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

Given that
- $\text{f}'(x) = 2x - \dfrac{k}{\sqrt{x}} + 1$ where $k$ is a constant
- $P$ is the point on $C$ where $x = 9$
- the normal to $C$ at $P$ has equation $x + 17y = 60$
(a) show that $k = 6$ $k = 6$

(3 marks)
(b) Hence find $\text{f}(x)$ $f (x)$ .

(4 marks)
Level 4 · 7 marksTry it
2
In this question you must show all stages of your working.

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

Given that
- $\text{f}'(x) = 15x^2 - kx\sqrt{x} - 4$ , where $k$ is a constant
- $C$ passes through the point $P(3, -90)$
- $\text{f}''(x) = 0$ at $P$
(a) find the exact value of $k$ $k$ ,

(4 marks)
(b) find $\text{f}(x)$ $f (x)$ , giving each term in simplest form.

(4 marks)
Level 4 · 8 marksTry it
3
The normal to the curve $y = a\sqrt{x} + \dfrac{b}{x}$ at the point with $x$ -coordinate $4$ has equation $x + 2y = 24$ , where $a$ and $b$ are constants.

Determine the values of $a$ and $b$ .

(8 marks)
Level 4 · 8 marksTry it
4
The diagram shows the point $A(4, 3)$ on the curve with equation $y = \tfrac12x^2 - 2x + 3$ . The normal to the curve at $A$ meets the curve at one other point, $B$ . Angle $AOB$ , where $O$ is the origin, is $\alpha$ .(a) Determine the coordinates of $B$ .

(6 marks)
(b) Hence determine the exact value of $\tan\alpha$ .

(3 marks)
Level 4 · 9 marksTry it
5
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

$y = (2x - 5)^{\frac{3}{2}} \qquad x \geqslant \dfrac{5}{2}$

The point $P(3, 1)$ lies on the curve. The normal to the curve at $P$ meets the $x$ -axis at the point $N$ .
(a) Use calculus to show that the normal to the curve at $P$ has equation $x + 3y = 6$

(5 marks)

The finite region $R$ , shown shaded in Figure 1, is bounded by the curve, the line $PN$ and the $x$ -axis.
(b) Use algebraic integration to find the exact area of $R$ .

(4 marks)
Level 4 · 9 marksTry it