Change

Exercise 1

Here you see a number of points in a graph.

a

Calculate the average slope of the secant line $A B$ .

b

Calculate the average slope of the secant line $C F$ .

c

Two of the points have a difference quotient of $0$ .
Which two points are these? (There are two possibilities!).

d

Point $F$ has a lower $y$ -value than $C$ . How do you know this from the difference quotient on the interval $[1, 4]$ ?

Exercise 2

You are given the following graph.
Calculate the difference quotient on the interval $[1, 3]$ .

Exercise 3

You are given the function $f (x) = x^{3} - 3 x^{2} + 6$ with domain $[2, 4]$ .

a

Calculate the difference quotient on the interval $[0, 2]$ .

b

Calculate the difference quotient on the interval $[- 1, 2]$ .

c

What did you notice in part b)? Can you explain this?

d

Take an interval where the graph of $f$ increases. Calculate the difference quotiënt on that interval.

Exercise 4

How quickly a cup of coffee cools down depends on the temperature of the coffee at the time of pouring and on the surrounding room temperature. The shape and material of the cup also play a role. The formula $T (t) = 20 + 70 \cdot 0, 82^{t}$ allows you to calculate the temperature of the coffee after a given time.

a

What was the temperature of the coffee at the time of pouring?

b

What is the average rate of cooling of the coffee during the first five minutes?

c

Now calculate the average rate of cooling over the next five minutes (rounded to one decimal).

d

The temperature of the coffee decreases more rapidly between $t = 0$ and $t = 5$ than between $t = 5$ and $t = 10$ . Explain how you can tell this is the case by looking at the difference quotients you calculated in b and c. Also try to give a physical explanation for this observation.

Exercise 5

You are given the function $f (x) = 3 x^{2}$ .
Show that for this function the difference quotient over every interval $[a, a + 1]$ is always equal to $6 \cdot a + 3$ .

Exercises