Differentiation rules

Exercise 1

Find the derivative of the following functions.

a

$f (x) = (x^{3} + 6) (4 x^{2} - 5 x)$

b

$g (x) = (10 - x) \cdot \sqrt{x}$

c

$R (t) = 3 t {(t + 5)}^{4}$

d

$y (x) = x \cdot \sqrt{5 + x^{2}}$

e

$y (x) = x - \sqrt{5 + x^{2}}$

f

$V (r) = (100 - \frac{5}{r}) {(20 - r)}^{2}$

Exercise 2

Here you see the graphs of the functions $y_{1} (x) = x^{2}$ and $y_{2} (x) = {(2 x - 8)}^{4}$ . The function $f (x) = y_{1} (x) \cdot y_{2} (x)$ is the product function of both.

a

The zeroes of $f$ can be deduced from the graph. What are the zeroes of the graph of $f$ ?

b

Show that $f' (x) = {(2 x - 8)}^{3} (12 x^{2} - 16 x)$

c

Use the derivative to find th extrema of $f$ .

d

For what values of $k$ does the equation $f (x) = k$ have exactly four solutions?

Exercise 3

Given is the function $f (x) = 4 x \sqrt{x} \cdot {(1 - x)}^{3}$ .

a

For what values of $x$ does the graph have a tangent parallell to the $x$ -axis?

b

This function has two extrema. What are they?

Exercise 4

Here you see the graph of the function $f (x) = x \cdot \sqrt{8 - x^{2}}$ the way the graphing calculator depicts it.

a

The graph is incomplete. You can tell this from the zeroes of this function. What zeroes does the graph of $f$ have?

b

Calculate the range of $f$ using differentiation .

c

Compose the equation for the tangent to the graph of $f$ in $(0, 0)$ .

Exercise 5

Given is the function $f (x) = 0,25 x^{2} - x \sqrt{x}$ .

a

Algebraically calculate the range of $f$ .

b

Calculate the coordinates of the inflection point of the graph of $f$ .

c

For what $p$ is the line with equation $y = 2 x + p$ a tangent to the graph of $f$ ?

Exercise 6

Somebody wishes to expand his house with a conservatory using four equally sized rectangular frames. The dimensions of each fo these frames are: height $2,5$ and width $3$ . He first studies the possible arrangements where two frames $A B$ and $D E$ are attached perpendicular to the wall. The other two frames $B C$ and $C D$ are placed in such a manner that the area of the floor is maximized.

a

The distance between two frames perpendicular to the wall is $2 x$ . Show that the area of the floor of the conservatory $A$ corresponds to: $A (x) = 6 x + x \cdot \sqrt{9 - x^{2}}$ .

b

Algebraically calculate the maximum floor area for this conservatory.

Exercises