A-level · Differentiation

Parametric differentiationExam-style questions

Finding the gradient of a curve whose x and y are both given in terms of a parameter. Why the gradient is dy/dt divided by dx/dt; finding the value of t at a point before substituting; tangents and normals through the point, not through t; horizontal and vertical tangents; and the second derivative, where the obvious shortcut fails.

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30 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
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 $P$ on $C$ 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 5 · 9 marksTry it
2
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 5 · 5 marksTry it
3
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 5 · 9 marksTry it
4
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 $M$ of $C$ . (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 5 · 10 marksTry it
5
The parametric equations of a curve are

$x = t^{2} + 1, \qquad y = 2t - \mathrm{e}^{-2t},$

where $t > 0$ . The tangent to the curve at the point $P$ , where $t = p$ , is parallel to the line $y = x$ .
(a) Show that $p$ satisfies the equation $p = 1 + \mathrm{e}^{-2p}$ .

(4 marks)
(b) Verify by calculation that $p$ lies between $1$ and $1.2$ .

(2 marks)
(c) Use the iterative formula $p_{n+1} = 1 + \mathrm{e}^{-2p_{n}}$ with an initial value of $1$ to find the value of $p$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(3 marks)
Level 5 · 9 marksTry it

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