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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Tangents, normals and stationary points

Equations of tangents and normals, increasing and decreasing functions, stationary points and their nature, optimisation problems.

12 original questions, 74 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

The curve C has equation y = x3 − 4x + 1

Find an equation of the tangent to C at the point where x = 2, giving your answer in the form y = mx + c

4 marks
Show answer
y = 8x - 15
2

The point P(3, 6) lies on the curve with equation y = 2x2 − 5x + 3

Find an equation of the normal to the curve at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

4 marks
Show answer
x + 7y - 45 = 0
3

f(x) = x3 + 3x2 − 24x + 5

Find the set of values of x for which f(x) is a decreasing function.

4 marks
Show answer
-4 < x < 2
4

The curve C has equation y = x3 − 6x2 + 9x + 2

  1. Find the coordinates of the stationary points of C
  2. Determine the nature of each stationary point, giving a reason for your answer.
6 marks
Show answer
a) (1, 6) and (3, 2) b) (1, 6) is a maximum, (3, 2) is a minimum
5

A curve has equation y = x2 + 16x,   x > 0

  1. Find (i) dydx   (ii) d2ydx2
  2. Verify that the curve has a stationary point when x = 2
  3. Determine the nature of this stationary point, giving a reason for your answer.
7 marks
Show answer
a) dydx = 2x - 16x-2; d2ydx2 = 2 + 32x-3 b) at x = 2, dydx = 4 - 4 = 0 c) d2ydx2 = 6 > 0 so it is a minimum
6

f(x) = x3 − 3x2 + 5x − 7

Prove that f(x) is an increasing function for all real values of x

4 marks
Show answer
f'(x) = 3(x - 1)2 + 2 > 0 for all x, so f is increasing
7

The curve C has equation y = 4√x,   x > 0. The point P(4, 8) lies on C.

  1. Find an equation of the tangent to C at P
  2. Find an equation of the normal to C at P
  3. The tangent and the normal meet the x-axis at A and B. Find the area of triangle APB
−6−4−22468101214−224681012OxyCP(4, 8)
7 marks
Show answer
a) y = x + 4 b) y = -x + 12 c) 64
8

The curve C has equation y = 2x3 − 9x2 + 12x − 1

Find equations of the two tangents to C that are parallel to the line y = 12x

5 marks
Show answer
y = 12x - 1 and y = 12x - 28
9

The curve with equation y = ax3 + bx2 − 4, where a and b are constants, has a stationary point at (2, −12)

  1. Find the value of a and the value of b
  2. Find the coordinates of the other stationary point and determine the nature of both stationary points.
7 marks
Show answer
a) a = 2, b = -6 b) (0, -4) is a maximum; (2, -12) is a minimum
10

A closed gift box is a cuboid with a square base of side x cm and a height of h cm. The volume of the box is 500 cm3.

  1. Show that the surface area, A cm2, of the box is given by A = 2x2 + 2000x
  2. Use calculus to find the minimum value of A, giving your answer to 3 significant figures.
  3. Justify that the value you found is a minimum.
9 marks
Show answer
a) h = 500x2, so A = 2x2 + 4xh = 2x2 + 2000x b) 378 cm2 (x = 7.94 cm) c) d2Adx2 = 12 > 0 so minimum
11

An open box is made from a rectangular sheet of metal 24 cm by 15 cm. A square of side x cm is cut from each corner and the sides are folded up.

  1. Show that the volume, V cm3, of the box is V = 4x3 − 78x2 + 360x
  2. State the range of possible values of x
  3. Use calculus to find the maximum volume of the box, justifying that it is a maximum.
8 marks
Show answer
a) V = x(24 - 2x)(15 - 2x) = 4x3 - 78x2 + 360x b) 0 < x < 7.5 c) 486 cm3 when x = 3
12

The curve C has equation y = x3 − 2x2 + 3. The point P on C has x-coordinate −1

  1. Show that the tangent to C at P has equation y = 7x + 7
  2. The tangent meets C again at the point Q. Show that the x-coordinates of P and Q satisfy x3 − 2x2 − 7x − 4 = 0 and hence find the coordinates of Q
  3. Find the exact length of PQ, giving your answer as a simplified surd.
−2−112345−1010203040OxyCP
9 marks
Show answer
a) y = 7x + 7 b) Q(4, 35) c) 25√2

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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