Differentiation rules > Derivative of trigonometric functions
123456Derivative of trigonometric functions

Exercises

Exercise 1

Differentiate the following functions.

a

f ( x ) = 3 sin ( 2 x )

b

g ( x ) = 16 - 20 cos ( 2 π 30 ( x 5 ) )

c

H ( t ) = sin 2 ( 440 π t )

d

y ( x ) = 16 + sin 2 ( x )

e

A ( r ) = 1 sin ( 2 r )

f

W ( p ) = 2 sin ( p ) cos ( 2 p )

Exercise 2

Look at the graph of f with f ( x ) = 3 cos ( 2 x ) + 1 op [ 0 , 2 π ] .

a

Algebraically calculate the extrema of f .

b

Algebraically solve: f ( x ) < 2,5 .

c

The line y = 2,5 intersects the graph of f in four points. calculate the slope in those points.

Exercise 3

In a tidal area the water level H (in meter) at time t (in hours) can be described by H ( t ) = 2 cos ( 2 π 12,25 ( t - 3 ) ) + 1 . Here t = 0 at 0:00 o'clock on a certain day.

a

Calculate the water level at 0:00 o'clock.

b

Using differentiation, calculate the times of the high waters that day. Why is it not really necessary to use differentiation?

c

Calculate H ' ( 4 ) . What meaning does this number have for the behaviour of the water level.

d

At what times does the water level change fastest?

Exercise 4

Here you see part of the graph of f ( x ) = 0,5 x + sin ( x ) with 0 x 2 π .

a

Use differentiation to find the two extrema of f .

b

Compose an equation for the tangent to the graph of f for x = 0 .

c

Calculate in which point of the graph of f the tangent has the smallest slope. What is this slope?

Exercise 5

Study the graph of function f with f ( x ) = sin ( x 2 ) .

How many extrema does this function have on the interval [ 2 π , 3 π ] ?

Exercise 6

The power of the alternating current P provided by power companies is proportional to the square of the voltage V (in Volt). The alternating current is delivered with an effective voltage of 230 V, a frequency of 50 Herz (that is 50 periods per second) and an amplitude of approximately 325 V. The voltage V as a function of time t is a true sinusoid.

a

Compose a formula for V ( t ) assuming that V ( 0 ) = 0 .

b

For the power P : P ( t ) = sin 2 ( 100 π t ) . What value does the proportionality constant have here?

c

Algebraically calculate the extrema and zeroes of P .

d

The graph of P is a true sinusoid. Give a formula for this sinusoid.

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