A-level · Differentiation

Tangents and normalsExam-style questions

The straight lines that touch a curve, and the ones at right angles to it. Where a tangent's point and gradient come from; writing its equation in the form asked for; the normal and why its gradient is the negative reciprocal; finding the point where a tangent has a given gradient; where tangents and normals meet the axes; and where a tangent meets the curve again.

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15 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A function is defined by $\mathrm{f}(x) = x^{2} + \dfrac{2}{x}$ for $x \neq 0$ . The graph of $y = \mathrm{f}(x)$ is shown in the diagram.
(a) Find the set of values of $x$ for which $\mathrm{f}(x)$ is decreasing.

(5 marks)
(b) The tangent to the curve at the point $P$ , where $x = 2$ , and the normal to the curve at $P$ form a triangle with the $x$ -axis.

Find the area of this triangle. Give your answer correct to 3 significant figures.

(8 marks)
Level 5 · 13 marksTry it
2
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A curve has equation $y = \text{f}(x)$ , $x > 0$ , where

$\text{f}'(x) = 6\sqrt{x} + \dfrac{k}{\sqrt{x}} - 22$

and $k$ is a constant.

The tangent to the curve at the point $P(16, 70)$ is parallel to the line with equation $12x - 2y + 7 = 0$
(a) (i) Show that $k = 16$

(ii) Find an equation of the tangent to the curve at $P$ , giving your answer in the form $y = mx + c$

(4 marks)
(b) Hence find the value of $\text{f}''(16)$ .

(3 marks)
(c) Find $\text{f}(x)$ .

(4 marks)
Level 5 · 11 marksTry it
3
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 $y = \text{f}(x)$ , $x > 0$ , where

$\text{f}'(x) = \dfrac{ax^2 + bx - 18}{x\sqrt{x}}$

and $a$ and $b$ are constants.

The point $P(9, 66)$ lies on $C$ and the gradient of $C$ at $P$ is $8$
(a) Show that $9a + b = 26$

(2 marks)

Given also that $a + b = 2$
(b) find $\text{f}(x)$ , giving each term in simplest form.

(7 marks)
(c) State the coordinates of the image of $P$ on the curve with equation $y = \text{f}(x + 6)$ .

(1 mark)
Level 5 · 10 marksTry it
4
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ , with equation $y = \text{f}(x)$ , $x > 0$ , passes through the point $P(9, -30)$ and has gradient function

$\text{f}'(x) = (\sqrt{x} + 3)(2\sqrt{x} - 5)$
(a) (i) Show that the gradient of $C$ at $P$ is $6$

(ii) Hence find an equation of the normal to $C$ at $P$ , writing it in the form $ax + by + c = 0$ , where $a$ , $b$ and $c$ are integers.

(4 marks)
(b) Find $\text{f}(x)$ .

(6 marks)
Level 5 · 10 marksTry it