Periodic functions > Periodicity

Exercise 1

Study the graph of this periodical function $f$.

a

Compute $f\left(25\right)$.

b

For what values of $x$ is $f\left(x\right)=10$?

c

Solve: $f\left(x\right)=5$ where $0\le x\le 9$.

Exercise 2

Point $A$ is situated on a wheel at distance $1$ from the axis. The height of point $A$ relative to the axis is called $h\left(t\right)$. Point $A$ starts at the right, so $h\left(0\right)=0$. The wheel turns around in $6$ seconds, counterclockwise. Point $A$ reaches the top after $\mathrm{1,5}$ seconds. So $h\left(\mathrm{1,5}\right)=1$.

a

Compute $h\left(\mathrm{4,5}\right)$, $h\left(\mathrm{10,5}\right)$ and $h\left(\mathrm{16,5}\right)$.

b

Compute the exact value of $h\left(\mathrm{0,75}\right)$.

c

Compute the exact value of $h\left(\mathrm{6,75}\right)$, $h\left(\mathrm{12,75}\right)$ and $h\left(\mathrm{-5,25}\right)$.

d

Solve: $h\left(t\right)=h\left(\mathrm{0,75}\right)$.

Exercise 3

The large arrow of a a church clock is $\mathrm{1,5}$ m long. Both arrows are connected to the axis of the clock at a height of $45$ m above the ground. Point $T$ represents the tip of the large arrow. The height $h$ in m above the ground of point $T$ depends on the angle of rotation $\alpha $ . Assume that $\alpha =0$ at 12 o'clock.

a

What is the height of $T$ at ten past two?

b

Sketch a graph of $h\left(\alpha \right)$.

c

There are two moments when $h\left(\alpha \right)=46$. The corresponding points for $T$ are $A$ and $B$. What is the distance between these points $A$ and $B$?

Exercise 4

A ball is shot into the air at $t=-1$ and falls back onto earth. It bounces completely elastically, so that it continues to bounce. Use the formula $h\left(t\right)=5-5{t}^{2}$ where $-1\le t\le 1$. $t$ is in seconds, $h$ is in meters.

a

Compute $h\left(0\right)$ and $h\left(\mathrm{0,5}\right)$.

b

Determine the period of this graph.

c

Compute $h\left(6\right)$ and $h\left(\mathrm{6,5}\right)$.

d

Compute $h\left(15\right)$ and $h\left(\mathrm{15,5}\right)$.

e

How realistic is this mathematical model?