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Exercise 1

You see a number of points on a graph.

a

Compute the slope of the chord $A B$ .

b

Compute the slope of the chord $C F$ .

c

There are two pairs of points for which the difference quotient is $0$ .
Which two pairs?

d

Point $F$ has a smaller $y$ -value than point $C$ . How can you tell from the difference quotient on the interval $[1, 4]$ ?

Exercise 2

Given is this graph.
Compute the difference quotient on the interval $[1, 3]$ .

Exercise 3

Given is the function $y = x^{3} - 3 x^{2} + 6$ on the interval $[2, 4]$ .

a

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

b

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

c

What do you notice at b? Can you explain what you see?

d

Give an interval on which the graph is increasing. And compute the difference quotient on that interval.

Exercise 4

The way a cup of coffee cools depends on the temperature of the coffee when it is poured and the room temperature. The form and the material the cup is made of also influence this process. The formula $T (t) = 20 + 70 \cdot 0, 82^{t}$ gives the temperature of the coffee minutes after pouring.

a

What is the temperature of the coffee at the moment of pouring?

b

What is the average temperature drop in the first 5 minutes?

c

Compute the average temperature drop in the next five minutes. Give your answer in one decimal.

d

From $t = 0$ to $t = 5$ the temperature of the coffee drops faster than from $t = 5$ to $t = 10$ . Explain how the difference quotients in b and c illustrate this. Also explain this from a physics' point of view.

Exercise 5

Give is the function $y = 3 x^{2}$ .
Show that on every interval $[a, a + 1]$ the difference quotient is equal to $6 \cdot a + 3$ .

Exercises