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Standard Level

11.19 Applications of the differentiation of trigonometric functions

Worksheet
Tangents and normals
1

Find the gradient of the normal to the curve y = x^{2} + \cos \left( \pi x\right), at the point where x = 1.

2

For the following functions, find the equation of the tangent at the given point:

a

y = - 5 x \sin x at \left(\pi, 0\right).

b

y = \cos 6 x at the point where x = \dfrac{\pi}{18}.

3

Consider the function f \left( x \right) = \cos 2 x - \sqrt{3} \sin 2 x.

a

Find the first positive value of x at which the curve intersects the x-axis.

b

Find the gradient of the tangent at this point.

c

Hence determine the positive angle, \theta, that the tangent makes with the positive \\x-axis at this point. Round your answer to the nearest degree.

4

Consider the function f \left( x \right) = - 4 \cos 3 x.

Find \theta, the angle that the tangent at x = \dfrac{\pi}{12} makes with the positive x-axis. Round your answer to the nearest degree.

5

For each of the following functions, find:

i

The derivative function.

ii

The gradient of the normal at the point \left(\dfrac{\pi}{4}, 1\right).

iii

The equation of the normal at the point \left(\dfrac{\pi}{4}, 1\right).

a
y = \sin 2 x
b
y = \sqrt{2}\sin x
c
y = \sqrt{2}\cos x
6

For the function y = \dfrac{\sin x}{x}, find:

a

The derivative function.

b

The gradient of the normal at the point \left( \dfrac{\pi}{2}, \dfrac{2}{\pi} \right).

c

The equation of the normal at the point \left( \dfrac{\pi}{2}, \dfrac{2}{\pi} \right).

Trigonometric graphs
7

Consider the function y = \dfrac{\sin x}{1 + \cos x}.

a

Find \dfrac{dy}{dx}.

b

Determine the number of turning points function y has.

8

Find the x-coordinate of each point of inflection on the curve y = 3 \cos \left( 2 x + \dfrac{\pi}{4}\right) for the interval 0\leq x\leq 2\pi.

9

Consider the function f \left( x \right) = - 4 \sin \left(x + \dfrac{\pi}{6}\right) on the interval 0 \leq x \leq 2 \pi.

a

Find the coordinates of any turning points.

b

Find an expression for f'' \left( x \right).

c

Hence classify the turning points.

10

For each of the following functions on the interval 0 \leq x \leq 2\pi:

i

Find the axes intercepts.

ii

Find an expression for f' \left( x \right).

iii

Determine the coordinates of any stationary points.

iv

Find an expression for f'' \left( x \right).

v

Hence classify the stationary points.

vi

Sketch the graph of f \left( x \right).

a
f \left( x \right) = 2 \sin x
b
f \left( x \right) = \cos 3x
c
f \left( x \right) = x\cos x
d
f \left( x \right) = e^{ - x } \cos x
e
f \left( x \right) = \sin ^{2}\left(x\right)
f
f \left( x \right) = \cos ^{3}\left(x\right)
g
f \left( x \right) = x + \sin x
Applications
11

Determine the possible values of k, given that y = \cos k x satisfies the following equation for all values of x: y'' + 25 y = 0

12

The displacement of an object is a measure of how far it is from a fixed origin point at time, \\t seconds. Displacement to the left of the origin point is negative and displacement to the right of the origin is positive. At time t seconds, velocity is given by v \left( t \right) = x'(t), and acceleration is given by a \left( t \right) = x''(t).

If the displacement of an object x about a fixed origin is given by: x \left( t \right) = - 5 \cos 2 t

a

Describe the initial position of the object, from the fixed origin point.

b

Write an expression for v \left( t \right).

c

Show that a \left( t \right) = - 4 x \left( t \right).

d

Find the acceleration of the object, when t = \dfrac{\pi}{2}.

13

A particle moves in a straight line and its displacement, x \text{ cm}, from a fixed origin point after t seconds is given by the function:

x \left( t \right) = \sin t - \sin t \cos t - 5 t
a

State the initial displacement of the particle.

b

The velocity is the rate at which displacement changes over time. Find an equation for v \left( t \right), in terms of \cos t.

c

Hence find the initial velocity, v \left( 0 \right).

14

The arm of a pendulum swings between its two extreme points A to the left and B to the right. Its horizontal displacement x \text{ cm} from the centre of the swing, at time t seconds after it starts swinging, is given by:

x \left( t \right) = 16 \sin 3 \pi t
a

Find the intial position of the pendulum.

b

Find the maximum distance of the pendulum from the central position of its swing.

c

Find an expression for the velocity function, v \left( t \right)= x'\left( t \right).

d

Find the first two times at which the pendulum comes to rest, v \left( t \right)= 0 .

e

Find the displacement of the pendulum when it first comes to rest.

f

Find the displacement of the pendulum when it comes to rest for the second time.

g

Hence determine the distance between the two points A and B.

15

The arm of a pendulum swings between its two extreme points A to the left and B to the right. Its horizontal displacement x \text{ cm} from the centre of the swing at time t seconds after it starts swinging is given by:

x \left( t \right) = 19 \sin 4 \pi t
a

Find an expression for the velocity function, v \left( t \right)=x'\left( t \right).

b

Find the maximum velocity of the pendulum.

c

Find the first two times, after it starts swinging, at which the pendulum reaches its maximum velocity, v' \left( t \right) = 0 and x' \left( t \right) < 0 .

d

Describe the position of the pendulum when it reaches its maximum velocity.

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