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
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 4 · 4 marksTry it
2
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 4 · 5 marksTry it
3
A curve has parametric equations $x = 3t - \dfrac{2}{t}$ and $y = t - \dfrac{4}{t}$ , for $t \ne 0$ .
(a) Find $\dfrac{dy}{dx}$ in terms of $t$ , simplifying your answer.

(4 marks)
(b) Explain why the gradient of the curve is positive at every point.

(2 marks)
(c) Determine a cartesian equation for the curve, in the form $ax^2 + bxy + cy^2 = d$ where $a$ , $b$ , $c$ and $d$ are integers.

(4 marks)
Level 4 · 10 marksTry it
4
A particle $P$ moves in a plane, starting when $t = 0$ . At time $t$ seconds, $P$ is at the point $(x, y)$ , where $x = 2e^{t} + 9e^{-t}, \qquad y = e^{2t} - 6e^{t} + 8.$
(a) Explain why $P$ is never on the $y$ -axis.

(1 mark)
(b) Determine, in exact form, the times at which $P$ is on the $x$ -axis.

(2 marks)
(c) Show that $\dfrac{dy}{dx} = \dfrac{2e^{2t}(e^{t} - 3)}{2e^{2t} - 9}$ .

(4 marks)
(d) Use your answer to part (c) to find the coordinates of $P$ when the direction of motion of $P$ is parallel to the $x$ -axis.

(3 marks)
Level 4 · 10 marksTry it
5
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 4 · 4 marksTry it

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