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Exercise 1

Here you see the graph of a function (in red) together with two different slope graphs (in short green dots and longer blue dots).
Which of the slope graphs corresponds to the red function graph?

Exercise 2

Choose one of the following functions:

$f (x) = - x^{2} + 4$
$g (x) = \sqrt{(x^{2} + 3)}$
$h (x) = \frac{4}{x}$
$k (x) = - x^{4} + 4 x$

a

For your chosen function, calculate the slope at $x = 1$ .

b

For your chosen function, draw a graph of the slope function.

c

Use the slope function to determine the extrema of your chosen function.

Exercise 3

Here you see the slope graph of a function $f$ .

a

Over which interval is the graph of the function $f$ increasing?

b

At what value(s) of $x$ does the function $f$ have a maximum?

c

Can you use the slope graph to determine the value of the function at this maximum?

d

Assume that $f (0) = 2$ . Now sketch a graph for $f$ .

Exercise 4

Here you see the sign scheme for the slope function of a function $g$ .
Sketch a possible graph for $g$ .

Exercise 5

A car starts as soon as the traffic light switches to green. The subsequent distance covered by the car is given by: $s (t) = 1, 6 t^{2}$ where $s$ is the distance in meters and $t$ is the time in seconds. Assume that there is no need to switch gears!

a

The speed of this car is expressed in meters per second. Draw the graph of this speed $v$ as a function of time $t$ .

b

If you answered a) correctly, then your speed diagram is a straight line. Now determine the a corresponding formula for $v (t)$ .

c

How many seconds does it take to reach a speed of $80$ km//h? Give your answer rounded to one decimal.

Exercise 6

Consider the following function $f (x) = 2 x^{3} - 6 x^{2} - 8 x$ .

a

You can use your graphing calculator to plot the graph of $f$ in such a way that all three x-axis intersects and both maxima are visible. Demonstrate that this graph intersects the $x$ -axis at point $(4, 0)$ .

b

Calculate the slope of the graph at this intersect.

c

Write down the equation for the tangent of the graph of $f$ at point $x = 4$

d

Draw a graph of the derivative function of $f$ .

e

With this derivative graph you can determine the extrema of $f$ . Use your graphing calculator to find the values of these extrema rounded to two decimals.

Exercise 7

Consider the following function $f (x) = 0, 5 x^{2} + 3 x$ .
Construct a rule for the slope function $f' (x)$ by first drawing up a table of values for $f'$ .

Exercises