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Exercise 1

A rock falls $500$ m down a perpendicular rock face. The distance covered by the rock $y$ (in m) is given by the formula $y (t) = 4, 9 t^{2}$ where $t$ is the time in seconds, while the rock is still falling and hasn't reached the ground yet.

a

Calculate the average velocity of the rock in the first $5$ seconds.

b

Calculate the velocity of the rock after exactly $5$ seconds. (Use sequential difference quotients to make this calculation, but check your result with the graphing calculator.)

c

Calculate the velocity of the rock at the moment that it hits the ground.

Exercise 2

Use your graphing calculator to look at the graph of the function $f (x) = 5 x^{2} - x^{3}$ .

a

Calculate the slope at $x = 2$ using a sequence of difference quotients.

b

You can already tell from the graph if the slope should be positive or negative. What feature of the graph provides that information?

c

Determine the equation describing the tangent at $x = 2$ for the graph of $f$ .

Exercise 3

For the interval $[- 5, 5]$ you are given the function rule $g (x) = \frac{4}{x}$ .

a

Calculate the rate of change of $g (x)$ at $x = 1$ .

b

There is one other point in the graph of $g$ where the slope of the curve is the same as at point $(1, 4)$ . Which point is that? Give an explanation for your answer.

c

At point $x = 0$ the function $g$ does not have a value. What then is the slope of the graph at that point? And what happens to the graph?

Exercise 4

The concentration $C$ of a certain substance that has been dissolved in water is decreasing over time according to the formula $C (t) = 10 \cdot 0, 9^{t}$ . In the formula, $C$ is in g/L (gram per liter) and $t$ in hours.

a

The amount of substance disappearing from the water is not the same every hour. Why is that?

b

How much of the substance on average disappears every hour during the first $5$ hours? (Give your answer rounded to two decimals.)

c

The rate of decay of this substance at time $t = 5$ is not equal to the average amount that has disappeared every hour up to that point. Calculate this rate of decay and give your answer rounded to two decimals.

Exercise 5

Here you see a graph of the growth of a tree (length in meters) over time (in years).

a

On average, how much longer does the tree get every year during the first $5$ years?

b

What is the speed of growth after exactly $5$ years? Give an estimate that is as precise as possible.

c

At which point do you see the fastest growth? Give an explanation.

d

What is the final speed of growth, presuming the tree remains healthy?

Exercise 6

A firework follows a parabolic trajectory until is explodes. This trajectory is given by the formula $h (x) = - x^{2} + 10 x$ with both $h$ and $x$ in meters.

a

What is the slope of the trajectory when the firework is first launched?

b

Using the slope found in part a) you can use trigonometry to calculate at what angle the firework has been launched. Calculate an exact value of this angle $α$ in degrees.

c

At what point of the trajectory do you see a slope of $0$ ?

d

The firework explodes after is has covered $8$ meter of horizontal distance. How high up in the air is it at that point, and what is the slope of the trajectory?

Exercises