Periodic functions

Periodic functions > Periodicity

Overview

123456Periodicity

Exercises

Exercise 1

Study the graph of this periodical function $f$ .

Compute $f (25)$ .

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

Solve: $f (x) = 5$ where $0 \leq x \leq 9$ .

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 $h (t)$ . Point $A$ starts at the right, so $h (0) = 0$ . The wheel turns around in $6$ seconds, counterclockwise. Point $A$ reaches the top after $1,5$ seconds. So $h (1,5) = 1$ .

Compute $h (4,5)$ , $h (10,5)$ and $h (16,5)$ .

Compute the exact value of $h (0,75)$ .

Compute the exact value of $h (6,75)$ , $h (12,75)$ and $h (-5,25)$ .

Solve: $h (t) = h (0,75)$ .

Exercise 3

The large arrow of a a church clock is $1,5$ 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 $α$ . Assume that $α = 0$ at 12 o'clock.

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

Sketch a graph of $h (α)$ .

There are two moments when $h (α) = 46$ . 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 $h (t) = 5 - 5 t^{2}$ where $-1 \leq t \leq 1$ . $t$ is in seconds, $h$ is in meters.

Compute $h (0)$ and $h (0,5)$ .

Determine the period of this graph.

Compute $h (6)$ and $h (6,5)$ .

Compute $h (15)$ and $h (15,5)$ .

How realistic is this mathematical model?

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