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Exercises

Exercise 1

This is a part of the graph of a function.
Use this graph to draw a diagram of increments with step size $1$ , starting at $x = - 2$ .

Exercise 2

Given is the function $y = 0, 5 \cdot x^{4} - 4 x^{2} + 8$ .

Using the graphic calculator you can look at the graph of this function and make a diagram of increments. Draw the diagram of increments on the interval $[- 3, 3]$ with a step size of $0,5$ .

There is exactly one interval where the curve is increasingly decreasing. What interval is that? And how can you tell from the diagram of increments?

How can you use the diagram of increments to conclude that the curve probably has three extreme values?

Exercise 3

In an amusement park the number of visitors entering and leaving the park have been counted every hour from the opening time at 8 am until closing time at 6 pm. The difference between both is the increment of the number of visitors. Here you see these increments.

Make a graph of the total number of vistors depending on the hour of this day, from 8 am until 6 pm.

Around what time did the park have the most visitors?

Can you determine the maximum number of visitors in the park at any time?
Explain your answer.

Exercise 4

Biologists keep track of the numbers of a certain type of butterflies in an isolated nature reserve. This table shows the gathered data.

Year	2000	2001	2002	2003	2004
Number of butterflies	2450	2050	1850	1665	1580
Difference w.r.t. last year		–400	–240	–145	–85

It appears that the difference $D$ with respect to the previous year changes exponentially with time $t$ in years. The relation $D = - 400 \cdot 0, 6^{(t - 1)}$ seems valid with $t = 0$ in the year 2000.

Check whether this formula corresponds to the found increments.

Assume that this formula remains valid in the years after 2004. Make a diagram of increments of the number of butterflies in this nature reserves with a step size of $1$ year.

Make a graph of the number of butterflies $N$ over the years.

What kind of decreasing behaviour do you see in the number of butterflies? How can you tell from the diagram of increments?

The number of butterflies in this nature reserve seems to stabilize. How can you tell from diagram of increments? What does this imply for the graph of the number of butterflies?

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