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123456Differentiation rules

Exercises

Exercise 1

Find the derivative of the following functions:

a

f ( x ) = 5 x 6 - 13 x 5 + 10 x - 25

b

f ( x ) = a x 2 + b x + c

c

P ( I ) = R I 2

d

y ( x ) = ( x 2 - 1 ) ( x 2 - 9 )

e

f ( x ) = -8 x 8 - 88

f

f ( x ) = 2 a x 3 - 3 a 2 x + a 3

g

A ( r ) = π r 2 + l 2 r

h

h ( x ) = 3 x 2 ( 10 - x ) 2

Exercise 2

Given is the function f ( x ) = 4 5 x 3 - 3 x 2 .

a

Algebraically find the extrema of f .

b

Calculate the slope of the graph of f for x = 5 .

c

Algebraically calculate the coördinates of the inflection point of the graph of f .

Exercise 3

The point ( 2 , 0 ) lies on the graph of the function y = x 3 - 5 x 2 + 7 x - 2 .

a

Calculate the slope of the tangent of the graph in this point.

b

In how many other points does the tangent have the same slope?

Exercise 4

A piece of cardboard of 20 by 60 centimeters is folded into a tray. Assume that the height of the tray is x cm.

a

The volume I of this tray only depends on x (if nothing is allowed to stick out above the edge). Compose a fitting function rule for the volume I ( x ) .

b

Algebraically calculate what value of x gives the maximum volume for the tray.

Exercise 5

Here you see a part of the graph of f ( x ) = x 3 ( x - 20 ) 2 .

a

In the visible part of the graph there are three points in which the tangent to the graph is parallell to the x -axis. Algebraically calculate the x -coordinates of those three points.

b

Why does the function f have only two (local) extrema?

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