Differentiation rules > Derivative of trigonometric functions

Exercise 1

Differentiate the following functions.

a

$f\left(x\right)=3sin\left(2x\right)$

b

$g\left(x\right)=16-20cos\left(\frac{2\pi}{30}(x-5)\right)$

c

$H\left(t\right)={sin}^{2}\left(440\pi t\right)$

d

$y\left(x\right)=\sqrt{16+{sin}^{2}\left(x\right)}$

e

$A\left(r\right)=\frac{1}{sin\left(2r\right)}$

f

$W\left(p\right)=2sin\left(p\right)cos\left(2p\right)$

Exercise 2

Look at the graph of $f$ with $f\left(x\right)=3cos\left(2x\right)+1$ op $[0,2\pi ]$.

a

Algebraically calculate the extrema of $f$.

b

Algebraically solve: $f\left(x\right)<\mathrm{2,5}$.

c

The line $y=\mathrm{2,5}$ 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 $H\left(t\right)=2cos\left(\frac{2\pi}{\mathrm{12,25}}(t-3)\right)+1$. 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 $H\text{'}\left(4\right)$. 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 $f\left(x\right)=\mathrm{0,5}x+sin\left(x\right)$ with $0\le x\le 2\pi $.

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 $f\left(x\right)=sin\left({x}^{2}\right)$.

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

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 $V\left(t\right)$ assuming that $V\left(0\right)=0$.

b

For the power $P$: $P\left(t\right)={sin}^{2}\left(100\pi t\right)$. 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.