Derivative functions

Derivative functions > Finding the derivative

Overview

1234Finding the derivative

Exercises

Exercise 1

Determine the derivative of each of the following functions. Also determine the slope of the graph at $(1, y)$ .

$f (x) = x^{3} - 4 x$

$g (x) = x^{4} + 2 x^{3} - 5 x^{2} + 12 x - 35$

$s (t) = 60 t - 4,9 t^{2}$

$H (t) = 2 (t^{2} - 4)$

$y = 5 - {(x - 3)}^{2}$

$P (x) = a x^{3} + b x^{2} + c x + d$

$T W (q) = 0,5 q^{3} - 6 q^{2} - 25 q + 112$

$K (x) = (3 x^{2} - 2 a) (a x - 1)$

Exercise 2

Determine the derivative of each of the following functions. Then determine those points of the corresponding graph where the tangent has a slope of $0$ . (Rounded to one decimal where necessary.)

$f (x) = 0,5 x^{4} - 4 x^{2}$

$T W (q) = - q^{3} + 3 q^{2} + 3 q + 6$

$v (t) = t {(t - 1)}^{2}$

$T W = 40 p - 0,02 p^{2}$

Exercise 3

Here you see the graph of the function $f (x) = (x^{2} - 4) (x^{2} - 9)$ .

Show how you can fin the x-intersects of the graph using the function rule $f$ .

What is the derivative function of $f$ ?

Determine the point where the tangents of the graph of $f$ at $x = -2$ and $x = 2$ intersect each other.

Solve: $f' (x) = 0$ .

What significance does your answer in d have for the graph of $f$ ?

Exercise 4

If you launch an object with a certain initial speed and angle, then its trajectory will be parabolic if you neglect any air resistance. An example of such a trajectory is the graph of the function $h (x) = 1,5 - 0,01 {(x - 10)}^{2}$ . In this function $h$ is the height of the object above ground (in m) and $x$ is the horizontal distance of the object from the launching point (in m).

At what height was the object released?

Determine $h' (0)$ .

What does is the significance of this number with respect to the trajectory?

Calculate the point of the trajectory where $h (x) = 0$ .

At the highest point of the trajectory, the value of the derivative is zero. The object, however, is still moving with a given speed at this point. How can you explain this?

Exercise 5

The total production costs of an article are given by: $T C = 1200 + 0,2 q^{2}$ . In this formula $q$ is the number of units of the article produced and $T C$ are the total costs in euro. The production costs per unit are given by $A T C = \frac{T C}{q}$ . You can also call this the average total costs.

Express the average total costs as a function of $q$ .

You can use your graphing calculator to look at the graph of $A T C$ bekijken. Which is the vertical asymptote of the graph of $A T C$ ? What is the economic significance of this asymptote?

You cannot determine a derivative of this function (yet). What you can do is let your graphing calculator draw (an approximation of) the slope function. Draw this slope function and use it to determine at what production size the average total costs would be as low as possible.

What value does the slope of the graph of $A T C$ approach at very large production sizes? What does that mean for the production costs per unit?

previous | next