Derivative functions

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Exercises

Exercise 1

Given the function $f (x) = x^{2} + 4 x$ .

Calculate the slope of the graph of $f$ for $x = 1$ using the difference quotient over the interval $[1, 1 + h]$ . Check your answer using your graphing calculator.

Write down the function rule for the derivative of $f$ .

You can now use $f' (x)$ to determine the slope at $x = 1$ a second time. Do so, and check if your answer is the same as in a.

The graph of $f'$ has one x-intersect. What does this point represent in the graph of $f$ ?

Determine the slope of the graph of $f$ at the points where it intersects the x-axis (the zeros).

The slope of the graph of $f$ has a value of $2$ at exactly one point. Calculate the coordinates of this point.

Exercise 2

The following formula is a suitable approximation for the movement of a free-falling object (such as a parachute jumper before he opens his parachute): $s (t) = 0,5 {g t}^{2}$ where $s$ is the distance (in m) covered after $t$ seconds. $g$ is a constant, the gravitational constant, which is about $9,8$ m/s².

What is the average speed during the first $10$ seconds of free fall?

The speed after $10$ seconds of free fall is bigger than the average speed during the first $10$ seconds. Show this using a calculation.

Write down a formula that shows speed as a function of $t$ .

After how many seconds of free fall has the object reached a speed of $120$ km/h?

Exercise 3

A constant function has the rule $f (x) = c$ .

Show that the derivative of this function always has a value of $0$ .

Exercise 4

A certain car manufacturer is the only one producing a small city car. The total profit from the sales of these cars is given by: $T P = 900 q - 60 q^{2}$ where $T P$ is given in thousands and $q$ is the planned production scale in hundreds per year. You can assume that all cars produced will be sold.

Write down the function rule for the derivative of this profit function.

What does $T P' (5)$ represent in the profit function?

The car manufacturer wants to know what the production scale has to be in order to achieve the maximum profit. Determine this scale using the derivative function. Check your answer with the graphing calculator.

Exercise 5

The amount of a certain toxic substance in a lake is decreasing as this substance is naturally degraded. The following formula holds: $H (t) = 20 \cdot {0,8}^{t}$ where $H$ is the amount of this substance in milligrams per litre and $t$ is the amount of time (in days) after the substance first entered the water.

How much of the substance (in milligrams per litre) has disappeared on average per day during the first four days?

The speed of degradation of this toxic substance is higher at $t = 0$ than at $t = 4$ . Determine both speeds with your graphing calculator and explain why they are different.

You should be able to determine the speed of degradation using a difference quotient. If you try to do this you will encounter a problem. Which problem?

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