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Exercises

Exercise 1

Here you see part of a graph of a function.
Construct a diagram of increments for this graph, using step size $1$ , starting at $x = - 2$ .

Exercise 2

You are given a function $f$ with rule $f (x) = 0, 5 x^{4} - 4 x^{2} + 8$ .

Use your graphing calculator to look at the graph of this function and to generate the table of increments. Draw a diagram of increments on the interval $[- 3, 3]$ with a step size of $0,5$ .

There is exactly one interval in which the curve shows an accelerating decrease. Which interval is this? How can you tell from the diagram of increments?

Why does the diagram of increments allow you to conclude that this function probably has three extrema?

Exercise 3

In an amusement park, hourly counts of arriving and leaving visitors are recorded from opening at 8.00 a.m. to closing at 6.00 p.m. The difference between people arriving and leaving represents the increase (or decrease) of the number of people in the park. The graph of increments shows these changes on a particular day.

Construct a graph of the total number of visitors per hour of the day, from 8.00 a.m. (8h) until 6.00 p.m (18h).

Around what time did the number of visitors in the park reach a peak?

Can you determine what the maximum number of visitors was at any given time? Give an explanation for your answer.

Exercise 4

Biologists register changes in the numbers of a particular butterfly in a closed-off nature reserve. The resulting counts are shown in the table below:

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

It appears that the difference $V$ w.r.t. the previous year changes exponentially with time $t$ (in years). The relationship seems to be $V = - 400 \cdot 0, 6^{(t - 1)}$ with $t = 0$ in the year 2000.

Check if this formula corresponds to the observed differences.

Assume that the formula continues to be valid after 2004. Draw a diagram of increments for the numbers of butterflies in this nature reserve with a step size of $1$ year.

Also draw a graph of the numbers of butterflies $N$ over time.

What type of decrease in butterfly numbers do you see? How can you see this in the diagram of increments?

The number of butterflies in the nature reserve appears to be stabilizing. How can you see this in the diagram of increments? How will this be reflected in the graph of butterfly numbers?

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