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
Find an equation of the tangent to the curve

$y = (x - 3)^5$

at the point where $x = 0$

(3 marks)
Level 2 · 3 marksTry it
2
(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 3 · 6 marksTry it
3
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 3 · 8 marksTry it
4
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 $C$ at $P$ . (3 marks)
Level 3 · 6 marksTry it
5
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 3 · 6 marksTry it

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