curve

y = \text{f}(x)

intersects the line

y = x

at the points

P

and

Q

as shown below.(a) State the value of

x

which is not in the domain of

\text{f}

(1 mark)
(b) Explain how you know that the function

\text{f}

is many-to-one.

(2 marks)
(c) (i) Show that the

x

-coordinates of

P

and

Q

satisfy the equation

x^2 + 2x - 7 = 0

(ii) Hence, find the exact

x

-coordinate of

P

and the exact

x

-coordinate of

Q

(3 marks)
(d) Show that

P

and

Q

are stationary points of the curve.

Fully justify your answer.

(5 marks)
(e) Using set notation, state the range of

\text{f}

(2 marks)

●●●●●Level 513 marksStart

Figure 4 shows part of the curve

C

with equation

y = \dfrac{\ln x}{x^2}, \qquad x > 0

Figure 4
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}

. (3 marks)
(b) Find the exact coordinates of the stationary point of

C

, and determine its nature. (5 marks)
(c) Find the exact

x

-coordinate of the point of inflection of

C

. (2 marks)

●●●●●Level 510 marksStart

A company makes open-topped containers for storing grain.

A container is modelled so that

the base of the container is a circle of radius $r$ cm
the curved surface of the container is perpendicular to the base
the container has no lid
the total area of the base and the curved surface is $300\pi$ cm $^2$

The volume of the container is

V

^3

.
(a) Show that

V = \frac{\pi}{2}\left(300r - r^3\right)

making your method clear.

(4 marks)

Using algebraic differentiation,
(b) find the value of

r

for which

V

has a stationary point.

(4 marks)
(c) Prove, by further differentiation, that this value of

r

gives the maximum volume of the container.

(2 marks)
(d) Hence find the maximum volume of the container, giving your answer in terms of

\pi

.

(1 mark)

●●●●●Level 511 marksStart

A curve has equation

y = \ln(3 + 6\sin x)

, where

x

is in radians.
(a) State the values of

x

, for

0 \le x \le 2\pi

, for which

\ln(3 + 6\sin x)

is not defined.

(2 marks)
(b)

P

is the stationary point on the curve that has the smallest positive

x

-coordinate.

Determine the exact coordinates of

P

.

(4 marks)
(c) (i) Show that

\dfrac{d^2y}{dx^2} + 3e^{-y} + 27e^{-2y} = 0

.

(ii) State what can be deduced about all of the stationary points on this curve, giving a reason for your answer.

(6 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.

The function f is defined by

f(x) = \sin 2x + 2\sin x

with domain

0 \le x \le 2\pi

.
(a) Solve the following equations.

(i)

f'(x) = 0

(ii)

f''(x) = 0

(7 marks)

The diagram shows the graph of the gradient function

y = f'(x)

for the domain

0 \le x \le 2\pi

. The points

A

C

and

E

are where the graph meets the

x

-axis, and

B

and

D

are minimum points.(b) Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points

A

B

C

D

and

E

.

(3 marks)
(c) (i) Explain how to use the graph of the gradient function to find the values of

x

for which

f(x)

is increasing.

(ii) Using set notation, write down the set of values of

x

for which

f(x)

is increasing in the domain

0 \le x \le 2\pi

.

(3 marks)

●●●●●Level 513 marksStart

In this question you must show detailed reasoning.

A curve has equation

y = \dfrac{x - 1}{x^2 + 3}

.
(a) State the coordinates of the points where the curve crosses the axes.

(2 marks)

The point

A

is where the curve crosses the

y

-axis. The point

B

on the curve has

x

-coordinate

-3

.
(b) Show that the line

AB

is parallel to the

x

-axis.

(2 marks)
(c) Determine the coordinates of the turning points of the curve.

(4 marks)
(d) Determine whether each turning point is a maximum or a minimum.

(5 marks)

●●●●●Level 513 marksStart

A manufacturer makes cylindrical food cans of radius

r

cm. Each can is closed at both ends and has a volume of

250\pi

^3

. The metal for the curved surface costs

0.2

pence per cm

^2

and the metal for the two ends costs

0.6

pence per cm

^2

.
(a) Show that the total cost,

C

pence, of the metal for one can is given by

C = 1.2\pi r^2 + 100\pi r^{-1}

●●●●●Level 511 marksStart

The diagram shows the curve

C

with parametric equations

x = e^{t}, \qquad y = 4t - t^2, \qquad \text{where } 0 \le t \le 4.

The maximum point on

C

is denoted by

P

.
(a) Determine the exact coordinates of

P

.

(4 marks)

The shaded region

R

is enclosed by the curve, the

x

-axis and the lines

x = e

and

x = e^3

.
(b) Show that the area of

R

is given by

\displaystyle\int_a^b (4t - t^2)e^{t}\,dt

, where

a

and

b

are constants to be determined.

(3 marks)
(c) Hence determine the exact area of

R

.

(5 marks)

●●●●●Level 512 marksStart

Figure 1In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve

C

with equation

y = 2x^{\frac{3}{2}} - 9x + 30 \qquad x \geqslant 0

(a) Find, using calculus, the range of values of

x

for which

y

is increasing.

(4 marks)

The point

P

lies on

C

and has coordinates

(16, 14)

.

The line

l

is the tangent to

C

at the point

P

.

The finite region

R

, shown shaded in Figure 1, is bounded by the curve

C

, the line

l

, the

x

-axis and the

y

-axis.
(b) Find, using calculus, the exact area of

R

.

(8 marks)

●●●●●Level 512 marksStart

An open rectangular tank is to be made with a square base of side

x

metres and vertical sides. It must hold

45

^{3}

of water.

The material for the base costs

10

dollars per square metre and the material for the four sides costs

3

dollars per square metre.

Find the least possible cost of the material for the tank, justifying that your value is a minimum. (9 marks)

●●●●●Level 59 marksStart

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.Figure 1

Figure 1 shows

the curve $C_1$ with equation $y = x^{3} - 3x + 4$
the circle $C_2$ with centre $T$

The point

T

is the minimum turning point of

C_1

Using Figure 1 and calculus,
(a) find the coordinates of

T

(3 marks)

The point

A

, with

x

coordinate

2

, lies on both

C_1

and

C_2

(b) Find an equation of the circle

C_2

(3 marks)

The line

l

, shown in Figure 1, is the tangent to circle

C_2

A

.
(c) Show that an equation of

l

x + 4y = 26

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by

C_1

l

and the

y

-axis.
(d) Find the exact area of

R

.

(3 marks)

●●●●●Level 512 marksStart

y = 10x - 2x^{\frac{3}{2}} \qquad x \geqslant 0

The curve has a turning point at the point

M

, as shown in Figure 1.
(a) Using calculus, find the coordinates of

M

.

(5 marks)

The curve crosses the

x

-axis at the point

P

, as shown in Figure 1.
(b) Use algebra to find the

x

coordinate of

P

.

(2 marks)

The finite region

R

, shown shaded in Figure 1, is bounded by the curve, the line through

M

parallel to the

x

-axis and the line through

P

parallel to the

y

-axis.
(c) Use algebraic integration to find the area of

R

, giving your answer to one decimal place.

(5 marks)

●●●●●Level 512 marksStart

Figure 1Figure 1 shows a sketch of an open container in the shape of a prism.

The faces

ABFE

and

CDEF

are rectangles.

The ends

ADE

and

BCF

are congruent right-angled triangles, with the right angles at

A

and

B

.

The top

ABCD

is open, and the container is made from sheet steel of negligible thickness.

The diagram is not drawn to scale.

Given that

$AE = BF = 3x$ metres
$AD = BC = 4x$ metres
$AB = DC = EF = L$ metres

and the capacity of the container is

6

^3

(a) show that the area of steel used to make the container,

S

^2

, is given by

S = Px^2 + \dfrac{Q}{x}

where

P

and

Q

are positive integers to be found.

(4 marks)

Given that

x

can vary,
(b) use algebraic calculus to find the minimum value of

S

, giving your answer to one decimal place.

(5 marks)
(c) Justify that the value of

S

found in part (b) is a minimum.

(2 marks)

●●●●●Level 511 marksStart

A square sheet of card of side

20

cm has a square of side

x

cm removed from each corner. The sides are folded up to make an open box.
(a) Show that the volume is

V = x(20 - 2x)^2

. (2 marks)
(b) Find the value of

x

that maximises

V

. (4 marks)
(c) Find the maximum volume. (2 marks)

●●●●●Level 58 marksStart

Differentiating trigonometric functions

31 questions

LessonNot started

Differentiate

y = \sin 4x + \cos 2x

. (2 marks)

●●●●●Level 22 marksStart

Differentiate

y = \cos(5x)

●●●●●Level 22 marksStart

Differentiate

y = \sin(3x^2)

●●●●●Level 22 marksStart

Differentiate

y = \tan(2x)

●●●●●Level 22 marksStart

(a) Differentiate the following with respect to

x

.

(i)

\ln(3x^2 + 5)

(ii)

x\cos 4x

(5 marks)
(b) Find

\displaystyle\int \sin 3x\,dx

.

(2 marks)
(c) The gradient of a curve is given by

\dfrac{dy}{dx} = 4e^{2x}

and the curve passes through the point

(0, 5)

.

Find the equation of the curve.

(2 marks)

●●●●●Level 29 marksStart

Differentiate

y = \sin(x^{2} + 1)

. (2 marks)

●●●●●Level 22 marksStart

Differentiate

y = \cos^{3} x

. (3 marks)

●●●●●Level 33 marksStart

The curve

C

has equation

y = 3\sin 2x + \cos 2x

.
(a) Find

\dfrac{dy}{dx}

. (2 marks)
(b) Find an equation of the tangent to

C

at the point where

x = \dfrac{\pi}{4}

. (3 marks)
(c) This tangent crosses the

x

-axis at the point

P

. Find the exact

x

-coordinate of

P

. (2 marks)

●●●●●Level 37 marksStart

Differentiate

y = \tan^{2} x

. (3 marks)

●●●●●Level 33 marksStart

Differentiate

y = x^2\cos x

●●●●●Level 32 marksStart

Differentiate

y = \sin^2 x

, and write your answer as a single trig term.

●●●●●Level 32 marksStart

Find

\dfrac{dy}{dx}

for
(a)

y = \sec x

and (b)

y = \cot x

, deriving each from the quotient rule.

●●●●●Level 34 marksStart

A curve has equation

y = x\sin x

. Show that

\dfrac{dy}{dx} = \sin x + x\cos x

, and find the gradient at the point where

x = \dfrac{\pi}{2}

●●●●●Level 33 marksStart

The curve

C

has equation

y = \sin x + \cos x

for

0 \leqslant x \leqslant \pi

.
(a) Find the coordinates of the stationary point of

C

, giving exact values. (4 marks)
(b) Determine its nature. (1 mark)

●●●●●Level 35 marksStart

The function f is defined by

\mathrm{f}(x) = x^3 + \sin 2x \quad \text{for } -\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}

Determine whether the curve with equation

y = \mathrm{f}(x)

has a point of inflection at the point where

x = 0

Fully justify your answer.

(4 marks)

●●●●●Level 34 marksStart

Find an equation of the tangent to the curve

y = 2\sin x

at the point where

x = \dfrac{\pi}{3}

, giving your answer exactly. (4 marks)

●●●●●Level 34 marksStart

Differentiate

\sin x

with respect to

x

from first principles.

(4 marks)

●●●●●Level 34 marksStart

Differentiate

y = \operatorname{cosec} x

from

y = (\sin x)^{-1}

●●●●●Level 33 marksStart

Differentiate

y = x\tan x

. (3 marks)

●●●●●Level 33 marksStart

\text{f}(x) = x\cos x + 2 \qquad 0 \leqslant x \leqslant \pi

where

x

is measured in radians.
(a) Find

\text{f}'(x)

(2 marks)

The curve with equation

y = \text{f}(x)

has a maximum turning point at

x = \alpha

(b) Show that

\alpha

lies in the interval

[0.8, 0.9]

(2 marks)
(c) Using

x_0 = 0.8

as a first approximation for

\alpha

, apply the Newton–Raphson method once to

\text{f}'(x)

to find a second approximation for

\alpha

, giving your answer to 3 decimal places.

(3 marks)

●●●●●Level 47 marksStart

\text{f}(x) = 5\sin 2x - 3x + 2 \qquad 0 \le x \le \pi

where

x

is measured in radians.

The curve with equation

y = \text{f}(x)

has a single maximum turning point in the interval

0 \le x \le \pi

(a) Using calculus, find the

x

coordinate of this maximum turning point, giving your answer to 3 decimal places.

(4 marks)

The equation

\text{f}(x) = 0

has a single root

\alpha

in the interval

0 \le x \le \pi

Given that

\text{f}(1.3) = 0.678

and

\text{f}(1.4) = -0.525

, to 3 decimal places,
(b) explain why

\alpha

must lie in the interval

[1.3,\ 1.4]

(1 mark)
(c) Using

x_0 = 1.4

as a first approximation for

\alpha

, apply the Newton–Raphson method once to obtain a second approximation for

\alpha

.

Show your method and give your answer to 3 significant figures.

(2 marks)

●●●●●Level 47 marksStart

A continuous curve has equation

y = x^3\cos 2x \qquad 0 \leqslant x \leqslant \dfrac{\pi}{4}

The curve has a stationary point at the point

Q

, where

x > 0

(a) Show, using calculus, that the

x

coordinate of

Q

is a solution of the equation

x = \dfrac{1}{2}\arctan\left(\dfrac{3}{2x}\right)

(4 marks)

Using the iteration formula

x_{n+1} = \dfrac{1}{2}\arctan\left(\dfrac{3}{2x_n}\right) \qquad x_1 = 0.6

(b) find, to 4 decimal places, the value of

(i)

x_2

(ii)

x_3

(3 marks)
(c) Using a suitable interval and a suitable function which should be stated, show that the

x

coordinate of

Q

0.596

correct to 3 decimal places.

(2 marks)

●●●●●Level 49 marksStart

A water company is investigating how the amount of water used in a town depends on the air temperature. The investigation runs for several weeks during the summer. A model is proposed for the amount of water used per day,

W

megalitres, and the mean daily temperature,

T

°C, at time

t

weeks after the start of the investigation.

In the model

W = 25e^{0.04t}

and

T = 15 + 6\sin(0.2t)

, where

0.2t

is in radians.

The model assumes that

W

and

T

can be treated as continuous variables.
(a) State the meaning of

\dfrac{dW}{dT}

.

(1 mark)
(b) Determine

\dfrac{dW}{dT}

when

t = 5

.

(4 marks)
(c) Suggest a reason why this model may not be valid for values of

t

greater than

15

.

(1 mark)

●●●●●Level 46 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of the curve with equation

y = \text{f}(x)

, where

\text{f}(x) = 3\cos x\,\text{e}^{2\sin x} \qquad 0 \leqslant x \leqslant \pi

The curve intersects the

x

-axis at the point

R

, as shown in Figure 1.
(a) State the coordinates of

R

(1 mark)

The curve has two turning points, at the point

P

and the point

Q

, also shown in Figure 1.
(b) Show that, at

P

and at

Q

a\sin^2 x + b\sin x + c = 0

where

a

b

and

c

are integers to be found.

(4 marks)
(c) Hence find the

x

coordinate of

Q

, giving your answer to 3 decimal places.

(2 marks)

●●●●●Level 47 marksStart

A flat metal weather vane is to be made in the shape shown in the diagram below.

The points

P

O

and

B

lie on a straight line with

PO = 2OB

.

The shape consists of the sector

OBD

of a circle with centre

O

and radius

OB

, together with the triangle

OPD

.

The angle

POD

\theta

radians, where

0 < \theta < \pi

The weather vane is to be made with

PB = 1.5

metres.
(a) Show that the area of the weather vane,

A

square metres, is given by

A = \frac{1}{8}\left(\pi - \theta + 2\sin\theta\right)

(4 marks)
(b) (i) Show that the maximum value of

A

occurs when

\theta = \dfrac{\pi}{3}

Fully justify your answer.

(ii) Find the exact maximum value of

A

(8 marks)
(c) Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if

PB

were increased to 3 metres.

(2 marks)

●●●●●Level 514 marksStart

(a) Given that

y = \sec\theta

(a) (i) Express

y

in terms of

\cos\theta

.

(ii) Hence, prove that

\frac{\mathrm{d}y}{\mathrm{d}\theta} = \sec\theta\tan\theta

(iii) Show that

\frac{\sqrt{y^2 - 1}}{y} = \sin\theta \quad \text{for } 0 < \theta < \frac{\pi}{2}

(7 marks)
(b) (i) Use the substitution

x = 4\sec u

to show that

\int \frac{1}{\left(x^2 - 16\right)^{\frac{3}{2}}} \, \mathrm{d}x \quad \text{for } x > 4

can be written as

k\int \operatorname{cosec} u \cot u \, \mathrm{d}u

where

k

is a constant to be found.

(ii) Hence, show that

\int \frac{1}{\left(x^2 - 16\right)^{\frac{3}{2}}} \, \mathrm{d}x = -\frac{x}{16\sqrt{x^2 - 16}} + c \quad \text{for } x > 4

where

c

is a constant.

(9 marks)

●●●●●Level 516 marksStart

The diagram shows the cross-section of a horizontal pipe, which is a circle with centre

O

and radius

50

cm. The pipe contains some water. In the cross-section, the surface of the water is the chord

AB

and the angle

AOB

\theta

radians. The water occupies the shaded segment bounded by the chord

AB

and the arc

AB

, which has area

900

^2

.(a) Show that

\theta = \sin\theta + 0.72

.

(3 marks)
(b) Let

\text{F}(\theta) = \sin\theta + 0.72

.

By considering the value of

\text{F}'(\theta)

where

\theta = 1.7

, explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that

1.7

is sufficiently close to the root.

(2 marks)
(c) Use the iterative formula

\theta_{n+1} = \sin\theta_n + 0.72

with a starting value of

1.7

to find the value of

\theta

correct to 4 significant figures.

You should show the result of each iteration.

(3 marks)
(d) Use a change of sign method to show that the value of

\theta

found in part (c) is correct to 4 significant figures.

(3 marks)

●●●●●Level 511 marksStart

A curve has equation

y = \ln(3 + 6\sin x)

, where

x

is in radians.
(a) State the values of

x

, for

0 \le x \le 2\pi

, for which

\ln(3 + 6\sin x)

is not defined.

(2 marks)
(b)

P

is the stationary point on the curve that has the smallest positive

x

-coordinate.

Determine the exact coordinates of

P

.

(4 marks)
(c) (i) Show that

\dfrac{d^2y}{dx^2} + 3e^{-y} + 27e^{-2y} = 0

.

(ii) State what can be deduced about all of the stationary points on this curve, giving a reason for your answer.

(6 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.

The function f is defined by

f(x) = \sin 2x + 2\sin x

with domain

0 \le x \le 2\pi

.
(a) Solve the following equations.

(i)

f'(x) = 0

(ii)

f''(x) = 0

(7 marks)

The diagram shows the graph of the gradient function

y = f'(x)

for the domain

0 \le x \le 2\pi

. The points

A

C

and

E

are where the graph meets the

x

-axis, and

B

and

D

are minimum points.(b) Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points

A

B

C

D

and

E

.

(3 marks)
(c) (i) Explain how to use the graph of the gradient function to find the values of

x

for which

f(x)

is increasing.

(ii) Using set notation, write down the set of values of

x

for which

f(x)

is increasing in the domain

0 \le x \le 2\pi

.

(3 marks)

●●●●●Level 513 marksStart

In this question you must show detailed reasoning.The diagram shows the curve

y = \tan x

, for

0 \le x < \tfrac12\pi

, and the normal to the curve at the point

P\left(\tfrac13\pi, \sqrt3\right)

.

Determine the exact area of the shaded region enclosed by the curve, the normal to the curve and the

y

-axis, giving your answer in the form

p\pi\sqrt3 + q\pi^2 + r\ln 2

, where

p

q

and

r

are rational numbers to be determined.

(10 marks)

●●●●●Level 510 marksStart

A curve has equation

y = k\cos x + x - x^2

, where

x

is in radians and

k

is a positive constant.
(a) Given that the curve has a point of inflection where

x = \tfrac23\pi

, show that

k = 4

.

(5 marks)

It is also given that the curve crosses the positive

x

-axis at exactly one point.
(b) Show by calculation that the

x

-coordinate of this point lies between

1.2

and

1.6

.

(2 marks)
(c) Starting with

x_0 = 1.4

, use the Newton-Raphson method to find this

x

-coordinate correct to 5 decimal places. Show the result of each iteration to 6 decimal places.

(3 marks)
(d) By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.

(1 mark)

●●●●●Level 511 marksStart

Differentiating exponentials and logarithms

55 questions

LessonNot started

\displaystyle\lim_{h \to 0} \dfrac{e^{3h} - 1}{h}

is equal to

Circle your answer.

A $0$
B $1$
C $3$
D $e^3$

●●●●●Level 11 markStart

Identify the gradient of the curve

y = \ln(3x)

at the point where

x = 2

.

Circle your answer.

A $\dfrac{3}{2}$
B $\dfrac{1}{6}$
C $\dfrac{1}{2}$
D $3$

●●●●●Level 11 markStart

The curve

y = e^{3x}

has gradient at

x = 0

equal to

Circle your answer.

A $0$
B $1$
C $3$
D $e^3$

●●●●●Level 11 markStart

Differentiate

y = \mathrm{e}^{5x} + \mathrm{e}^{-2x}

. (2 marks)

●●●●●Level 22 marksStart

Differentiate

y = \ln(x^{3})

, simplifying your answer. (2 marks)

●●●●●Level 22 marksStart

Differentiate

y = e^{3x}

●●●●●Level 22 marksStart

Differentiate

y = \ln(3x^2 + 1)

●●●●●Level 22 marksStart

Differentiate

y = e^{2x^2 - 1}

●●●●●Level 22 marksStart

A curve has equation

y = 6e^{-2x}

Find the gradient of the curve at the point where

y = 15

(3 marks)

●●●●●Level 23 marksStart

Differentiate

(x^2 + 4x - 1)e^{-x}

with respect to

x

, giving your answer in the form

e^{-x}(ax^2 + bx + c)

, where

a

b

and

c

are integers to be determined.

(3 marks)

●●●●●Level 23 marksStart

Differentiate the following with respect to

x

.
(a)

\dfrac{6}{x^3} + 4\sqrt{x}

(2 marks)
(b)

x^2e^{3x}

(2 marks)
(c)

\dfrac{2x + 1}{x - 3}

(3 marks)

●●●●●Level 27 marksStart

(a) Differentiate the following with respect to

x

.

(i)

\ln(3x^2 + 5)

(ii)

x\cos 4x

(5 marks)
(b) Find

\displaystyle\int \sin 3x\,dx

.

(2 marks)
(c) The gradient of a curve is given by

\dfrac{dy}{dx} = 4e^{2x}

and the curve passes through the point

(0, 5)

.

Find the equation of the curve.

(2 marks)

●●●●●Level 29 marksStart

Differentiate

y = 2^x

, giving your answer exactly.

●●●●●Level 32 marksStart

Find an equation of the tangent to the curve

y = \ln x

at the point where

x = \mathrm{e}

. (3 marks)

●●●●●Level 33 marksStart

Differentiate

y = (\mathrm{e}^{2x} + 1)^{3}

. (3 marks)

●●●●●Level 33 marksStart

(a) Differentiate

y = 3^{x}

, giving your answer exactly. (2 marks)
(b) Hence find the exact gradient of the curve at

x = 2

. (1 mark)

●●●●●Level 33 marksStart

The mass

m

mg of a radioactive isotope in a sample after

t

days is modelled by

m = 240e^{-kt}

,

where

k

is a positive constant. When

t = 8

, the mass has fallen to

180

mg.
(a) Show that

k = \frac{1}{8}\ln\frac{4}{3}

, and evaluate

k

to 3 significant figures. (3 marks)
(b) Find the mass predicted by the model after 30 days. (1 mark)
(c) Find, to 3 significant figures, the number of days it takes for the mass to halve. (2 marks)
(d) Find the rate at which the mass is decreasing when

t = 10

. (2 marks)

●●●●●Level 38 marksStart

The curve

C

has equation

y = x^2 e^x

. Find the coordinates of the stationary points of

C

●●●●●Level 34 marksStart

The curve

C

has equation

y = \ln(x^{2} + 4)

.
(a) Find

\dfrac{dy}{dx}

. (2 marks)
(b) Hence find the coordinates of the stationary point of

C

and determine its nature. (2 marks)

●●●●●Level 34 marksStart

The function f is defined by

\mathrm{f}(x) = e^{2x} + e^{-2x} - 4x^2 \quad \text{for } x \in \mathbb{R}

Determine whether the curve with equation

y = \mathrm{f}(x)

has a point of inflection at the point where

x = 0

Fully justify your answer.

(4 marks)

●●●●●Level 34 marksStart

Differentiate

y = \dfrac{\ln x}{x}

, and hence find the

x

-coordinate of the stationary point.

●●●●●Level 33 marksStart

Show that the curve

y = x\ln x - x

has gradient

\ln x

, and hence find the coordinates of its stationary point.

●●●●●Level 34 marksStart

The curve

C

has equation

y = 4e^{3x} - 2x

The point

P(0, 4)

lies on

C

.

Find an equation of the normal to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 34 marksStart

Differentiate

y = \dfrac{e^x}{x}

●●●●●Level 32 marksStart

\text{f}(x) = e^{\frac{1}{2}x} \qquad x > 0

\text{g}(x) = 40\ln x \qquad x > 0

(a) Find

(i)

\text{f}'(x)

(ii)

\text{g}'(x)

(2 marks)

Given that

\text{f}'(x) = \text{g}'(x)

x = \alpha

(b) show that

\alpha

satisfies the equation

xe^{\frac{1}{2}x} - 80 = 0

(2 marks)

The iterative formula

x_{n+1} = 2\ln\left(\frac{80}{x_n}\right)

is used with

x_1 = 5

to find an approximate value for

\alpha

.
(c) Calculate, giving each answer to 4 decimal places,

(i) the value of

x_2

(ii) the value of

\alpha

(3 marks)

●●●●●Level 47 marksStart

f(x) = e^{2x} - 8e^x + 12

.
(a) Solve the equation

f(x) = 0

, giving your answers exactly. (4 marks)
(b) Find the exact coordinates of the stationary point of the curve

y = f(x)

, and state its nature. (4 marks)
(c) Show that

\displaystyle\int_0^{\ln 2} f(x)\; dx = 12\ln 2 - \frac{13}{2}

. (3 marks)

●●●●●Level 411 marksStart

A cup of coffee is left to cool in a room. Its temperature,

\theta

°C,

t

minutes after it is made, is modelled by

\theta = 20 + 55e^{-0.08t}

\quad t \ge 0

.
(a) State the initial temperature of the coffee. (1 mark)
(b) Find the temperature of the coffee after

10

minutes. (2 marks)
(c) Find, to the nearest tenth of a minute, the time at which the temperature reaches

30

°C. (3 marks)
(d) Find the rate at which the coffee is cooling after

10

minutes. (3 marks)
(e) State, with a reason, the significance of the constant

20

in the model. (1 mark)

●●●●●Level 410 marksStart

A particle is moving in a straight line through the origin

O

The displacement of the particle,

r

metres, from

O

, at time

t

seconds is given by

r = p + 1.5t - q\text{e}^{-0.4t}

where

p

and

q

are constants.

When

t = 3

, the acceleration of the particle is

-0.5\ \text{m s}^{-2}

(a) Show that

q \approx 10.4

(5 marks)
(b) The particle has an initial displacement of

-6

metres.

Find the value of

p

Give your answer to two significant figures.

(2 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = x^2 e^{-x}

.
(a) Show that

\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = (x^2 - 4x + 2)e^{-x}

. (3 marks)
(b) Hence find the exact

x

-coordinates of the points of inflection of

C

. Fully justify your answer. (4 marks)

●●●●●Level 47 marksStart

Two colonies of insects are being monitored. At time

t

days after monitoring began, the number of insects in colony

A

is modelled by

N_A = 1200e^{0.08t}

and the number of insects in colony

B

is modelled by

N_B = 4000e^{-0.05t}

(a) Find the value of

t

at which the two colonies contain the same number of insects, giving your answer to

2

decimal places. (3 marks)
(b) Find the rate, in insects per day, at which colony

A

is growing at this value of

t

, giving your answer to the nearest integer. (3 marks)

●●●●●Level 46 marksStart

The curve

C

has equation

y = \mathrm{e}^{3x} - 3x

.
(a) Find the coordinates of the stationary point of

C

. (3 marks)
(b) Determine its nature. (2 marks)

●●●●●Level 45 marksStart

A curve has equation

xe^{2y} + y^2 - 3x^2 = k

where

k

is a constant.

The curve has a stationary point at which

y = \ln 3

Determine the exact value of

k

(7 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = (2x - 1)e^{3x}

(a) Show that

\frac{dy}{dx} = (6x - 1)e^{3x}

(2 marks)
(b) Hence find the exact coordinates of the stationary point of

C

.

(3 marks)
(c) Use further differentiation to determine the nature of this stationary point.

(2 marks)

●●●●●Level 47 marksStart

The curve

C

has parametric equations

x = t^2 - 4t - 12 \qquad\qquad y = 8\ln(t - 2) \qquad\qquad t > 2

(a) Show that a Cartesian equation for

C

y = A\ln(x + B) \qquad\qquad x > -B

where

A

and

B

are integers to be found.

(3 marks)

The curve

C

cuts the

y

-axis at the point

P

.
(b) Show that the equation of the normal to

C

P

can be written in the form

ax + by = c\ln 2

where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 47 marksStart

\text{f}(x) = x - 3\ln x \qquad x > 0

(a) Show that the equation

\text{f}(x) = 0

has a root

\alpha

in the interval

[4.5,\ 5]

(2 marks)
(b) Find

\text{f}'(x)

(2 marks)
(c) Using

x_0 = 5

as a first approximation for

\alpha

, apply the Newton–Raphson method once to obtain a second approximation for

\alpha

.

Give your answer to 3 decimal places.

(2 marks)

A student attempts to use the Newton–Raphson method with

x_0 = 3

as a first approximation for

\alpha

.
(d) Explain why the method fails.

(1 mark)

●●●●●Level 47 marksStart

The curve

C

has parametric equations

x = t^2 - 14t + 40 \qquad\qquad y = 12\ln(t - 7) \qquad\qquad t > 7

(a) Show that a Cartesian equation for

C

y = A\ln(x + B) \qquad\qquad x > -B

where

A

and

B

are integers to be found.

(3 marks)

The curve

C

cuts the

y

-axis at the point

P

.
(b) Show that the equation of the tangent to

C

P

can be written in the form

ax + by = c\ln 3

where

a

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = x^2 \ln x

\quad x > 0

.
(a) Show that

\dfrac{dy}{dx} = x(2\ln x + 1)

. (3 marks)
(b) Hence find the exact coordinates of the stationary point of

C

. (3 marks)
(c) Determine the nature of this stationary point. (2 marks)
(d) Find the range of values of

x

for which

C

is convex. (1 mark)

●●●●●Level 49 marksStart

(a) Differentiate

(x^2 - 2x + 4)e^{3x}

with respect to

x

.

(3 marks)
(b) Hence show that

(x^2 - 2x + 4)e^{3x}

is increasing for all values of

x

.

(4 marks)

●●●●●Level 47 marksStart

A curve has equation

y = 2^{x^2 - 6x}

.
(a) Determine the coordinates of any stationary points on the curve.

(4 marks)
(b) Show that the curve is convex for all values of

x

.

(5 marks)

●●●●●Level 49 marksStart

W

megalitres, and the mean daily temperature,

T

°C, at time

t

weeks after the start of the investigation.

In the model

W = 25e^{0.04t}

and

T = 15 + 6\sin(0.2t)

, where

0.2t

is in radians.

The model assumes that

W

and

T

can be treated as continuous variables.
(a) State the meaning of

\dfrac{dW}{dT}

.

(1 mark)
(b) Determine

\dfrac{dW}{dT}

when

t = 5

.

(4 marks)
(c) Suggest a reason why this model may not be valid for values of

t

greater than

15

.

(1 mark)

●●●●●Level 46 marksStart

A young fish is released into a lake.

The length,

L

cm, of the fish

t

years after it is released into the lake is modelled by the equation

L = 64 - 52\mathrm{e}^{-0.3t} \qquad t \geqslant 0

Use the equation of the model to answer parts (a) to (d).
(a) Find the length of the fish when it is released into the lake.

(1 mark)
(b) Sketch the graph of

L

against

t

for

t \geqslant 0

On your sketch, state the equation of the asymptote to the curve.

(2 marks)
(c) Find, according to the model, the time taken for the length of the fish to reach

48

cm. Give your answer in years to 2 decimal places.

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)
(d) Show by differentiation that

\frac{dL}{dt} = a + bL

where

a

and

b

are constants to be found.

(3 marks)

●●●●●Level 49 marksStart

(a) The mass of substance A,

m

grams, at time

t

minutes decays exponentially. Find the missing values of

m

and

t

in the following table.

$t$	$0$	$6$	$12$
$m$	$800$	$600$		$337.5$

(2 marks)
(b) Substance B is also decaying exponentially. Its mass after

t

minutes is

m

grams, where

m = 500e^{-0.04t}

.

(i) Determine the time at which the mass of substance B has fallen to half of its initial value.

(ii) Determine the rate of decay of substance B when

t = 10

.

(6 marks)
(c) State whether substance A or substance B is decaying at a faster rate, giving a reason for your answer.

(1 mark)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A kiln is used to fire pottery. The kiln is switched off and the pottery is left inside it to cool.

The temperature,

\theta\ ^\circ

C, of the pottery

t

minutes after the kiln is switched off is modelled by the equation

\theta = 20 + A\mathrm{e}^{-kt}

where

A

and

k

are positive constants.

Given that the temperature of the pottery was

1100\ ^\circ

C when the kiln was switched off,
(a) find the value of

A

.

(2 marks)

Given also that it takes

150

minutes for the temperature of the pottery to decrease from

1100\ ^\circ

C to

380\ ^\circ

C,
(b) find the value of

k

, giving your answer to 3 significant figures.

(3 marks)

At time

T

minutes, the temperature of the pottery is decreasing at a rate of

2\ ^\circ

C per minute.
(c) Find the value of

T

according to the model, giving your answer to the nearest minute.

(3 marks)

●●●●●Level 48 marksStart

An ecologist is studying the numbers of two species of seabird on an island.

The number of puffins,

N

, on the island is modelled by the equation

N = A\mathrm{e}^{kt} \qquad t \geqslant 0

where

A

and

k

are positive constants and

t

is the time in years from the start of the study.

Given that

there were $640$ puffins on the island $3$ years after the start of the study
there were $1000$ puffins on the island $6$ years after the start of the study

(a) find the exact value of

A

and the value of

k

to 4 significant figures.

(4 marks)

The number of gulls,

G

, on the island is modelled by the equation

G = 3000\mathrm{e}^{-0.08t} \qquad t \geqslant 0

where

t

is the time in years from the start of the study.
(b) Find the rate of decrease of the number of gulls on the island exactly

5

years after the start of the study. Give your answer to 3 significant figures.

(2 marks)

When

t = T

, the number of puffins on the island was the same as the number of gulls on the island.
(c) Find the value of

T

, giving your answer to 3 significant figures.

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of the curve with equation

y = \text{f}(x)

, where

\text{f}(x) = 3\cos x\,\text{e}^{2\sin x} \qquad 0 \leqslant x \leqslant \pi

The curve intersects the

x

-axis at the point

R

, as shown in Figure 1.
(a) State the coordinates of

R

(1 mark)

The curve has two turning points, at the point

P

and the point

Q

, also shown in Figure 1.
(b) Show that, at

P

and at

Q

a\sin^2 x + b\sin x + c = 0

where

a

b

and

c

are integers to be found.

(4 marks)
(c) Hence find the

x

coordinate of

Q

, giving your answer to 3 decimal places.

(2 marks)

●●●●●Level 47 marksStart

A patient is given a dose of drug P. The concentration of drug P in the patient's blood,

C

mg per litre, decreases exponentially with the time,

t

hours, after the dose.
(a) Find the missing values of

C

and

t

in the following table.

$t$	$0$	$2$		$8$
$C$	$50$	$40$	$32$

(2 marks)

A second patient is given a dose of drug Q. The concentration of drug Q in this patient's blood,

C

mg per litre,

t

hours after the dose is modelled by

C = 60e^{-0.15t}

.
(b) (i) Determine the time taken for the concentration of drug Q to fall to one quarter of its initial value.

(ii) Determine the rate at which the concentration of drug Q is decreasing when

t = 3

.

(6 marks)
(c) State whether the concentration of drug P or of drug Q is decreasing at a faster rate, giving a reason for your answer.

(1 mark)

●●●●●Level 49 marksStart

The diagram shows part of the curve

y = \text{f}(x)

, where

\text{f}(x) = 3\ln x + 5 - 2x

. The equation

\text{f}(x) = 0

has a positive root

\alpha

close to

x = 0.2

.(a) Explain why using the sign change method with

x = 0.1

and

x = 6

will fail to locate

\alpha

.

(1 mark)
(b) Show that the equation

\text{f}(x) = 0

can be written as

x = \text{e}^{\frac{1}{3}(2x - 5)}

.

(2 marks)
(c) Use the iterative formula

x_{n+1} = \text{e}^{\frac{1}{3}(2x_n - 5)}

with a starting value of

x_1 = 0.2

to find the value of

\alpha

correct to 4 significant figures, showing the result of each iteration.

(3 marks)
(d) An alternative iterative formula is

x_{n+1} = \text{F}(x_n)

, where

\text{F}(x) = 1.5\ln x + 2.5

.

By considering

\text{F}'(0.2)

explain why this iterative formula will not find

\alpha

.

(3 marks)

●●●●●Level 49 marksStart

The curve

y = (x + 3)\text{e}^{-x} + x^2

has one stationary point for

x > 0

, which is close to

x = 0.7

.
(a) Show that the

x

-coordinate of this stationary point satisfies the equation

2x\text{e}^{x} - x - 2 = 0

.

(2 marks)
(b) Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a) can be written in the form

x_{n+1} = \frac{2x_n^2\text{e}^{x_n} + 2}{2(x_n + 1)\text{e}^{x_n} - 1}.

(4 marks)
(c) Apply the Newton-Raphson formula with initial value

x_1 = 0.7

to find

x_2

and

x_3

.

(1 mark)
(d) Find the coordinates of this stationary point, giving each coordinate correct to 3 decimal places.

(2 marks)

●●●●●Level 49 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x^2}, \qquad x > 0

(a) Find the exact coordinates of the stationary point of

C

, and determine its nature. (6 marks)
(b) The finite region

R

is bounded by

C

, the

x

-axis and the line

x = \mathrm{e}

.

Find the exact area of

R

. (4 marks)

●●●●●Level 510 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x}, \qquad x > 0.

(a) Determine the exact coordinates of the stationary point of

C

, and show that it is a maximum point. (5 marks)
(b) The equation

\ln x = kx,

where

k

is a constant, has exactly two distinct real roots. Determine the set of possible values of

k

, justifying your answer fully. (4 marks)

●●●●●Level 59 marksStart

Figure 4 shows part of the curve

C

with equation

y = \dfrac{\ln x}{x^2}, \qquad x > 0

Figure 4
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}

. (3 marks)
(b) Find the exact coordinates of the stationary point of

C

, and determine its nature. (5 marks)
(c) Find the exact

x

-coordinate of the point of inflection of

C

. (2 marks)

●●●●●Level 510 marksStart

Figure 1A frog jumps from a point

O

on horizontal ground. Figure 1 shows the path of the frog from

O

until it lands, a horizontal distance of

d

cm from

O

.

When the frog has travelled a horizontal distance of

x

cm, its height above the ground is

h

cm.

The frog is modelled as a particle travelling in a vertical plane above horizontal ground.

The path of the frog is modelled by the equation

h = x - 0.25x\text{e}^{0.04x} \qquad 0 \leqslant x \leqslant d

Use the model to answer parts (a),
(b) and (c).
(a) Find the value of

d

, giving your answer to 2 decimal places.

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)
(b) Show that the maximum value of

h

occurs when

x = 25\ln\left(\dfrac{100}{25 + x}\right)

(4 marks)

Using the iteration formula

x_{n+1} = 25\ln\left(\dfrac{100}{25 + x_n}\right) \qquad x_1 = 15

(c) (i) find the value of

x_2

to 2 decimal places,

(ii) find, by repeated iteration, the horizontal distance travelled by the frog before it reaches its maximum height. Give your answer to 2 decimal places.

(3 marks)

●●●●●Level 510 marksStart

A curve has equation

y = \ln(3 + 6\sin x)

, where

x

is in radians.
(a) State the values of

x

, for

0 \le x \le 2\pi

, for which

\ln(3 + 6\sin x)

is not defined.

(2 marks)
(b)

P

is the stationary point on the curve that has the smallest positive

x

-coordinate.

Determine the exact coordinates of

P

.

(4 marks)
(c) (i) Show that

\dfrac{d^2y}{dx^2} + 3e^{-y} + 27e^{-2y} = 0

.

(ii) State what can be deduced about all of the stationary points on this curve, giving a reason for your answer.

(6 marks)

●●●●●Level 512 marksStart

Figure 1

A group of red squirrels is released into a forest and the number of red squirrels in the forest is then monitored.

The number of red squirrels,

N

, in the forest is modelled by the equation

N = 180 + 300\mathrm{e}^{-0.1t} - 420\mathrm{e}^{-0.3t}

where

t

months is the time after the squirrels are released.

Figure 1 is a sketch of

N

against

t

.

Use the equation of the model to answer parts (a) to (e).
(a) State the number of red squirrels released into the forest.

(1 mark)

In the long term, the number of red squirrels in the forest approaches

L

.
(b) State the value of

L

.

(1 mark)

The number of red squirrels in the forest reaches its maximum value after

T

months.
(c) Find the value of

T

, giving your answer to 3 decimal places.

(Solutions based entirely on calculator technology are not acceptable.)

(5 marks)

The number of red squirrels in the forest is

250

for the second time after

M

months.
(d) Show that

M

is a solution of the equation

t = 10\ln\left(\frac{30}{7 + 42\mathrm{e}^{-0.3t}}\right)

(2 marks)

Using the iteration formula

t_{n+1} = 10\ln\left(\frac{30}{7 + 42\mathrm{e}^{-0.3t_n}}\right) \qquad \text{with } t_1 = 13

(e) (i) find, to 4 decimal places, the value of

t_2

(ii) find, to 4 decimal places, the value of

M

(3 marks)

●●●●●Level 512 marksStart

A substance X is decaying exponentially. Its mass,

M

grams, at time

t

minutes is given by

M = 120e^{-0.03t}

.
(a) Find the time taken for the mass to decrease to half of its original value.

(3 marks)

A second substance Y is also decaying exponentially. Initially its mass was 90 grams and, after 6 minutes, its mass was 40 grams.
(b) Determine the time at which both substances are decaying at the same rate.

(8 marks)

●●●●●Level 511 marksStart

The chain rule

37 questions

LessonNot started

Identify the gradient of the curve

y = \ln(3x)

at the point where

x = 2

.

Circle your answer.

A $\dfrac{3}{2}$
B $\dfrac{1}{6}$
C $\dfrac{1}{2}$
D $3$

●●●●●Level 11 markStart

Differentiate

y = (3x^2 - 1)^5

●●●●●Level 22 marksStart

Differentiate

y = (2x + 5)^4

●●●●●Level 22 marksStart

Differentiate

y = (5x - 2)^{3}

. (2 marks)

●●●●●Level 22 marksStart

Find an equation of the tangent to the curve

y = (x - 3)^5

at the point where

x = 0

(3 marks)

●●●●●Level 23 marksStart

(a) Differentiate the following with respect to

x

.

(i)

\ln(3x^2 + 5)

(ii)

x\cos 4x

(5 marks)
(b) Find

\displaystyle\int \sin 3x\,dx

.

(2 marks)
(c) The gradient of a curve is given by

\dfrac{dy}{dx} = 4e^{2x}

and the curve passes through the point

(0, 5)

.

Find the equation of the curve.

(2 marks)

●●●●●Level 29 marksStart

Differentiate

y = (x^{2} + 3x)^{5}

. (3 marks)

●●●●●Level 33 marksStart

The radius of a circular oil slick is increasing at a constant rate of

0.5

cm s

^{-1}

.

Find the rate at which the area is increasing when the radius is

10

cm, giving your answer in terms of

\pi

. (4 marks)

●●●●●Level 34 marksStart

Differentiate

y = \dfrac{1}{(4 - 3x)^{2}}

. (3 marks)

●●●●●Level 33 marksStart

The curve

C

has equation

y = (2x - 3)^{5}

.

Find an equation of the tangent to

C

at the point where

x = 2

, giving your answer in the form

y = mx + c

. (4 marks)

●●●●●Level 34 marksStart

In this question you must show detailed reasoning.

The curve

C

has equation

y = \sqrt{2x+1}

\quad x \ge -\dfrac{1}{2}

,

and meets the

x

-axis at the point

A

. The tangent to

C

at the point

P

has gradient

\dfrac{1}{3}

.
(a) Find the coordinates of

P

. (3 marks)

The normal to

C

P

meets the

x

-axis at the point

Q

. The region

R

is bounded by the arc

AP

of the curve, the line segment

PQ

and the

x

-axis.
(b) Determine the exact area of

R

. (5 marks)

●●●●●Level 38 marksStart

A curve is given by

x = y^3 + y

. Find

\dfrac{dy}{dx}

in terms of

y

, and its value at the point where

y = 1

●●●●●Level 33 marksStart

Differentiate

y = \sqrt{1 + \sqrt{x}}

●●●●●Level 33 marksStart

Given

y = \sqrt{x^{2} + 9}

:
(a) find

\dfrac{dy}{dx}

; (3 marks)
(b) evaluate the gradient at

x = 4

. (1 mark)

●●●●●Level 34 marksStart

The curve

C

has equation

y = (3x - 2)^4

. Find an equation of the tangent to

C

at the point where

x = 1

●●●●●Level 34 marksStart

The curve

C

has equation

y = (x^{2} - 4x)^{3}

.

Find the

x

-coordinates of the stationary points of

C

. (4 marks)

●●●●●Level 34 marksStart

A curve has equation

y = (x^2 - 5)^3

. Find the gradient of the curve at the point where

x = 3

●●●●●Level 33 marksStart

Differentiate

y = \dfrac{1}{(2x + 1)^3}

●●●●●Level 32 marksStart

Differentiate

y = \sqrt{4x - 3}

, giving your answer in surd form.

●●●●●Level 32 marksStart

A snowball is rolled through fresh snow. The snowball is modelled as a sphere with radius

r

cm.

The surface area of the snowball,

S

cm², is increasing at a constant rate,

k

cm² per second, where

k

is a constant.

Given that

S = 4\pi r^2

, show that the rate of increase of the radius of the snowball with respect to time is inversely proportional to the radius.

(4 marks)

●●●●●Level 34 marksStart

(a) Find

\dfrac{d}{dx}\left(5 - 2x^2\right)^4

(2 marks)
(b) Hence show that

\int_0^1 x\left(5 - 2x^2\right)^3 dx = R

where

R

is an integer to be found.

(Solutions relying on calculator technology are not acceptable.)

(3 marks)

●●●●●Level 35 marksStart

\text{f}(x) = e^{\frac{1}{2}x} \qquad x > 0

\text{g}(x) = 40\ln x \qquad x > 0

(a) Find

(i)

\text{f}'(x)

(ii)

\text{g}'(x)

(2 marks)

Given that

\text{f}'(x) = \text{g}'(x)

x = \alpha

(b) show that

\alpha

satisfies the equation

xe^{\frac{1}{2}x} - 80 = 0

(2 marks)

The iterative formula

x_{n+1} = 2\ln\left(\frac{80}{x_n}\right)

is used with

x_1 = 5

to find an approximate value for

\alpha

.
(c) Calculate, giving each answer to 4 decimal places,

(i) the value of

x_2

(ii) the value of

\alpha

(3 marks)

●●●●●Level 47 marksStart

Figure 5 shows a container in the shape of a right circular cone with its vertex pointing downwards. The cone has radius

6

cm and height

18

cm.Figure 5 (not accurately drawn)

Water is poured into the container at a constant rate of

8

cm³ s⁻¹. At time

t

seconds the depth of the water is

h

cm and the radius of the water surface is

r

cm.
(a) Show that the volume

V

cm³ of water in the container is given by

V = \dfrac{\pi h^3}{27}

. (3 marks)
(b) Find the rate at which the depth of the water is increasing at the instant when

h = 6

, giving your answer to

3

significant figures. (4 marks)
(c) Find the rate at which the area of the water surface is increasing at that same instant, giving your answer to

3

significant figures. (3 marks)

[The volume of a cone of radius

r

and height

h

\tfrac13\pi r^2 h

●●●●●Level 410 marksStart

Figure 1 shows the curve

C

with equation

y = x^2\sqrt{3 - 2x}, \qquad x < \dfrac32

Figure 1
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{x(6 - 5x)}{\sqrt{3 - 2x}}

. (4 marks)
(b) Find the exact coordinates of the stationary points of

C

. (3 marks)

●●●●●Level 47 marksStart

A cube is expanding. At time

t

seconds the length of each edge is

x

cm, the volume is

V

cm³ and the total surface area is

A

cm².

The volume is increasing at the constant rate of

24

cm³ s⁻¹.
(a) Find the rate at which

x

is increasing at the instant when

x = 4

. (4 marks)
(b) Find the rate at which

A

is increasing at that same instant. (3 marks)
(c) Show that, at any instant, the rate at which the surface area is increasing is inversely proportional to

x

. (2 marks)

●●●●●Level 49 marksStart

A curve has equation

y = x + \dfrac{4}{x-2}

, for

x > 2

.

Find the coordinates of the stationary point of the curve and determine its nature. (6 marks)

●●●●●Level 46 marksStart

A container is in the shape of an inverted right circular cone, with its vertex at the bottom. The container has height

60

cm and the radius of its circular top is

20

cm.

Water is poured into the empty container at a constant rate of

45

cm³ s⁻¹. At time

t

seconds the depth of the water is

h

cm and the volume of water in the container is

V

cm³.
(a) Show that

V = \dfrac{\pi h^3}{27}

. (2 marks)
(b) Find the rate at which the depth of the water is increasing at the instant when

h = 15

, giving your answer in exact form. (3 marks)
(c) Find the rate at which the area of the circular surface of the water is increasing at the same instant. (2 marks)

[The volume of a cone of base radius

r

and height

h

V = \frac{1}{3}\pi r^2 h

●●●●●Level 47 marksStart

A spherical balloon is being inflated so that its volume increases at the constant rate of

200

cm³ s⁻¹.
(a) Find the rate at which the radius is increasing at the instant when the radius is

5

cm, giving your answer exactly. (4 marks)
(b) Find the rate at which the surface area is increasing at the same instant. (3 marks)

[The volume and surface area of a sphere are

V = \frac{4}{3}\pi r^3

and

S = 4\pi r^2

●●●●●Level 47 marksStart

A garden water feature is being filled with water.

The container is modelled as a hollow cone with its vertex pointing vertically downwards, where

the depth of the container is $24$ cm
the radius of the circular top of the container is $8$ cm
water is poured into the container at a constant rate of $30$ cm $^3$ per second

At time

t

seconds after the water starts to be poured in, the depth of the water in the container is

h

cm and the volume of water in the container is

V

^3

.

Using the model,
(a) show that

V = \frac{\pi h^3}{27}

(2 marks)
(b) find the rate at which the depth of the water is increasing at the instant when

h = 12

, giving your answer to 3 significant figures.

(4 marks)
(c) Find the rate at which the radius of the surface of the water is increasing at this instant.

(1 mark)
(d) State a limitation of the model.

(1 mark)

●●●●●Level 48 marksStart

[For a cone with base radius

r

, height

h

and slant height

l

, the following formulae are given.

Curved surface area,

S = \pi rl

Volume,

V = \tfrac13\pi r^2h

]

A sweet shop makes paper cones, open at the top, to hold sweets. Each cone has base radius

r

cm and holds a volume of

V

^3

when full. Each cone uses

25\pi

^2

of paper.
(a) Show that

V = \tfrac13\pi r\sqrt{625 - r^4}

.

(4 marks)
(b) In this question you must show detailed reasoning.

It is given that

V

has a maximum value for a certain value of

r

.

Find the maximum value of

V

, giving your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 49 marksStart

A curve has equation

y = 2^{x^2 - 6x}

.
(a) Determine the coordinates of any stationary points on the curve.

(4 marks)
(b) Show that the curve is convex for all values of

x

.

(5 marks)

●●●●●Level 49 marksStart

W

megalitres, and the mean daily temperature,

T

°C, at time

t

weeks after the start of the investigation.

In the model

W = 25e^{0.04t}

and

T = 15 + 6\sin(0.2t)

, where

0.2t

is in radians.

The model assumes that

W

and

T

can be treated as continuous variables.
(a) State the meaning of

\dfrac{dW}{dT}

.

(1 mark)
(b) Determine

\dfrac{dW}{dT}

when

t = 5

.

(4 marks)
(c) Suggest a reason why this model may not be valid for values of

t

greater than

15

.

(1 mark)

●●●●●Level 46 marksStart

Figure 1Figure 1 shows a sketch of the curve

C

with equation

y = \text{f}(x)

, where

\text{f}(x) = \dfrac{2x + 12}{\sqrt{2x + 3}} \qquad x > -\dfrac{3}{2}

(a) Show that

\text{f}'(x) = \dfrac{k(x - 3)}{(2x + 3)^{\frac{3}{2}}}

where

k

is a constant to be found.

(4 marks)

The point

P

, shown in Figure 1, is the minimum turning point on

C

.
(b) Find the coordinates of

P

.

(2 marks)
(c) Hence find the range of the function g defined by

\text{g}(x) = -3\text{f}(x) \qquad x > -\dfrac{3}{2}

(2 marks)

The point

Q

with coordinates

(-1, 10)

lies on

C

.
(d) Find the coordinates of the point to which

Q

is mapped when

C

is transformed to the curve with equation

y = \dfrac{1}{2}\text{f}(2x - 5) + 4

(2 marks)

●●●●●Level 410 marksStart

y = \sqrt{3x - 11} \qquad x \geqslant \dfrac{11}{3}

The line

l

, shown in Figure 1, is the normal to the curve at the point

P(9, 4)

(a) Use calculus to show that an equation of

l

8x + 3y - 84 = 0

(5 marks)

The region

R

, shown shaded in Figure 1, is bounded by the curve, the

x

-axis and

l

.
(b) Use algebraic integration to find the exact area of

R

.

(4 marks)

●●●●●Level 49 marksStart

The diagram shows the curve

M

with equation

y = (\ln x)^2

, for

x > 0

.
(a) Show that

M

has a point of inflection at the point

P

where

x = e

.

(5 marks)

The shaded region

R

is enclosed by

M

, the

x

-axis and the line through

P

parallel to the

y

-axis.
(b) Determine the exact area of

R

, giving your answer in the form

ae + b

, where

a

and

b

are integers to be determined.

(6 marks)

●●●●●Level 511 marksStart

The diagram shows a laboratory container in the shape of an inverted cone. The height of the container is

36

cm and the radius of its circular top is

12

cm. Initially the container is full of liquid, and the depth of the liquid is

36

cm.Liquid drips out through a small hole at the vertex of the container. The rate at which the liquid drips out is modelled by

\dfrac{dV}{dt} = -4\sqrt{h}

, where

V

^3

is the volume of liquid remaining and

h

cm is the depth of liquid in the container

t

seconds after the liquid begins to drip out.

Determine the exact time taken for the container to become empty.

[For a cone with base radius

r

and height

h

the volume

V

is given by

\tfrac13\pi r^2h

.]

(7 marks)

●●●●●Level 57 marksStart

A curve has equation

y = \ln(3 + 6\sin x)

, where

x

is in radians.
(a) State the values of

x

, for

0 \le x \le 2\pi

, for which

\ln(3 + 6\sin x)

is not defined.

(2 marks)
(b)

P

is the stationary point on the curve that has the smallest positive

x

-coordinate.

Determine the exact coordinates of

P

.

(4 marks)
(c) (i) Show that

\dfrac{d^2y}{dx^2} + 3e^{-y} + 27e^{-2y} = 0

.

(ii) State what can be deduced about all of the stationary points on this curve, giving a reason for your answer.

(6 marks)

●●●●●Level 512 marksStart

The product rule

40 questions

LessonNot started

Differentiate

y = x^2(2x + 1)^3

, giving your answer in fully factorised form.

●●●●●Level 23 marksStart

Differentiate

y = (x + 1)(x - 3)^4

, giving your answer in factorised form.

●●●●●Level 23 marksStart

Differentiate

(x^2 + 4x - 1)e^{-x}

with respect to

x

, giving your answer in the form

e^{-x}(ax^2 + bx + c)

, where

a

b

and

c

are integers to be determined.

(3 marks)

●●●●●Level 23 marksStart

Differentiate the following with respect to

x

.
(a)

\dfrac{6}{x^3} + 4\sqrt{x}

(2 marks)
(b)

x^2e^{3x}

(2 marks)
(c)

\dfrac{2x + 1}{x - 3}

(3 marks)

●●●●●Level 27 marksStart

(a) Differentiate the following with respect to

x

.

(i)

\ln(3x^2 + 5)

(ii)

x\cos 4x

(5 marks)
(b) Find

\displaystyle\int \sin 3x\,dx

.

(2 marks)
(c) The gradient of a curve is given by

\dfrac{dy}{dx} = 4e^{2x}

and the curve passes through the point

(0, 5)

.

Find the equation of the curve.

(2 marks)

●●●●●Level 29 marksStart

A curve has equation

y = x^2(x + 3)^4

. Find the gradient of the curve at the point where

x = 1

●●●●●Level 33 marksStart

The curve

C

has equation

y = (x - 1)^2(x + 2)^3

. Find the

x

-coordinates of the stationary points of

C

●●●●●Level 34 marksStart

Differentiate

y = (2x + 1)^{3}(x - 2)

, giving your answer in fully factorised form. (4 marks)

●●●●●Level 34 marksStart

Differentiate

y = (x^{2} + 1)\sqrt{x}

, giving your answer as a single fraction. (4 marks)

●●●●●Level 34 marksStart

The curve

C

has equation

y = x\ln x

\; x > 0

.
(a) Find

\dfrac{dy}{dx}

. (2 marks)
(b) Hence find the exact

x

-coordinate of the stationary point of

C

. (2 marks)

●●●●●Level 34 marksStart

Find an equation of the tangent to the curve

y = x\mathrm{e}^{x}

at the point where

x = 1

, giving your answer in terms of

\mathrm{e}

. (4 marks)

●●●●●Level 34 marksStart

Differentiate

y = (3x - 1)\sqrt{x}

, giving your answer as a single fraction.

●●●●●Level 33 marksStart

Differentiate

y = (2x^2 + 1)(x - 4)^3

, giving your answer in factorised form.

●●●●●Level 33 marksStart

The curve

C

has equation

y = x^3(x - 2)^5

. Find the

x

-coordinates of the stationary points of

C

●●●●●Level 34 marksStart

Differentiate

y = x\sqrt{2x + 1}

, giving your answer as a single fraction.

●●●●●Level 33 marksStart

The curve

C

has equation

y = x^{2}\mathrm{e}^{x}

.

Find the

x

-coordinates of the stationary points of

C

. (4 marks)

●●●●●Level 34 marksStart

The curve

C

has equation

y = x^{2}\sin x

.

Find the exact gradient of

C

at the point where

x = \pi

. (4 marks)

●●●●●Level 34 marksStart

A curve has equation

2xy^3 + ye^{3x} = 7

Find an expression for

\dfrac{dy}{dx}

in terms of

x

and

y

(4 marks)

●●●●●Level 34 marksStart

Differentiate

y = x^{3}(x + 2)^{2}

, giving your answer in fully factorised form. (3 marks)

●●●●●Level 33 marksStart

Figure 1 shows the curve

C

with equation

y = x^2\sqrt{3 - 2x}, \qquad x < \dfrac32

Figure 1
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{x(6 - 5x)}{\sqrt{3 - 2x}}

. (4 marks)
(b) Find the exact coordinates of the stationary points of

C

. (3 marks)

●●●●●Level 47 marksStart

A particle is moving such that its position vector,

\mathbf{r}

metres, at time

t

seconds, is given by

\mathbf{r} = \mathrm{e}^{2t}\cos t\,\mathbf{i} + \mathrm{e}^{2t}\sin t\,\mathbf{j}

Show that the magnitude of the acceleration of the particle,

a\ \text{m s}^{-2}

, is given by

a = 5\mathrm{e}^{2t}

Fully justify your answer.

(7 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = x^2 e^{-x}

.
(a) Show that

\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = (x^2 - 4x + 2)e^{-x}

. (3 marks)
(b) Hence find the exact

x

-coordinates of the points of inflection of

C

. Fully justify your answer. (4 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = x^3 e^{-x}

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = x^2 e^{-x}(3 - x)

. (2 marks)
(b) Find the coordinates of the stationary points of

C

. (2 marks)
(c) Determine the nature of each stationary point. (3 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

2x^2 - 3xy + 2y^2 = 14

The points

P

and

Q

lie on

C

. The tangent to

C

at each of the points

P

and

Q

is parallel to the

x

-axis.

Find the coordinates of

P

and

Q

, showing all your working. (7 marks)

●●●●●Level 47 marksStart

A curve has equation

2y^3e^{3x} + 5y + 48x = k

where

k

is a constant.

The curve has a stationary point on the

y

-axis.

Determine the value of

k

(7 marks)

●●●●●Level 47 marksStart

In a model of a sprint race, the speed of an athlete,

v

m s

^{-1}

t

seconds after the start of the race, is modelled by the equation

v = \frac{1}{2}(18 - t)\ln(t + 1) \qquad 0 \leqslant t \leqslant 18

Using the model,
(a) show that the speed of the athlete is a maximum when

t

satisfies the equation

(t + 1)\ln(t + 1) + t - 18 = 0

(4 marks)

The equation in part (a) has exactly one root,

\alpha

, in the interval

0 \leqslant t \leqslant 18

(b) Show that

\alpha

lies in the interval

[5, 6]

(2 marks)
(c) Using

t_0 = 6

as a first approximation for

\alpha

, apply the Newton–Raphson method once to find a second approximation for

\alpha

, giving your answer to 3 decimal places.

(2 marks)

●●●●●Level 48 marksStart

The curve

C

has equation

y = (2x - 1)e^{3x}

(a) Show that

\frac{dy}{dx} = (6x - 1)e^{3x}

(2 marks)
(b) Hence find the exact coordinates of the stationary point of

C

.

(3 marks)
(c) Use further differentiation to determine the nature of this stationary point.

(2 marks)

●●●●●Level 47 marksStart

A continuous curve has equation

y = x^3\cos 2x \qquad 0 \leqslant x \leqslant \dfrac{\pi}{4}

The curve has a stationary point at the point

Q

, where

x > 0

(a) Show, using calculus, that the

x

coordinate of

Q

is a solution of the equation

x = \dfrac{1}{2}\arctan\left(\dfrac{3}{2x}\right)

(4 marks)

Using the iteration formula

x_{n+1} = \dfrac{1}{2}\arctan\left(\dfrac{3}{2x_n}\right) \qquad x_1 = 0.6

(b) find, to 4 decimal places, the value of

(i)

x_2

(ii)

x_3

(3 marks)
(c) Using a suitable interval and a suitable function which should be stated, show that the

x

coordinate of

Q

0.596

correct to 3 decimal places.

(2 marks)

●●●●●Level 49 marksStart

The curve

C

has equation

y = x^2 \ln x

\quad x > 0

.
(a) Show that

\dfrac{dy}{dx} = x(2\ln x + 1)

. (3 marks)
(b) Hence find the exact coordinates of the stationary point of

C

. (3 marks)
(c) Determine the nature of this stationary point. (2 marks)
(d) Find the range of values of

x

for which

C

is convex. (1 mark)

●●●●●Level 49 marksStart

[For a cone with base radius

r

, height

h

and slant height

l

, the following formulae are given.

Curved surface area,

S = \pi rl

Volume,

V = \tfrac13\pi r^2h

]

A sweet shop makes paper cones, open at the top, to hold sweets. Each cone has base radius

r

cm and holds a volume of

V

^3

when full. Each cone uses

25\pi

^2

of paper.
(a) Show that

V = \tfrac13\pi r\sqrt{625 - r^4}

.

(4 marks)
(b) In this question you must show detailed reasoning.

It is given that

V

has a maximum value for a certain value of

r

.

Find the maximum value of

V

, giving your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 49 marksStart

(a) Differentiate

(x^2 - 2x + 4)e^{3x}

with respect to

x

.

(3 marks)
(b) Hence show that

(x^2 - 2x + 4)e^{3x}

is increasing for all values of

x

.

(4 marks)

●●●●●Level 47 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of the curve with equation

y = \text{f}(x)

, where

\text{f}(x) = 3\cos x\,\text{e}^{2\sin x} \qquad 0 \leqslant x \leqslant \pi

The curve intersects the

x

-axis at the point

R

, as shown in Figure 1.
(a) State the coordinates of

R

(1 mark)

The curve has two turning points, at the point

P

and the point

Q

, also shown in Figure 1.
(b) Show that, at

P

and at

Q

a\sin^2 x + b\sin x + c = 0

where

a

b

and

c

are integers to be found.

(4 marks)
(c) Hence find the

x

coordinate of

Q

, giving your answer to 3 decimal places.

(2 marks)

●●●●●Level 47 marksStart

The curve

y = (x + 3)\text{e}^{-x} + x^2

has one stationary point for

x > 0

, which is close to

x = 0.7

.
(a) Show that the

x

-coordinate of this stationary point satisfies the equation

2x\text{e}^{x} - x - 2 = 0

.

(2 marks)
(b) Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a) can be written in the form

x_{n+1} = \frac{2x_n^2\text{e}^{x_n} + 2}{2(x_n + 1)\text{e}^{x_n} - 1}.

(4 marks)
(c) Apply the Newton-Raphson formula with initial value

x_1 = 0.7

to find

x_2

and

x_3

.

(1 mark)
(d) Find the coordinates of this stationary point, giving each coordinate correct to 3 decimal places.

(2 marks)

●●●●●Level 49 marksStart

The curve

C

has equation

x^2 + 2xy + 3y^2 = 6

(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{x + y}{x + 3y}

(3 marks)
(b) Hence find the exact coordinates of the two points on

C

at which the tangent is parallel to the

x

-axis. (4 marks)
(c) Show that there are exactly two points on

C

at which

\dfrac{\mathrm{d}y}{\mathrm{d}x}

is not defined, and find their coordinates. (3 marks)

●●●●●Level 510 marksStart

\mathrm{f}(x) = (2x - 1)\mathrm{e}^{-x}

(a) Show that

\mathrm{f}'(x) = (3 - 2x)\mathrm{e}^{-x}

, and hence find the exact coordinates of the stationary point of the curve

y = \mathrm{f}(x)

. (5 marks)
(b) Determine the nature of the stationary point. (3 marks)
(c) Find the exact value of

\displaystyle\int_0^1 (2x - 1)\mathrm{e}^{-x}\;\mathrm{d}x

. (3 marks)

●●●●●Level 511 marksStart

The curve

C

has equation

2x^3 + y^2 - 6xy = 0

C

passes through the origin

O

and has a stationary point at

P

, where

P

is not

O

, as shown in the diagram below.(a) Show that the

x

-coordinate of any stationary point of

C

satisfies the equation

x^3(x - 4) = 0

(7 marks)
(b) Hence, find the coordinates of

P

(2 marks)

●●●●●Level 59 marksStart

In this question you must show detailed reasoning.

Figure 1 shows the curve

C

with equation

y = x\sqrt{4 - x}, \qquad 0 \leqslant x \leqslant 4

The region

R

, shown shaded, is bounded by

C

and the

x

-axis.Figure 1
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{8 - 3x}{2\sqrt{4 - x}}

. (4 marks)
(b) Find the exact coordinates of the maximum point of

C

. (3 marks)
(c) Using the substitution

u = 4 - x

, show that the area of

R

\dfrac{128}{15}

. (5 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = \frac{2\ln x}{\sqrt{x}} \qquad x > 0

The curve

C

has a maximum turning point at the point

M

.
(a) Use calculus to find the exact coordinates of

M

.

(3 marks)

The finite region

R

is bounded by the curve

C

, the

x

-axis and the line with equation

x = 16

(b) Use the substitution

u = \sqrt{x}

to show that the area of

R

is given by

8\int_1^{4} \ln u\,du

(4 marks)
(c) Hence use algebraic integration to show that the exact area of

R

a\ln 2 + b

where

a

and

b

are integers to be found.

(4 marks)

●●●●●Level 511 marksStart

Figure 1A frog jumps from a point

O

on horizontal ground. Figure 1 shows the path of the frog from

O

until it lands, a horizontal distance of

d

cm from

O

.

When the frog has travelled a horizontal distance of

x

cm, its height above the ground is

h

cm.

The frog is modelled as a particle travelling in a vertical plane above horizontal ground.

The path of the frog is modelled by the equation

h = x - 0.25x\text{e}^{0.04x} \qquad 0 \leqslant x \leqslant d

Use the model to answer parts (a),
(b) and (c).
(a) Find the value of

d

, giving your answer to 2 decimal places.

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)
(b) Show that the maximum value of

h

occurs when

x = 25\ln\left(\dfrac{100}{25 + x}\right)

(4 marks)

Using the iteration formula

x_{n+1} = 25\ln\left(\dfrac{100}{25 + x_n}\right) \qquad x_1 = 15

(c) (i) find the value of

x_2

to 2 decimal places,

(ii) find, by repeated iteration, the horizontal distance travelled by the frog before it reaches its maximum height. Give your answer to 2 decimal places.

(3 marks)

●●●●●Level 510 marksStart

(a) Use the substitution

u = 4 - x

to show that

\int x\sqrt{4 - x}\,dx = -\tfrac{2}{15}(3x + 8)(4 - x)^{\frac32} + c.

(5 marks)
(b) In this question you must show detailed reasoning.The diagram shows part of the curve

y = x\sqrt{4 - x}

, for

0 \le x \le 4

, and the normal to the curve at the point

P(3, 3)

. The shaded region is enclosed by the curve, the normal to the curve at

P

and the

x

-axis.

Find the exact area of the shaded region.

(7 marks)

●●●●●Level 512 marksStart

The quotient rule

32 questions

LessonNot started

Differentiate

y = \dfrac{x}{x + 1}

●●●●●Level 22 marksStart

Differentiate

y = \dfrac{3x - 1}{x + 2}

, simplifying your answer. (3 marks)

●●●●●Level 23 marksStart

Differentiate the following with respect to

x

.
(a)

\dfrac{6}{x^3} + 4\sqrt{x}

(2 marks)
(b)

x^2e^{3x}

(2 marks)
(c)

\dfrac{2x + 1}{x - 3}

(3 marks)

●●●●●Level 27 marksStart

The function f is defined by

\text{f}(x) = \dfrac{2x^2 + 7x + 3}{x^2 - x - 12} - \dfrac{6}{x - 4} \qquad x > 4

(a) Show that

\text{f}(x) = \dfrac{2x - 5}{x - 4} \qquad x > 4

(3 marks)
(b) Hence find

\text{f}'(x)

giving your answer in simplest form.

(2 marks)

●●●●●Level 25 marksStart

Differentiate

y = \dfrac{x - 2}{x + 2}

●●●●●Level 32 marksStart

Differentiate

y = \dfrac{x^3 - 5x}{x^2}

, choosing the most efficient method.

●●●●●Level 32 marksStart

Differentiate

y = \dfrac{x^{2}}{x + 3}

, giving your answer in factorised form. (4 marks)

●●●●●Level 34 marksStart

Differentiate

y = \dfrac{3x^2}{2x - 1}

, giving your answer in factorised form.

●●●●●Level 33 marksStart

A curve has equation

y = \dfrac{x^2 + 1}{x - 1}

. Find the gradient of the curve at the point where

x = 2

●●●●●Level 33 marksStart

Differentiate

y = \dfrac{\sin x}{x}

\; x \neq 0

. (3 marks)

●●●●●Level 33 marksStart

Differentiate

y = \dfrac{2x + 1}{x^2 - 3}

●●●●●Level 33 marksStart

Differentiate

y = \dfrac{x}{\sqrt{x + 4}}

●●●●●Level 33 marksStart

The curve

C

has equation

y = \dfrac{2x - 3}{x^2 + 1}

. Find the exact

x

-coordinates of the stationary points of

C

●●●●●Level 34 marksStart

Differentiate

y = \dfrac{\mathrm{e}^{x}}{x^{2}}

, giving your answer in the form

\dfrac{\mathrm{e}^{x}f(x)}{x^{3}}

. (4 marks)

●●●●●Level 34 marksStart

The curve

C

has equation

y = \dfrac{x}{x^{2} + 1}

.

Find the coordinates of the stationary points of

C

. (5 marks)

●●●●●Level 35 marksStart

The curve

C

has equation

y = \dfrac{2x + 1}{x - 3}

.

Verify that the point

(4, 9)

lies on

C

, and find the gradient of

C

there. (3 marks)

●●●●●Level 33 marksStart

The curve

C

has equation

y = \frac{x^2 + 5}{x + 2} \qquad x \in \mathbb{R},\ x \neq -2

(a) Find

\dfrac{dy}{dx}

, giving your answer as a single fraction in its simplest form.

(3 marks)
(b) Hence find the

x

coordinates of the points on

C

at which the tangent to

C

is parallel to the

x

-axis.

(2 marks)

●●●●●Level 35 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A curve has equation

y = \dfrac{3x - 5}{(x - 1)^2} \qquad x \neq 1,\ x \in \mathbb{R}

Use calculus to find the range of values of

x

for which

y

is increasing.

(6 marks)

●●●●●Level 36 marksStart

f(x) = \dfrac{2x - 1}{x^2 + 3}

.
(a) Show that

f'(x) = \dfrac{-2x^2 + 2x + 6}{(x^2 + 3)^2}

. (4 marks)
(b) Hence find the exact range of values of

x

for which

f

is an increasing function. (4 marks)

●●●●●Level 48 marksStart

A curve has equation

y = \dfrac{\ln x}{x^2}

\quad x > 0

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}

. (3 marks)
(b) Determine the exact coordinates of the stationary point of the curve, and show that it is a maximum. (4 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = \dfrac{x}{x^2 + 1}

(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - x^2}{\left(x^2 + 1\right)^2}

. (3 marks)
(b) Find the coordinates of the two stationary points of

C

, and determine the nature of each. (4 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x^{2}}

\; x > 0

.
(a) Show that

\dfrac{dy}{dx} = \dfrac{1 - 2\ln x}{x^{3}}

. (3 marks)
(b) Hence find the exact

x

-coordinate of the stationary point of

C

. (1 mark)

●●●●●Level 44 marksStart

Figure 1Figure 1 shows a sketch of the curve

C

with equation

y = \text{f}(x)

, where

\text{f}(x) = \dfrac{2x + 12}{\sqrt{2x + 3}} \qquad x > -\dfrac{3}{2}

(a) Show that

\text{f}'(x) = \dfrac{k(x - 3)}{(2x + 3)^{\frac{3}{2}}}

where

k

is a constant to be found.

(4 marks)

The point

P

, shown in Figure 1, is the minimum turning point on

C

.
(b) Find the coordinates of

P

.

(2 marks)
(c) Hence find the range of the function g defined by

\text{g}(x) = -3\text{f}(x) \qquad x > -\dfrac{3}{2}

(2 marks)

The point

Q

with coordinates

(-1, 10)

lies on

C

.
(d) Find the coordinates of the point to which

Q

is mapped when

C

is transformed to the curve with equation

y = \dfrac{1}{2}\text{f}(2x - 5) + 4

(2 marks)

●●●●●Level 410 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x^2}, \qquad x > 0

(a) Find the exact coordinates of the stationary point of

C

, and determine its nature. (6 marks)
(b) The finite region

R

is bounded by

C

, the

x

-axis and the line

x = \mathrm{e}

.

Find the exact area of

R

. (4 marks)

●●●●●Level 510 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x}, \qquad x > 0.

(a) Determine the exact coordinates of the stationary point of

C

, and show that it is a maximum point. (5 marks)
(b) The equation

\ln x = kx,

where

k

is a constant, has exactly two distinct real roots. Determine the set of possible values of

k

, justifying your answer fully. (4 marks)

●●●●●Level 59 marksStart

The function

\text{f}

is defined by

\text{f}(x) = \frac{x^2 + 7}{2x + 2}

where

\text{f}

has its maximum possible domain.

The curve

y = \text{f}(x)

intersects the line

y = x

at the points

P

and

Q

as shown below.(a) State the value of

x

which is not in the domain of

\text{f}

(1 mark)
(b) Explain how you know that the function

\text{f}

is many-to-one.

(2 marks)
(c) (i) Show that the

x

-coordinates of

P

and

Q

satisfy the equation

x^2 + 2x - 7 = 0

(ii) Hence, find the exact

x

-coordinate of

P

and the exact

x

-coordinate of

Q

(3 marks)
(d) Show that

P

and

Q

are stationary points of the curve.

Fully justify your answer.

(5 marks)
(e) Using set notation, state the range of

\text{f}

(2 marks)

●●●●●Level 513 marksStart

In this question you must show detailed reasoning.

The diagram shows the curve with equation

y = \dfrac{2x - 5}{(x + 1)^2}

for

x > -1

. The tangent to the curve at the point

P

has gradient

1

.(a) Show that the

x

-coordinate of

P

satisfies the equation

x^3 + 3x^2 + 5x - 11 = 0

.

(5 marks)
(b) Show by calculation that the

x

-coordinate of

P

lies between

1

and

1.2

.

(2 marks)
(c) Show that the iteration

x_{n+1} = \frac{11 - x_n^3 - 3x_n^2}{5}

cannot converge to the

x

-coordinate of

P

whatever starting value is used.

(2 marks)
(d) Use the Newton-Raphson method, with initial value

1

, to determine the coordinates of

P

correct to 5 decimal places.

(5 marks)

●●●●●Level 514 marksStart

Figure 4 shows part of the curve

C

with equation

y = \dfrac{\ln x}{x^2}, \qquad x > 0

Figure 4
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}

. (3 marks)
(b) Find the exact coordinates of the stationary point of

C

, and determine its nature. (5 marks)
(c) Find the exact

x

-coordinate of the point of inflection of

C

. (2 marks)

●●●●●Level 510 marksStart

The function f is defined by

\text{f}(x) = \dfrac{3x^2 + 13x + 4}{x^2 + 3x - 4} - \dfrac{2}{x - 1} \qquad x \in \mathbb{R},\ x > 1

(a) Show that

\text{f}(x) = \dfrac{3x - 1}{x - 1}

(3 marks)
(b) Show, using calculus, that f is a decreasing function. You must make your reasoning clear.

(3 marks)

The function g is defined by

\text{g}(x) = 4\ln(x + 2) - 1 \qquad x \geqslant -1

(c) Find

\text{g}^{-1}

(3 marks)
(d) Find the exact value of

a

for which

\text{gf}(a) = 7

(Solutions relying on calculator technology are not acceptable.)

(4 marks)

●●●●●Level 513 marksStart

A curve has equation

y = \ln(3 + 6\sin x)

, where

x

is in radians.
(a) State the values of

x

, for

0 \le x \le 2\pi

, for which

\ln(3 + 6\sin x)

is not defined.

(2 marks)
(b)

P

is the stationary point on the curve that has the smallest positive

x

-coordinate.

Determine the exact coordinates of

P

.

(4 marks)
(c) (i) Show that

\dfrac{d^2y}{dx^2} + 3e^{-y} + 27e^{-2y} = 0

.

(ii) State what can be deduced about all of the stationary points on this curve, giving a reason for your answer.

(6 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.

A curve has equation

y = \dfrac{x - 1}{x^2 + 3}

.
(a) State the coordinates of the points where the curve crosses the axes.

(2 marks)

The point

A

is where the curve crosses the

y

-axis. The point

B

on the curve has

x

-coordinate

-3

.
(b) Show that the line

AB

is parallel to the

x

-axis.

(2 marks)
(c) Determine the coordinates of the turning points of the curve.

(4 marks)
(d) Determine whether each turning point is a maximum or a minimum.

(5 marks)

●●●●●Level 513 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with equation

y = \text{f}(x)

where

\text{f}(x) = \dfrac{x^2 + 4x - 5}{x + 1} \qquad x > -1

(a) Find

\text{f}'(x)

, giving the answer in simplest form.

(3 marks)

The line

l

is the normal to

C

at the point

P(3, 4)

(b) Show that an equation for

l

2x + 3y = 18

(3 marks)
(c) Write

\text{f}(x)

in the form

Ax + B + \dfrac{D}{x + 1}

where

A

B

and

D

are constants.

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by

C

l

and the

x

-axis.
(d) Use algebraic integration to find the exact area of

R

, giving your answer in the form

p + q\ln 2

, where

p

and

q

are rational constants.

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 514 marksStart

Parametric differentiation

41 questions

LessonNot started

A curve has parametric equations

x = t^2,\ y = t^3

. Find

\dfrac{dy}{dx}

in terms of

t

●●●●●Level 22 marksStart

A curve has parametric equations

x = t^2 + 1,\ y = t^3 - 3t

. Find

\dfrac{dy}{dx}

in terms of

t

, and its value at

t = 2

●●●●●Level 23 marksStart

A curve has parametric equations

x = 4t, \; y = t^{2} - 1

.

Find

\dfrac{dy}{dx}

in terms of

t

. (2 marks)

●●●●●Level 22 marksStart

A curve has parametric equations

x = t^{3}, \; y = t^{2} + 2t

.

Find the gradient of the curve at the point where

t = 1

. (3 marks)

●●●●●Level 33 marksStart

A curve has parametric equations

x = t^{2}, \; y = 4t

.

Find an equation of the tangent to the curve at the point where

t = 1

, giving your answer in the form

y = mx + c

. (4 marks)

●●●●●Level 34 marksStart

A designer is making a curved metal bracket to hold a hanging flower basket.

The edge of the bracket can be modelled by the parametric equations

x = 10t - \frac{5}{t} \qquad y = 3t^2 + \frac{12}{t}

for

1 \leq t \leq 3

, as shown in the diagram below, where

x

and

y

are measured in centimetres.Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(3 marks)

●●●●●Level 33 marksStart

An ellipse has parametric equations

x = 2\cos t,\ y = 3\sin t

. Find

\dfrac{dy}{dx}

in terms of

t

, and its exact value at

t = \dfrac{\pi}{6}

●●●●●Level 33 marksStart

A curve has parametric equations

x = t + \dfrac{1}{t},\ y = t - \dfrac{1}{t}

t \ne 0

. Show that

\dfrac{dy}{dx} = \dfrac{t^2 + 1}{t^2 - 1}

●●●●●Level 33 marksStart

A curve has parametric equations

x = 3t^2,\ y = 2t^3

. Find an equation of the tangent to the curve at the point where

t = 1

●●●●●Level 34 marksStart

The curve

C

has parametric equations

x = 5t - t^2 \qquad\qquad y = t^3 + 2t \qquad\qquad t \in \mathbb{R}

(a) Show that

\frac{dy}{dx} = \frac{3t^2 + 2}{5 - 2t} \qquad t \neq \tfrac{5}{2}

(2 marks)
(b) Hence find the gradient of

C

at the point where

t = 4

, giving your answer as an exact fraction.

(2 marks)

●●●●●Level 34 marksStart

A curve

C

has parametric equations

x = 2\cos t

\quad y = \cos 2t

\quad 0 \le t \le \pi

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\cos t

. (3 marks)
(b) Find the Cartesian equation of

C

, stating the range of values of

x

for which the curve is defined. (3 marks)
(c) Verify that differentiating your Cartesian equation gives a result consistent with part (a). (1 mark)

●●●●●Level 37 marksStart

The curve

C

has parametric equations

x = \dfrac{2}{t+1}

\quad y = \dfrac{3t}{t+2}

\quad t > 0

.
(a) Show that a Cartesian equation of

C

y = \dfrac{6 - 3x}{x + 2}

. (3 marks)
(b) Find

\dfrac{dy}{dx}

in terms of

t

, and hence find the gradient of

C

at the point where

t = 1

. (3 marks)
(c) Find the range of values of

x

for points on

C

. (2 marks)

●●●●●Level 48 marksStart

A curve has parametric equations

x = 2\cos t, \; y = 2\sin t

.

Find the exact gradient of the curve at the point where

t = \dfrac{\pi}{6}

. (4 marks)

●●●●●Level 44 marksStart

The curve

C

has parametric equations

x = 4\cos t, \qquad y = 3\sin t, \qquad 0 \leqslant t < 2\pi

(a) Find

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

. (3 marks)
(b) The point

P

C

has parameter

t = \dfrac{\pi}{3}

.

Find an equation of the normal to

C

P

, giving your answer in the form

y = mx + c

, where

m

and

c

are exact constants. (6 marks)

●●●●●Level 49 marksStart

A curve has parametric equations

x = t^2,\ y = t^3

. Show that

\dfrac{d^2y}{dx^2} = \dfrac{3}{4t}

●●●●●Level 43 marksStart

A curve has parametric equations

x = 2t, \; y = t^{2}

.

Find an equation of the normal to the curve at the point where

t = 2

. (4 marks)

●●●●●Level 44 marksStart

A curve has parametric equations

x = t^{2} + 1, \; y = t^{3} - 12t

.

Find the coordinates of the stationary points of the curve. (5 marks)

●●●●●Level 45 marksStart

A design for a wooden serving platter is shown in Figure 1.

Figure 1The platter is symmetrical about its length.

The curve of the top half of the platter can be modelled by the parametric equations

x = 5t^2 + 6t \qquad y = 9t - t^3

for

0 \leq t \leq 3

, as shown in Figure 2, where

x

and

y

are measured in centimetres.

Figure 2(a) Find the length of the platter.

(2 marks)
(b) (i) Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(ii) Hence, show that the width of the platter is approximately one third of its length.

(7 marks)

●●●●●Level 49 marksStart

A furniture maker is designing a rocking chair, shown in Figure 1.

Figure 1The curve of each runner of the rocking chair can be modelled by the parametric equations

x = 20t \qquad y = 2t^2 + \frac{32}{t}

for

1 \leq t \leq 4

, as shown in Figure 2, where

x

and

y

are measured in centimetres.

P

is the point where

t = 1

and

Q

is the point where

t = 4

Figure 2(a) Show that a Cartesian equation of the curve is

y = \frac{x^2}{200} + \frac{640}{x}

(2 marks)
(b) (i) Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(ii) The chair rests on horizontal ground, with each runner touching the ground at its lowest point.

Hence, find the height of

Q

above the ground.

(6 marks)
(c) Suggest one limitation of the model.

(1 mark)

●●●●●Level 49 marksStart

A curve

C

has parametric equations

x = t^2, \qquad y = 2t, \qquad 0 \leqslant t \leqslant 3

(a) Show that the Cartesian equation of

C

y^2 = 4x

. (2 marks)
(b) Find the area of the region bounded by

C

, the

x

-axis and the line

x = 9

. (4 marks)
(c) Find an equation of the normal to

C

at the point where

t = 2

. (4 marks)

●●●●●Level 410 marksStart

The curve

C

has parametric equations

x = 2t^2 + 3 \qquad\qquad y = 3t^2 - t^3 \qquad\qquad 0 \leqslant t \leqslant 3

The point

P

with coordinates

(5, 2)

lies on

C

.
(a) Using parametric differentiation, show that the tangent to

C

P

has equation

3x - 4y = 7

(5 marks)

The curve

C

is used to model the profile of a humpback bridge. Units are in metres, with

y

being the height of the bridge above the level ground on either side of it.
(b) Find, according to the model, the greatest height of the bridge above the ground.

(1 mark)

●●●●●Level 46 marksStart

The curve

C

has parametric equations

x = t^3 + 6t \qquad\qquad y = 18 - 2t^2 \qquad\qquad -2 \leqslant t \leqslant 2

The point

P

with coordinates

(7, 16)

lies on

C

.
(a) Using parametric differentiation, show that the tangent to

C

P

has equation

4x + 9y = 172

(5 marks)

The curve

C

is used to model the profile of a section of a roller coaster track. Units are in metres, with

y

being the height of the track above the ground.
(b) Find, according to the model, the greatest height of the track above the ground.

(1 mark)

●●●●●Level 46 marksStart

The curve

C

has parametric equations

x = 4\tan t \qquad\qquad y = 5\sin 2t \qquad\qquad 0 \leqslant t < \frac{\pi}{2}

The region

R

is bounded by

C

, the

x

-axis and the line with equation

x = 4

.
(a) Show that the area of

R

is given by

\int_{0}^{a} 40\tan t\,dt

where

a

is a constant to be found.

(5 marks)
(b) Hence find the area of

R

, giving your answer to 3 significant figures.

(1 mark)

●●●●●Level 46 marksStart

Figure 1Figure 1 shows a sketch of the curve

C

with parametric equations

x = 5 - 2\cos 2\theta \qquad y = 6\sin\theta \qquad 0 \leqslant \theta \leqslant \dfrac{\pi}{2}

(a) Show that

\dfrac{dy}{dx} = k\operatorname{cosec}\theta \qquad \theta \neq 0

where

k

is a constant to be found.

(3 marks)

The point

P

lies on

C

where

\theta = \dfrac{\pi}{6}

(b) Find the equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

where

a

b

and

c

are integers.

(3 marks)
(c) Show that

C

has Cartesian equation

y^2 = 9(x - q) \qquad 3 \leqslant x \leqslant 7

where

q

is a constant to be found.

(3 marks)

●●●●●Level 49 marksStart

A particle moves in the

x

y

plane so that at time

t

seconds, where

t \ge 0

, its coordinates are given by

x = e^{2t} - 7e^{t} + 10, \qquad y = e^{t} + 4e^{-t}.

(a) Explain why the path of the particle never crosses the

x

-axis.

(1 mark)
(b) Determine the exact values of

t

when the path of the particle intersects the

y

-axis.

(2 marks)
(c) Show that

\dfrac{dy}{dx} = \dfrac{e^{2t} - 4}{e^{2t}(2e^{t} - 7)}

.

(4 marks)
(d) Hence find the coordinates of the particle when its path is parallel to the

y

-axis.

(3 marks)

●●●●●Level 410 marksStart

A curve has parametric equations

x = \dfrac{2t^2 + 1}{t}

and

y = \dfrac{t^2 - 3}{t}

, for

t \ne 0

.
(a) Find

\dfrac{dy}{dx}

in terms of

t

, giving your answer in its simplest form.

(4 marks)
(b) Explain why the curve has no stationary points.

(2 marks)
(c) By considering

x - 2y

, or otherwise, find a cartesian equation of the curve, giving your answer in a form not involving fractions or brackets.

(4 marks)

●●●●●Level 410 marksStart

A curve has parametric equations

x = t^2,\ y = t^3 - t

. Find the exact coordinates of the points where the tangent is parallel to the

x

-axis.

●●●●●Level 44 marksStart

A curve

C

has parametric equations

x = t^2 + 1, \qquad y = t^3 - 4t, \qquad t \in \mathbb{R}

(a) Find

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

. (3 marks)
(b) Find the exact coordinates of the two points on

C

at which the tangent is parallel to the

x

-axis. (4 marks)
(c) Show that

C

crosses itself, and find the coordinates of the point at which it does so. (2 marks)

●●●●●Level 59 marksStart

The curve

C

has parametric equations

x = t + \dfrac{1}{t}

y = t - \dfrac{1}{t}

t > 0

(a) Show that a Cartesian equation of

C

x^2 - y^2 = 4

. (2 marks)
(b) Find an equation of the normal to

C

at the point where

t = 2

, giving your answer in the form

ax + by = c

, where

a

b

and

c

are integers. (4 marks)
(c) The finite region

R

is bounded by

C

, the

x

-axis and the normal found in part (b).

Find the exact area of

R

. (5 marks)

●●●●●Level 511 marksStart

The curve

C

has parametric equations

x = 6\sin t, \qquad y = 2\sin 2t, \qquad 0 \leqslant t \leqslant \dfrac{\pi}{2}

(a) The point

P

C

is the point at which the tangent to

C

is parallel to the

x

-axis.

Find the exact coordinates of

P

. (4 marks)
(b) The finite region

R

is bounded by

C

and the

x

-axis.

Use integration to find the exact area of

R

. (5 marks)

●●●●●Level 59 marksStart

The curve

C

has parametric equations

x = 3\sin\theta

\qquad y = \sin 2\theta

\qquad 0 \le \theta \le \dfrac{\pi}{2}

.
(a) Find the exact coordinates of the maximum point

M

C

. (4 marks)

The finite region

R

is bounded by

C

and the

x

-axis.
(b) Find the exact area of

R

. (3 marks)
(c) Show that the line through

M

parallel to the

y

-axis divides

R

into two regions whose areas are in the ratio

\left(2\sqrt{2} - 1\right) : 1

. (3 marks)

●●●●●Level 510 marksStart

The curve

C

has parametric equations

x = 4\cos t, \qquad y = 2\sin 2t, \qquad 0 \leqslant t \leqslant \dfrac{\pi}{2}

The point

P

lies on

C

where

t = \dfrac{\pi}{6}

.
(a) Show that the tangent to

C

P

has equation

x + y = 3\sqrt{3}

. (4 marks)

The tangent to

C

P

meets the

x

-axis at the point

Q

, and

C

meets the positive

x

-axis at the point

A

.

The finite region

R

is bounded by the arc of

C

from

A

P

, the line segment

PQ

and the line segment

QA

.
(b) Find the exact area of

R

. (6 marks)

●●●●●Level 510 marksStart

A curve has parametric equations

x = t^{2}, \; y = t^{2} + t

\; t \neq 0

.
(a) Show that

\dfrac{dy}{dx} = 1 + \dfrac{1}{2t}

. (2 marks)
(b) Find

\dfrac{d^{2}y}{dx^{2}}

in terms of

t

. (3 marks)

●●●●●Level 55 marksStart

A company that builds adventure playgrounds is designing a rope bridge between two platforms, as shown in Figure 1.

Figure 1The walkway of the bridge can be modelled by the parametric equations

x = t^2 + 2t \qquad y = 0.2t^3 - 1.6t + 6

for

0 \leq t \leq 3

, as shown in Figure 2, where

x

and

y

are measured in metres and the

x

-axis represents the horizontal ground.

P

is the point where

t = 0

and

Q

is the point where

t = 3

Figure 2(a) So that children can step on and off the bridge safely, the difference in height between

P

and

Q

must be less than

1

metre.

Show that the bridge meets this safety requirement.

(3 marks)
(b) (i) Find an expression for

\dfrac{\mathrm{d}y}{\mathrm{d}x}

in terms of

t

(ii) A vertical support,

RS

, is to be added between the ground and the lowest point on the bridge, as shown in Figure 3 below.

Figure 3Find the length of

RS

(iii) Find the acute angle the bridge makes with the horizontal at

Q

Give your answer to the nearest degree.

(9 marks)

●●●●●Level 512 marksStart

The curve

C

has parametric equations

x = 12\sin^2 t \qquad\qquad y = 4\sin 2t + \sin t \qquad\qquad 0 \leqslant t \leqslant \frac{\pi}{2}

The region

R

is bounded by

C

, the

x

-axis and the line with equation

x = 9

.
(a) Show that the area of

R

is given by

\int_{0}^{a} \left(48\sin^2 2t + 24\sin^2 t\cos t\right) dt

where

a

is a constant to be found.

(5 marks)
(b) Hence, using algebraic integration, find the exact area of

R

.

(4 marks)

●●●●●Level 59 marksStart

The curve

C

has parametric equations

x = \tan t + 5 \qquad\qquad y = \sec^2 t + 2 \qquad\qquad -\frac{\pi}{2} < t < \frac{\pi}{4}

The line

l

is the normal to

C

at the point

P

where

t = -\dfrac{\pi}{4}

(a) Using parametric differentiation, show that an equation for

l

y = \tfrac{1}{2}x + 2

(5 marks)
(b) Show that all points on

C

satisfy the equation

y = (x - 5)^2 + 3

(2 marks)

The straight line with equation

y = \tfrac{1}{2}x + k

, where

k

is a constant, intersects

C

at two distinct points.
(c) Find the range of possible values for

k

.

(5 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of the curve

C

with parametric equations

x = \cos 2t \qquad y = \sin t\cos 2t \qquad -\dfrac{\pi}{2} < t < \dfrac{\pi}{2}

where

t

is a parameter.

The point

P

lies on

C

where

t = \dfrac{\pi}{6}

The line

l

, shown in Figure 1, is the tangent to

C

P

.
(a) Use parametric differentiation to show that

(i)

\dfrac{dy}{dx} = p\sin t + q\operatorname{cosec} t

where

p

and

q

are constants to be found

(ii) an equation for

l

2x - 8y + 1 = 0

(6 marks)

The line

l

intersects the curve

C

again at the point

Q

.
(b) Using algebra and showing detailed reasoning, find the exact coordinates of

Q

.

(6 marks)

●●●●●Level 512 marksStart

A curve has parametric equations

x = 2t^2

y = t^3

. The point

P

on the curve has parameter

p

, where

p > 0

.
(a) Show that the equation of the tangent at

P

can be written as

4y = 3px - 2p^3

.

(4 marks)

The tangent to this curve at

P

crosses the

x

-axis at the point

A

and the normal to this curve at

P

crosses the

x

-axis at the point

B

.
(b) Show that the ratio

PA : PB

4 : 3p

.

(8 marks)

●●●●●Level 512 marksStart

The diagram shows the curve

C

with parametric equations

x = e^{t}, \qquad y = 4t - t^2, \qquad \text{where } 0 \le t \le 4.

The maximum point on

C

is denoted by

P

.
(a) Determine the exact coordinates of

P

.

(4 marks)

The shaded region

R

is enclosed by the curve, the

x

-axis and the lines

x = e

and

x = e^3

.
(b) Show that the area of

R

is given by

\displaystyle\int_a^b (4t - t^2)e^{t}\,dt

, where

a

and

b

are constants to be determined.

(3 marks)
(c) Hence determine the exact area of

R

.

(5 marks)

●●●●●Level 512 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with parametric equations

x = t^2 + 2t \qquad y = 2t - \dfrac{2}{t} \qquad t > 0

The curve

C

intersects the

x

-axis at the point

Q

.
(a) Find the

x

coordinate of

Q

.

(1 mark)

The line

l

is the normal to

C

at the point

P

as shown in Figure 1.

Given that

t = 2

P

(b) write down the coordinates of

P

(1 mark)
(c) Using calculus, show that an equation of

l

12x + 5y = 111

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by the curve

C

, the line

l

and the

x

-axis.
(d) Using algebraic integration, find the exact volume of the solid of revolution formed when the region

R

is rotated through

2\pi

radians about the

x

-axis. Give your answer in the form

\pi(p + q\ln 2)

where

p

and

q

are rational constants.

(7 marks)

●●●●●Level 512 marksStart

A curve has parametric equations

x = \sin t,\ y = \sin 2t

. Find

\dfrac{dy}{dx}

in terms of

t

, and its exact value at

t = \dfrac{\pi}{6}

●●●●●Level 53 marksStart

Implicit differentiation

38 questions

LessonNot started

The curve

C

has equation

x^2 + y^2 - 2x - 6y = 0

. Find the coordinates of the points on

C

where the tangent is horizontal.

●●●●●Level 34 marksStart

The curve

C

has equation

x^{2} + 2y^{2} = 9

.

Find the gradient of

C

at the point

(1, 2)

. (3 marks)

●●●●●Level 33 marksStart

The curve

C

has equation

x^{2} + xy + y^{2} = 13

.

Find the gradient of

C

at the point

(1, 3)

. (4 marks)

●●●●●Level 34 marksStart

A circle has equation

x^2 + y^2 = 25

. Find

\dfrac{dy}{dx}

in terms of

x

and

y

, and the gradient at the point

(3, 4)

●●●●●Level 33 marksStart

Find

\dfrac{dy}{dx}

for the curve

x^2 + 3xy + y^2 = 11

, and the gradient at the point

(1, 2)

●●●●●Level 34 marksStart

The curve

C

has equation

x^3 + y^3 = 9xy

and passes through the point

(2, 4)

. Find the gradient of

C

at this point.

●●●●●Level 34 marksStart

The curve

C

has equation

x^2 - xy + y^2 = 7

. Find the gradient of

C

at the point

(3, 2)

●●●●●Level 34 marksStart

A curve has equation

xy = 8

. Find

\dfrac{dy}{dx}

in terms of

x

and

y

, and the gradient at the point

(2, 4)

●●●●●Level 33 marksStart

The curve

C

has equation

x^2 y + x y^2 = 30

. The point

P(2, 3)

lies on

C

. Find the exact gradient of

C

P

●●●●●Level 35 marksStart

The circle

C

has equation

x^{2} + y^{2} = 20

.

Find the gradient of

C

at the point

(2, 4)

. (3 marks)

●●●●●Level 33 marksStart

The curve

C

has equation

x^{2} + y^{2} + 2x - 4y = 20

.

Find the coordinates of the points on

C

where the tangent is parallel to the

x

-axis. (5 marks)

●●●●●Level 35 marksStart

The curve

C

has equation

x^{2} + y^{2} + 2x - 4y = 20

.

Find the coordinates of the points on

C

where the tangent is parallel to the

y

-axis. (3 marks)

●●●●●Level 33 marksStart

The curve

C

has equation

y^{2} = 4x

.

Find an equation of the tangent to

C

at the point

(1, 2)

. (4 marks)

●●●●●Level 34 marksStart

A curve has equation

2xy^3 + ye^{3x} = 7

Find an expression for

\dfrac{dy}{dx}

in terms of

x

and

y

(4 marks)

●●●●●Level 34 marksStart

Figure 1The curve

C

, shown in Figure 1, has equation

2y^2 - x^2y + 12x = k \qquad y > 0

where

k

is a constant.
(a) Find

\dfrac{dy}{dx}

in terms of

x

and

y

(5 marks)

The point

P(p, 3)

, where

p

is a constant, lies on

C

.

Given that

P

is the minimum turning point on

C

,
(b) find

(i) the value of

p

(ii) the value of

k

(4 marks)

●●●●●Level 49 marksStart

The curve

C

has equation

x^2 + xy + y^2 = 12

Find the coordinates of the two points on

C

at which the tangent is parallel to the

x

-axis. (7 marks)

●●●●●Level 47 marksStart

A curve has equation

x^{2} + xy + y^{2} = 12

.

Find the coordinates of the two points on the curve at which the tangent is parallel to the

x

-axis. (7 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

2x^2 + xy + y^2 = 16

.
(a) Show that

\dfrac{dy}{dx} = -\dfrac{4x + y}{x + 2y}

. (4 marks)
(b) The point

P(2, 2)

lies on

C

. Find an equation of the tangent to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers. (2 marks)
(c) Find a relationship between

x

and

y

that holds at any point of

C

where the tangent is parallel to the

x

-axis. (1 mark)

●●●●●Level 47 marksStart

The curve

C

has equation

x^2 - xy + y^2 = 7

. The point

A(3, 2)

lies on

C

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{y - 2x}{2y - x}

. (3 marks)
(b) Find the equation of the tangent to

C

A

. (2 marks)
(c) The normal to

C

A

meets the curve again at the point

B

. Determine the coordinates of

B

. (3 marks)

●●●●●Level 48 marksStart

The ellipse

x^2 + 4y^2 = 8

passes through

(2, 1)

. Find an equation of the tangent to the ellipse at this point.

●●●●●Level 44 marksStart

The curve

C

has equation

x^2 + 3xy - y^2 = 3

The point

P(1, 2)

lies on

C

.

Find an equation of the tangent to

C

P

. (7 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

2x^2 - 3xy + 2y^2 = 14

The points

P

and

Q

lie on

C

. The tangent to

C

at each of the points

P

and

Q

is parallel to the

x

-axis.

Find the coordinates of

P

and

Q

, showing all your working. (7 marks)

●●●●●Level 47 marksStart

A curve has equation

xe^{2y} + y^2 - 3x^2 = k

where

k

is a constant.

The curve has a stationary point at which

y = \ln 3

Determine the exact value of

k

(7 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

xy = 12

.

Find an equation of the tangent to

C

at the point

(3, 4)

, giving your answer in the form

ax + by = c

with integer coefficients. (4 marks)

●●●●●Level 44 marksStart

A curve has equation

2y^3e^{3x} + 5y + 48x = k

where

k

is a constant.

The curve has a stationary point on the

y

-axis.

Determine the value of

k

(7 marks)

●●●●●Level 47 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A curve

C

has equation

2x^2 + xy + y^2 = 44

(a) Find

\dfrac{dy}{dx}

in terms of

x

and

y

.

(4 marks)

The point

P(4, 2)

lies on

C

.
(b) Find an equation of the normal to

C

P

, giving your answer in the form

ax + by + c = 0

, where

a

b

and

c

are integers to be found.

(3 marks)

●●●●●Level 47 marksStart

A curve

C

has equation

x^2 + xy + y^2 = 12

(a) Show that

\frac{dy}{dx} = -\frac{2x + y}{x + 2y}

(4 marks)
(b) Hence find the coordinates of the two points on

C

at which the tangent to

C

is parallel to the

x

-axis.

(4 marks)

●●●●●Level 48 marksStart

A curve has equation

x^3 - 6xy + 3y^2 = -7

.

Show that there are no points on this curve where the tangent is parallel to

y = x

.

(8 marks)

●●●●●Level 48 marksStart

The diagram shows the curve with equation

x^2 - 4xy + 5y^2 + 2y - k = 0

, where

k

is a positive integer.

At the points

P

and

Q

on the curve, the tangents to the curve are parallel to the

y

-axis.

Given that the difference in the

y

-coordinates of

P

and

Q

4

, determine the

x

-coordinates of

P

and

Q

.

(7 marks)

●●●●●Level 47 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with equation

2^x + xy^2 - 5y = 1

The point

P

has coordinates

(2, 1)

.
(a) Verify that

P

lies on

C

.

(1 mark)
(b) Find

\dfrac{dy}{dx}

in terms of

x

and

y

.

(5 marks)

The normal to

C

P

crosses the

x

-axis at a point

Q

.
(c) Find the

x

coordinate of

Q

, giving your answer in the form

a + b\ln 2

where

a

and

b

are integers.

(3 marks)

●●●●●Level 49 marksStart

The curve

C

has equation

x^2 + 2xy + 3y^2 = 6

(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{x + y}{x + 3y}

(3 marks)
(b) Hence find the exact coordinates of the two points on

C

at which the tangent is parallel to the

x

-axis. (4 marks)
(c) Show that there are exactly two points on

C

at which

\dfrac{\mathrm{d}y}{\mathrm{d}x}

is not defined, and find their coordinates. (3 marks)

●●●●●Level 510 marksStart

The curve

C

has equation

x^2 - 4xy + 5y^2 = 4

.
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2y - x}{5y - 2x}

. (3 marks)
(b) Find the exact coordinates of the two points on

C

at which the tangent is parallel to the

y

-axis. (3 marks)
(c) The point

P(4, 2)

lies on

C

. The normal to

C

P

meets

C

again at the point

Q

. Find the coordinates of

Q

. (3 marks)

●●●●●Level 59 marksStart

The curve

C

has equation

2x^3 + y^2 - 6xy = 0

C

passes through the origin

O

and has a stationary point at

P

, where

P

is not

O

, as shown in the diagram below.(a) Show that the

x

-coordinate of any stationary point of

C

satisfies the equation

x^3(x - 4) = 0

(7 marks)
(b) Hence, find the coordinates of

P

(2 marks)

●●●●●Level 59 marksStart

In this question you must show detailed reasoning.

The curve

C

has equation

x^2 - 2xy + 2y^2 = 8

The tangent to

C

at the point

P

is parallel to the

x

-axis, and the tangent to

C

at the point

Q

is parallel to the

y

-axis. Both

P

and

Q

lie in the first quadrant.

Determine the exact area of triangle

OPQ

, where

O

is the origin. (7 marks)

●●●●●Level 57 marksStart

The curve

C

has equation

x^2 - xy + y^2 = 12

(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2x - y}{x - 2y}

. (3 marks)
(b) Find the coordinates of the two points on

C

at which the tangent is parallel to the

y

-axis. (4 marks)
(c) The line with equation

y = x + k

, where

k

is a constant, is a tangent to

C

.

Find the possible values of

k

. (3 marks)

●●●●●Level 510 marksStart

The diagram below shows the approximate shape of the vertical cross section of a natural rock arch above a horizontal footpath.

The inside of the arch meets the footpath at the points

O

and

P

Rhiannon models the shape of the inside of the arch using the equation

x^2 + 2y^2 = a\sqrt{x}

where

a

is a constant, and

x

and

y

are the horizontal and vertical distances respectively, in metres, measured from

O

(a) The distance

OP

5

metres.

Find the exact value of

a

that Rhiannon should use in the model.

(2 marks)
(b) Show that the maximum height of the inside of the arch above the footpath is approximately

2.43

metres.

(6 marks)
(c) Suggest one limitation of the model Rhiannon has used.

(1 mark)

●●●●●Level 59 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y^3 + x^2y = 10

(a) Find

\dfrac{dy}{dx}

in terms of

x

and

y

.

(5 marks)

The point

P(3, 1)

lies on

C

.
(b) Show that the normal to

C

P

has equation

y = 2x - 5

(2 marks)
(c) Prove that the normal to

C

P

does not meet

C

again.

You should use algebra for your proof and make your reasoning clear.

(5 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.The diagram shows the curve with equation

(3x - y)^2 = 12x + 4y

.
(a) Show that

\dfrac{dy}{dx} = \dfrac{9x - 3y - 6}{3x - y + 2}

.

(3 marks)

At the point

P

on the curve the tangent to the curve is parallel to the

y

-axis and at the point

Q

on the curve the tangent to the curve is parallel to the

x

-axis.
(b) Show that the distance

PQ

k\sqrt{10}

, where

k

is a rational number to be determined.

(8 marks)

●●●●●Level 511 marksStart

Convex, concave and points of inflection

37 questions

LessonNot started

The curve

y = x^3 - 6x^2 + 5

has a point of inflection where

x

is equal to

Circle your answer.

A $0$
B $2$
C $4$
D $-2$

●●●●●Level 11 markStart

Find the set of values of

x

for which the curve

y = x^{3} - 12x

is convex. (2 marks)

●●●●●Level 22 marksStart

Find the set of values of

x

for which the curve

y = x^3 - 6x^2 + 5

is convex.

●●●●●Level 23 marksStart

\text{f}(x) = x^3 - 3x^2 - 24x + 7

(a) Find

\text{f}''(x)

(2 marks)
(b) (i) Solve

\text{f}''(x) = 0

(ii) Hence find the range of values of

x

for which

\text{f}

is concave.

(2 marks)

●●●●●Level 24 marksStart

The curve

C

has equation

y = x^3 - 3x^2 + 5

.
(a) Show that

C

has a point of inflection at

(1, 3)

. (4 marks)
(b) State the range of values of

x

for which

C

is convex. (1 mark)
(c) Determine whether the point of inflection is a stationary point of

C

. (2 marks)

●●●●●Level 37 marksStart

Find the coordinates of the point of inflection of the curve

y = x^3 - 3x^2 - 9x + 1

●●●●●Level 34 marksStart

Explain why the curve

y = x^4

does not have a point of inflection at the origin, even though

\dfrac{d^2y}{dx^2} = 0

there.

●●●●●Level 33 marksStart

Find the set of values of

x

for which the curve

y = x^4 - 6x^2 + 5

is concave, and state the

x

-coordinates of its points of inflection.

●●●●●Level 34 marksStart

Find the exact coordinates of the point of inflection of the curve

y = x e^x

●●●●●Level 34 marksStart

The function f is defined by

\mathrm{f}(x) = e^{2x} + e^{-2x} - 4x^2 \quad \text{for } x \in \mathbb{R}

Determine whether the curve with equation

y = \mathrm{f}(x)

has a point of inflection at the point where

x = 0

Fully justify your answer.

(4 marks)

●●●●●Level 34 marksStart

Find the coordinates of the point of inflection of the curve

y = x^3 - 3x^2 + 4

●●●●●Level 34 marksStart

Show that the curve

y = x^3 + 2

has a stationary point of inflection, and state its coordinates.

●●●●●Level 34 marksStart

The curve

C

has equation

y = x^4 - 4x^3

. Find the coordinates of the points of inflection of

C

, and the set of values of

x

for which

C

is convex.

●●●●●Level 35 marksStart

The curve

C

has equation

y = x^{4} - 24x^{2}

.

Find the coordinates of the points of inflection of

C

. (5 marks)

●●●●●Level 35 marksStart

The curve

C

has equation

y = x^{4} - 12x^{2}

.
(a) Find the set of values of

x

for which

C

is convex. (3 marks)
(b) Find the exact coordinates of the points of inflection of

C

. (2 marks)

●●●●●Level 35 marksStart

In this question you must show detailed reasoning.

A curve has equation

y = x^3 - 6x^2 + 15x - 2

.
(a) Show that the curve has no stationary points.

(2 marks)
(b) Show that the curve has exactly one point of inflection.

(2 marks)

●●●●●Level 34 marksStart

The function f is defined by

\mathrm{f}(x) = x^3 + \sin 2x \quad \text{for } -\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}

Determine whether the curve with equation

y = \mathrm{f}(x)

has a point of inflection at the point where

x = 0

Fully justify your answer.

(4 marks)

●●●●●Level 34 marksStart

The curve

C

has equation

y = x^{3} - 6x^{2} + 9x + 1

.

Find the coordinates of the point of inflection of

C

. (4 marks)

●●●●●Level 34 marksStart

y = 4x^{\frac{3}{2}} - 3x^2 + 5 \qquad x > 0

(a) Find, in simplest form,

(i)

\dfrac{dy}{dx}

(ii)

\dfrac{d^2y}{dx^2}

(3 marks)
(b) Hence find the exact value of

x

for which

\dfrac{d^2y}{dx^2} = 0

(2 marks)

●●●●●Level 35 marksStart

Show that the curve

y = x^{4} + 2x^{2}

has no point of inflection. (3 marks)

●●●●●Level 33 marksStart

The diagram shows part of the curve

y = \ln(1 + x^2)

.(a) Use the trapezium rule with 4 intervals of equal width to find an estimate for

\displaystyle\int_0^2 \ln(1 + x^2)\,dx

. Give your answer correct to 3 significant figures.

(4 marks)
(b) Explain how the trapezium rule could be used to obtain a more accurate estimate for

\displaystyle\int_0^2 \ln(1 + x^2)\,dx

.

(1 mark)
(c) Explain why it is not clear from the diagram whether the value from part (a) is an under-estimate or an over-estimate for

\displaystyle\int_0^2 \ln(1 + x^2)\,dx

.

(2 marks)

●●●●●Level 37 marksStart

The diagram shows part of the curve

y = \dfrac{1}{1 + \sqrt{x}}

.(a) Use the trapezium rule with 4 intervals to find an estimate for

\displaystyle\int_0^4 \frac{1}{1 + \sqrt{x}}\,dx

. Give your answer correct to 3 significant figures.

(4 marks)
(b) State whether the value from part (a) is an under-estimate or an over-estimate, giving a reason for your answer.

(1 mark)
(c) Explain how the trapezium rule could be used to obtain a more accurate estimate.

(1 mark)

●●●●●Level 36 marksStart

The curve

C

has equation

y = x^{3} + 3x^{2}

.

Find the coordinates of the point of inflection of

C

, justifying your answer. (4 marks)

●●●●●Level 34 marksStart

Find the set of values of

x

for which the curve

y = 2x^{3} - 9x^{2} + 12x

is concave. (3 marks)

●●●●●Level 33 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y = 2x^{\frac{3}{2}} + \dfrac{96}{\sqrt{x}} - 15 \qquad x > 0

(a) Find

\dfrac{dy}{dx}

, giving each term in simplest form.

(2 marks)
(b) Hence find the

x

coordinate of the stationary point of

C

.

(2 marks)
(c) (i) Find

\dfrac{d^2y}{dx^2}

, giving each term in simplest form.

(ii) Hence determine the nature of the stationary point of

C

, giving a reason for your answer.

(2 marks)
(d) State the range of values of

x

for which

y

is decreasing.

(1 mark)

●●●●●Level 37 marksStart

I = \int_1^3 \sqrt{1 + x^3}\;\mathrm{d}x

Figure 1 shows the region whose area is given by

I

.Figure 1
(a) Use the trapezium rule with four strips of equal width to find an estimate for

I

, giving your answer to

4

decimal places. (4 marks)
(b) State, with a reason, whether your answer to part (a) is an overestimate or an underestimate of

I

. (2 marks)

●●●●●Level 46 marksStart

The curve

C

has equation

y = x^2 e^{-x}

.
(a) Show that

\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = (x^2 - 4x + 2)e^{-x}

. (3 marks)
(b) Hence find the exact

x

-coordinates of the points of inflection of

C

. Fully justify your answer. (4 marks)

●●●●●Level 47 marksStart

The curve

C

has equation

y = x^2 \ln x

\quad x > 0

.
(a) Show that

\dfrac{dy}{dx} = x(2\ln x + 1)

. (3 marks)
(b) Hence find the exact coordinates of the stationary point of

C

. (3 marks)
(c) Determine the nature of this stationary point. (2 marks)
(d) Find the range of values of

x

for which

C

is convex. (1 mark)

●●●●●Level 49 marksStart

A curve has equation

y = 2^{x^2 - 6x}

.
(a) Determine the coordinates of any stationary points on the curve.

(4 marks)
(b) Show that the curve is convex for all values of

x

.

(5 marks)

●●●●●Level 49 marksStart

Figure 1Figure 1 shows an open-topped tray used for growing seedlings.

The tray is in the shape of a cuboid and is made of thin sheet plastic.

The base of the tray is a rectangle

5x

cm by

2x

cm.

The height of the tray is

h

cm, as shown in Figure 1.

The diagram is not drawn to scale.

Given that the capacity of the tray is

1500

^3

(a) show that the area

S

^2

of the plastic used to make the tray is given by

S = Px^2 + \dfrac{Q}{x}

where

P

and

Q

are positive constants to be found.

(4 marks)
(b) Use calculus to find the value of

x

for which

S

has a stationary value, giving your answer to 3 significant figures.

(3 marks)
(c) Find

\dfrac{d^2S}{dx^2}

and hence show that the value of

x

found in part (b) gives the minimum value of

S

.

(2 marks)

●●●●●Level 49 marksStart

(a) The graph of the function

y = f(x)

passes through the point

P

with coordinates

(-3, 8)

, and is a one-one function. State the coordinates of the point corresponding to

P

on each of the following curves.

(i)

y = f(x - 4)

(ii)

y = 3f(x) + 2

(iii)

y = f^{-1}(x)

(4 marks)
(b) The diagram shows part of the graph of

y = g'(x)

. This is the graph of the gradient function of

y = g(x)

. The graph of

y = g'(x)

meets the

x

-axis at

x = -3

and

x = 3

only, and has a maximum point where

x = -1

.

(i) State the

x

-coordinate of any stationary points on the graph of

y = g(x)

.

(ii) State the set of values of

x

for which

y = g(x)

is a decreasing function.

(iii) State the

x

-coordinate of any points of inflection on the graph of

y = g(x)

.

(3 marks)

●●●●●Level 47 marksStart

The diagram shows the curve

M

with equation

y = (\ln x)^2

, for

x > 0

.
(a) Show that

M

has a point of inflection at the point

P

where

x = e

.

(5 marks)

The shaded region

R

is enclosed by

M

, the

x

-axis and the line through

P

parallel to the

y

-axis.
(b) Determine the exact area of

R

, giving your answer in the form

ae + b

, where

a

and

b

are integers to be determined.

(6 marks)

●●●●●Level 511 marksStart

Figure 4 shows part of the curve

C

with equation

y = \dfrac{\ln x}{x^2}, \qquad x > 0

Figure 4
(a) Show that

\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}

. (3 marks)
(b) Find the exact coordinates of the stationary point of

C

, and determine its nature. (5 marks)
(c) Find the exact

x

-coordinate of the point of inflection of

C

. (2 marks)

●●●●●Level 510 marksStart

A company makes open-topped containers for storing grain.

A container is modelled so that

the base of the container is a circle of radius $r$ cm
the curved surface of the container is perpendicular to the base
the container has no lid
the total area of the base and the curved surface is $300\pi$ cm $^2$

The volume of the container is

V

^3

.
(a) Show that

V = \frac{\pi}{2}\left(300r - r^3\right)

making your method clear.

(4 marks)

Using algebraic differentiation,
(b) find the value of

r

for which

V

has a stationary point.

(4 marks)
(c) Prove, by further differentiation, that this value of

r

gives the maximum volume of the container.

(2 marks)
(d) Hence find the maximum volume of the container, giving your answer in terms of

\pi

.

(1 mark)

●●●●●Level 511 marksStart

In this question you must show detailed reasoning.

The function f is defined by

f(x) = \sin 2x + 2\sin x

with domain

0 \le x \le 2\pi

.
(a) Solve the following equations.

(i)

f'(x) = 0

(ii)

f''(x) = 0

(7 marks)

The diagram shows the graph of the gradient function

y = f'(x)

for the domain

0 \le x \le 2\pi

. The points

A

C

and

E

are where the graph meets the

x

-axis, and

B

and

D

are minimum points.(b) Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points

A

B

C

D

and

E

.

(3 marks)
(c) (i) Explain how to use the graph of the gradient function to find the values of

x

for which

f(x)

is increasing.

(ii) Using set notation, write down the set of values of

x

for which

f(x)

is increasing in the domain

0 \le x \le 2\pi

.

(3 marks)

●●●●●Level 513 marksStart

Figure 1Figure 1 shows a sketch of an open container in the shape of a prism.

The faces

ABFE

and

CDEF

are rectangles.

The ends

ADE

and

BCF

are congruent right-angled triangles, with the right angles at

A

and

B

.

The top

ABCD

is open, and the container is made from sheet steel of negligible thickness.

The diagram is not drawn to scale.

Given that

$AE = BF = 3x$ metres
$AD = BC = 4x$ metres
$AB = DC = EF = L$ metres

and the capacity of the container is

6

^3

(a) show that the area of steel used to make the container,

S

^2

, is given by

S = Px^2 + \dfrac{Q}{x}

where

P

and

Q

are positive integers to be found.

(4 marks)

Given that

x

can vary,
(b) use algebraic calculus to find the minimum value of

S

, giving your answer to one decimal place.

(5 marks)
(c) Justify that the value of

S

found in part (b) is a minimum.

(2 marks)

●●●●●Level 511 marksStart

A curve has equation

y = k\cos x + x - x^2

, where

x

is in radians and

k

is a positive constant.
(a) Given that the curve has a point of inflection where

x = \tfrac23\pi

, show that

k = 4

.

(5 marks)

It is also given that the curve crosses the positive

x

-axis at exactly one point.
(b) Show by calculation that the

x

-coordinate of this point lies between

1.2

and

1.6

.

(2 marks)
(c) Starting with

x_0 = 1.4

, use the Newton-Raphson method to find this

x

●●●●●Level 511 marksStart

Connected rates of change

25 questions

LessonNot started

The radius of a circular oil slick increases at a rate of

0.2\text{ m s}^{-1}

. Find the rate at which its area is increasing at the instant when the radius is

10\text{ m}

●●●●●Level 23 marksStart

Water is draining from a conical filter funnel.

In a model of this situation,

the surface of the water in the funnel is always a circle whose radius is half the depth of the water
the rate of decrease of the volume of the water in the funnel is proportional to the area of the surface of the water

At time

t

seconds after the water starts to drain, the depth of the water in the funnel is

h

cm.
(a) Show that

\frac{dh}{dt} = -\lambda

where

\lambda

is a positive constant.

(3 marks)

When the water started to drain, the depth of the water in the funnel was

12

cm. The funnel emptied completely

80

seconds later.
(b) Find the value of

\lambda

.

(1 mark)

●●●●●Level 34 marksStart

Water drains from an inverted cone whose height always equals its radius, so

V = \tfrac13\pi r^3

, at

8\text{ cm}^3\text{s}^{-1}

. Find the rate at which the radius is decreasing when

r = 4\text{ cm}

●●●●●Level 33 marksStart

A cylindrical rainwater tank has a radius of

0.8

m and is being filled with water from a downpipe.

In a model of this situation,

the walls of the tank have negligible thickness
the rate of increase of the volume of water in the tank is inversely proportional to the depth of the water

At time

t

minutes after the tank starts to fill, the depth of the water in the tank is

h

metres.
(a) Show that

\frac{dh}{dt} = \frac{\lambda}{h}

where

\lambda

is a positive constant.

(3 marks)

When the depth of the water is

0.5

m, the depth is increasing at a rate of

0.04

m per minute.
(b) Write down the value of

\lambda

.

(1 mark)

●●●●●Level 34 marksStart

Air is pumped into a spherical balloon at

50\text{ cm}^3\text{s}^{-1}

. Find the rate at which the radius is increasing when

r = 5\text{ cm}

\left(V = \tfrac43\pi r^3\right)

●●●●●Level 33 marksStart

5\text{ m}

ladder leans against a vertical wall. Its foot slides away from the wall at

0.5\text{ m s}^{-1}

. Find the rate at which the top of the ladder slides down when the foot is

3\text{ m}

from the wall.

●●●●●Level 34 marksStart

The side of a cube increases at

0.1\text{ cm s}^{-1}

. Find the rate at which its volume is increasing when the side is

5\text{ cm}

●●●●●Level 33 marksStart

A spherical balloon is inflated at

100\text{ cm}^3\text{s}^{-1}

. Show that its surface area

S = 4\pi r^2

increases at

\dfrac{200}{r}\text{ cm}^2\text{s}^{-1}

, and find this rate when

r = 5\text{ cm}

●●●●●Level 34 marksStart

Water is poured into a cylinder of radius

10\text{ cm}

20\pi\text{ cm}^3\text{s}^{-1}

. Find the rate at which the water level rises.

\left(V = \pi r^2 h\right)

●●●●●Level 33 marksStart

A spherical balloon is inflated so that its volume increases at a constant rate of

50\ \text{cm}^3\,\text{s}^{-1}

. The volume is

V = \tfrac43\pi r^3

.
(a) Find

\dfrac{\mathrm{d}V}{\mathrm{d}r}

when

r = 5

. (2 marks)
(b) Hence find the rate at which the radius is increasing when

r = 5

, to 3 s.f. (4 marks)

●●●●●Level 36 marksStart

An empty garden pond has vertical sides and a horizontal base of area

4

square metres.

The pond is filled with water from a hose at a rate of

0.06

cubic metres per minute.

At time

t

minutes after the hose is switched on, the depth of water in the pond is

h

metres and water soaks away through the base of the pond at a rate of

0.02h

cubic metres per minute.

At time

t

minutes after the hose is switched on, the volume of water in the pond is

V

cubic metres.

Show that

\frac{dV}{dt} = \frac{12 - V}{200}

(4 marks)

●●●●●Level 34 marksStart

A snowball is rolled through fresh snow. The snowball is modelled as a sphere with radius

r

cm.

The surface area of the snowball,

S

cm², is increasing at a constant rate,

k

cm² per second, where

k

is a constant.

Given that

S = 4\pi r^2

, show that the rate of increase of the radius of the snowball with respect to time is inversely proportional to the radius.

(4 marks)

●●●●●Level 34 marksStart

Water is poured at

4\text{ cm}^3\text{s}^{-1}

into an inverted cone of radius

6\text{ cm}

and height

12\text{ cm}

. Find the rate at which the water level rises when the depth is

2\text{ cm}

●●●●●Level 35 marksStart

The edge length

x

cm of a cube increases at a constant rate of

0.2\ \text{cm s}^{-1}

. The volume is

V = x^3

.
(a) Find

\dfrac{\mathrm{d}V}{\mathrm{d}x}

when

x = 4

. (2 marks)
(b) Hence find the rate of increase of the volume when

x = 4

. (3 marks)

●●●●●Level 35 marksStart

Figure 5 shows a container in the shape of a right circular cone with its vertex pointing downwards. The cone has radius

6

cm and height

18

cm.Figure 5 (not accurately drawn)

Water is poured into the container at a constant rate of

8

cm³ s⁻¹. At time

t

seconds the depth of the water is

h

cm and the radius of the water surface is

r

cm.
(a) Show that the volume

V

cm³ of water in the container is given by

V = \dfrac{\pi h^3}{27}

. (3 marks)
(b) Find the rate at which the depth of the water is increasing at the instant when

h = 6

, giving your answer to

3

significant figures. (4 marks)
(c) Find the rate at which the area of the water surface is increasing at that same instant, giving your answer to

3

significant figures. (3 marks)

[The volume of a cone of radius

r

and height

h

\tfrac13\pi r^2 h

●●●●●Level 410 marksStart

A cube is expanding. At time

t

seconds the length of each edge is

x

cm, the volume is

V

cm³ and the total surface area is

A

cm².

The volume is increasing at the constant rate of

24

cm³ s⁻¹.
(a) Find the rate at which

x

is increasing at the instant when

x = 4

. (4 marks)
(b) Find the rate at which

A

is increasing at that same instant. (3 marks)
(c) Show that, at any instant, the rate at which the surface area is increasing is inversely proportional to

x

. (2 marks)

●●●●●Level 49 marksStart

A container is in the shape of an inverted right circular cone, with its vertex at the bottom. The container has height

60

cm and the radius of its circular top is

20

cm.

Water is poured into the empty container at a constant rate of

45

cm³ s⁻¹. At time

t

seconds the depth of the water is

h

cm and the volume of water in the container is

V

cm³.
(a) Show that

V = \dfrac{\pi h^3}{27}

. (2 marks)
(b) Find the rate at which the depth of the water is increasing at the instant when

h = 15

r

and height

h

V = \frac{1}{3}\pi r^2 h

●●●●●Level 47 marksStart

A spherical balloon is being inflated so that its volume increases at the constant rate of

200

cm³ s⁻¹.
(a) Find the rate at which the radius is increasing at the instant when the radius is

5

cm, giving your answer exactly. (4 marks)
(b) Find the rate at which the surface area is increasing at the same instant. (3 marks)

[The volume and surface area of a sphere are

V = \frac{4}{3}\pi r^3

and

S = 4\pi r^2

●●●●●Level 47 marksStart

Resin is poured into an empty conical mould at a constant rate of 18 cm³/s

The mould is a hollow inverted cone. After

t

seconds the depth of the resin in the inverted cone is

h

cm, as shown in the diagram below.When the depth of the resin in the inverted cone is

h

cm, the volume,

V

cm³, is given by

V = \dfrac{1}{12}\pi h^3

(a) Show that when

t = 4

\dfrac{dV}{dh} = 18\sqrt[3]{2\pi}

(4 marks)
(b) Hence, find the rate at which the depth is increasing when

t = 4

Give your answer to three significant figures.

(3 marks)

●●●●●Level 47 marksStart

Sand is poured onto level ground at a constant rate of

0.5

^3

per minute. The sand forms a pile in the shape of a right circular cone whose height

h

metres is always equal to twice the base radius.
(a) Show that the volume

V

^3

of the pile is given by

V = \dfrac{\pi h^3}{12}

. (2 marks)
(b) Determine the exact rate at which the height of the pile is increasing at the instant when

h = 3

. (4 marks)
(c) Find the rate at which the base radius is increasing at the same instant. (2 marks)

●●●●●Level 48 marksStart

A garden water feature is being filled with water.

The container is modelled as a hollow cone with its vertex pointing vertically downwards, where

the depth of the container is $24$ cm
the radius of the circular top of the container is $8$ cm
water is poured into the container at a constant rate of $30$ cm $^3$ per second