iGCSE (2021 Edition)

# 11.04 Tangents and normals

Worksheet
Tangents
1

Find the gradient of the tangent to f \left( x \right) = x^{3} + 5 x at the point \left(2, 18\right).

2

Consider the function y = \dfrac{1}{3} x^{3} + \dfrac{1}{6} x^{6} + 2 x.

a

Find \dfrac{dy}{dx}.

b

Evaluate \dfrac{dy}{dx} when x = - 7.

3

Find the gradient of f \left( x \right) = \dfrac{11}{x} + \dfrac{10}{x^{2}} at the point \left(4, \dfrac{27}{8}\right).

4

Consider the parabola f \left( x \right) = x^{2} + 3 x - 10.

a

Find the x-intercepts.

b

Find the gradient of the tangent at the positive x-intercept.

5

Consider the function y = 4 x^{2} - 5 x + 2.

a

Find \dfrac{dy}{dx}.

b

Find the value of x at which the tangent to the parabola is parallel to the x-axis.

6

Consider the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).

a

Find the gradient of the function f \left( x \right) = - x^{3} at the point.

b

Hence, find the equation of the tangent to the curve at the point.

7

Consider the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).

a

Find the gradient of the function f \left( x \right) = 5 \sqrt{x} at the point.

b

Hence, find the equation of the tangent to the curve at the point.

8

Consider the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).

a

Find the gradient of the function f \left( x \right) = 6 \sqrt{x} at the point.

b

Hence, find the equation of the tangent to the curve at the point.

9

Consider the tangent to the curve f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.

a

Find the gradient of the function f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.

b

Find the y-coordinate of the point of intersection between the tangent line and the curve.

c

Hence, find the equation of the tangent to the curve at x = - 2.

10

Consider the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at x = 1.

a

Find the y-coordinate of the point of intersection between the tangent line and the curve.

b

Find the equation of the tangent to the curve at x = 1.

11

Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.

a

Describe the steps required to find the equation of the tangent line at x = 2.

b

Hence, find the equation of the tangent to the curve.

12

Consider the function f \left( x \right) = \dfrac{9 x + 4}{3 x}.

a

Find the y-coordinate of the point on the curve at x = - 1.

b

Hence, find the equation of the tangent to the curve at x = - 1.

13

Consider the function g \left( x \right) = \dfrac{8 x^{7} - 6 x^{6} + 4 x^{5} + 7}{2 x^{2}}.

a

Find the derivative of the function.

b

Find the equation in of the tangent to the curve at x = 1.

14

Find the equation of the tangent to f \left( x \right) = x^{2} + x at the point \left(2, 6\right).

15

Find the equation of the tangent to f \left( x \right) = \left( 3 x - 1\right) \left( 2 x - 5\right) at the point \left(2, - 5 \right).

16

Consider the function f \left( x \right) = x^{3} - 6 x^{2}. Find the coordinates of the points on the curve where the gradient is 495.

17

Consider the function f \left( x \right) = x^{3} + 3 x^{2} - 19 x + 2. Find the coordinates of the points on the curve where the gradient is 5.

18

Consider the function f \left( x \right) = 2 x^{2} - 216 \sqrt{x}. Find the coordinates of the point on the curve where the gradient is 0.

19

Consider the function f \left( x \right) = \dfrac{4 x^{3}}{3} + \dfrac{5 x^{2}}{2} - 3 x + 7. Find the x-coordinates of the points on the curve whose tangent is parallel to the line y = 3 x + 7.

20

At point M, the equation of the tangent to the curve y = x^{2} is given by y = 4 x - 4. Find the coordinates of M.

21

At point M, the equation of the tangent to the curve y = x^{3} is given by y = 12 x - 16. Find the coordinates of M.

22

Find the equation of the tangent to the parabola y = 2 x^{2} + 8 x - 5 at the point where the gradient is 0.

23

Consider the curve given by the function f \left( x \right) = x^{2} - 4 x + 2.

a

Find the gradient of the tangent to the curve at the point \left(3, -1\right).

b

Find the coordinates of the vertex of the parabola f \left( x \right).

c

Sketch the graph of the curve and the tangent at the point \left(3, -1\right).

24

Consider the curve given by the function f \left( x \right) = x^{2}-1.

a

Find the gradient of the tangent to the curve at the point \left(1, 0\right).

b

Sketch the graph of the curve and the tangent at the point \left(1, 0\right).

25

Consider the curve y = x^{3} - x^{2} and the line 4 x - y = 11.

a

Find the x-coordinates of the points on the curve at which the tangents are perpendicular to the line 4 x - y = 11.

b

Find the equation of the tangent to the curve at each of the x-coordinates found in part (a).

26

The curve f \left( x \right) = k \sqrt{x} - 5 x has a gradient of 0 at x = 16. Find k.

27

In the figure, the straight line y = \dfrac{1}{10} x + b is tangent to the graph of f \left( x \right) = 6 \sqrt{x} at \\ x = a.

a

Find a.

b

Find b.

28

5 x + y + 2 = 0 is the tangent line to the curve y = x^{2} + b x + c at the point \left(9, - 47 \right).

a

Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.

b

State the gradient of the tangent to the curve at x = 9.

c

Find the value of b.

d

Find the value of c.

29

The curve y = a x^{3} + b x^{2} + 2 x - 17 has a gradient of 58 at the point \left(2, 31\right).

a

Use the given information to write two equations expressing b in terms of a.

b

Hence, find the value of a and b.

30

The graph of y = a x^{3} + b x^{2} + c x + d has the following properties:

• It intersects the y-axis at y = - 28, where the tangent is parallel to the x-axis.
• It intersects the x-axis at \left(2, 0\right), where it has a gradient of 36.
a

Use the information given in the first point above to find the value of c and d.

b

Use the information given in the second point above to write two equations expressing b in terms of a.

c

Hence, find the value of a and b.

31

From an external point \left(3, 2\right), two tangents L_{1} and L_{2} are drawn to the curve y = x^{2} - 6.

a

Let the gradient of a tangent line to the curve be m. Find the two possible values of m that correspond to the gradients of L_{1} and L_{2}.

b

If L_{1} is the tangent with the larger gradient, find the equation of L_{1}.

c

Find the equation of L_{2}.

Normals
32

Consider the function y = 10 x^{5} - 2 x^{4} + 8 x^{3} - 6 x^{2} + 4 x - 164.

a

Find \dfrac{dy}{dx}.

b

Find the gradient of the tangent to the curve at the point (3, 2278).

c

Find the gradient of the normal to the curve at the point (3, 2278).

33

The gradient of the curve f \left( x \right) at x = 3 is 2.

a

Find the gradient of the tangent line at x = 3.

b

Find the gradient of the normal line at x = 3.

34

Consider the curve f \left( x \right) = x^{2} + 8 x + 15 at the point \left(4, 63\right).

a

Find f' \left( x \right).

b

Find the gradient of the tangent to the curve at the point.

c

Find the equation of the tangent to the curve at the point.

d

Find the gradient of the normal to the curve at the point.

e

Find the equation of the normal to the curve at the point.

35

Consider the curve f \left( x \right) = 4 x + \dfrac{64}{x} at the point \left(4, 32\right).

a

Find the equation of the tangent to the curve at the point.

b

Find the equation of the normal to the curve at the point.

36

- 4 x + y + 1 = 0 is the normal to the curve y = x^{2} + b x + c at the point \left( - 8 , - 32 \right).

a

Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.

b

State the gradient of the normal to the curve at x = - 8.

c

Find the value of b.

d

Find the value of c.

Angles of inclination
37

Find the angle of inclination of a line to the x-axis, whose gradient is 4. Write your answer correct to the nearest degree.

38

Consider the function y = x^{2} - 3 x + 4. Find the x-coordinate of the point on the curve where the tangent makes an angle of 45 \degree with the x-axis.

39

The given line has an angle of inclination of \theta with the positive x-axis:

a

Find the gradient of the line.

b

Hence, find \theta. Give your answer in degrees to two decimal places.

40

Consider the graph of the line y = \dfrac{1}{2} x + 1 given and the triangle whose vertices are the origin, the x-intercept and the y-intercept. \theta is the acute angle in the triangle that the line makes with the x-axis.

a

State the gradient of the line.

b

Find the value of \tan \theta.

41

A line with equation y = 6 x + 2 has an angle of inclination of \theta with the positive x-axis. Find the value of \theta to the nearest degree.

42

If a line has a gradient of - 4 and an angle of inclination of \theta with the positive x-axis, which of the following is true of the value of \theta?

A

-90\degree < \theta < 0\degree

B

0\degree < \theta < 90\degree

C

90\degree < \theta < 180\degree

D

180\degree < \theta < 270\degree

43

The given line passing through the origin makes an angle of \theta with the positive x-axis. The point on the line has coordinates of \left(5.2, 5.72\right).

Find \theta. Give your answer in degrees, correct to four decimal places.

44

A line with gradient m has an angle of inclination of 65 \degree with the positive x-axis, such that \tan 65 \degree = m. Without solving for m, find the value of \theta such that \tan \theta = - \dfrac{1}{m}.

45

A line with equation 5 x + 6 y - 2 = 0 has an angle of inclination of \theta with the positive x-axis. Find the value of \theta, correct to the nearest degree.

46

A line is drawn on the Cartesian plane at an angle of inclination \theta with the positive x-axis. Given that \sin \theta = \dfrac{12}{13}, find the gradient m of the line.

47

A line passing through the points \left(2, - 3 \right) and \left(x, 9\right) makes an angle of 120 \degree with the positive x-axis. Find the exact value of x.

48

Consider the curve y = x^{3} + 2 x^{2} + 4 x - 21 at the point \left(8, 651 \right).

a

Find \dfrac{dy}{dx}.

b

Find the gradient of the tangent to the curve at the point.

c

Find the gradient of the normal to the curve at the point.

d

Find the angle of the inclination, \alpha, of the tangent to the curve at the point, correct to two decimal places.

e

Find the angle of the inclination, \beta, of the normal to the curve at the point, correct to two decimal places.

49

Two curves C_{1}: y = 3 x^{2} and C_{2}: y = \dfrac{24}{x} intersect at point P. Tangents are drawn to both curves at point P such that the tangent to C_{1} makes an angle of \theta degrees with the positive x-axis, and the tangent to C_{2} makes an angle of \alpha degrees with the positive x-axis.

a

Find the x-coordinate of point P.

b

Find exact value of \theta.

c

Find exact value of \alpha.

d

Find the angle between the two tangents at point P, correct to one decimal place.

Applications
50

The steepness of a ski run is defined by the angle of inclination of the run with the horizontal ground.

By how many degrees is Run A steeper than Run B? Round your answer to one decimal place.

51

Consider the curve f \left( x \right) = 36 - x^{2} at the point P\left(3, 27\right).

a

Find f' \left( x \right).

b

Find the gradient of the tangent to the curve at the point.

c

Find the equation of the tangent to the curve at the point.

d

Find the gradient of the normal to the curve at the point.

e

Find the equation of the normal to the curve at the point.

f

The tangent line at \left(3, 27\right) meets the x-axis at point A. Find the x-coordinate of A.

g

The normal line at \left(3, 27\right) meets the x-axis at point B. Find the x-coordinate of B.

h

Find the length of AB.

i

Find the exact area of \triangle ABP.

52

The normal to the curve y = x \left(x - 5\right)^{2} at the point A \left(6, 6\right) cuts the x-axis at B.

a

b

Find the gradient of the tangent at A.

c

Find the gradient of the normal at A.

d

Hence, find the x-coordinate of point B.

e

Find the area of triangle whose vertices are the origin, point A and point B.