1.1 Exercises

From Förberedande kurs i matematik 2

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       Theory          Exercises      

Exercise 1.1:1

The graph for \displaystyle f(x) is shown in the figure.

a) What are the signs of \displaystyle f^{\,\prime}(-4) and \displaystyle f^{\,\prime}(1)?
b) For what values of \displaystyle x is \displaystyle f^{\,\prime}(x)=0?
c) In which interval(s) is \displaystyle f^{\,\prime}(x) negative?

(Each square in the grid of the figure has width and height 1.)

[Image]

Answer
Answer 1.1:1
Solution a
Solution 1.1:1a
Solution b
Solution 1.1:1b
Solution c
Solution 1.1:1c

Exercise 1.1:2

Determine the derivative \displaystyle f^{\,\prime}(x) when

a) \displaystyle f(x) = x^2 -3x +1 b) \displaystyle f(x)=\cos x -\sin x c) \displaystyle f(x)= e^x-\ln x
d) \displaystyle f(x)=\sqrt{x} e) \displaystyle f(x) = (x^2-1)^2 f) \displaystyle f(x)= \cos (x+\pi/3)
Answer
Answer 1.1:2
Solution a
Solution 1.1:2a
Solution b
Solution 1.1:2b
Solution c
Solution 1.1:2c
Solution d
Solution 1.1:2d
Solution e
Solution 1.1:2e
Solution f
Solution 1.1:2f

Exercise 1.1:3

A small ball, that is released from a height of \displaystyle h=10m above the ground at time \displaystyle t=0, is at a height \displaystyle h(t)=10-\displaystyle\frac{9{,}82}{2}\,t^2 at time \displaystyle t (measured in seconds) What is the speed of the ball when it hits the grounds?

Answer
Answer 1.1:3
Solution
Solution 1.1:3

Exercise 1.1:4

Determine the equation for the tangent and normal to the curve \displaystyle y=x^2 at the point \displaystyle (1,1).

Answer
Answer 1.1:4
Solution
Solution 1.1:4

Exercise 1.1:5

Determine all the points on the curve \displaystyle y=-x^2 which have a tangent that goes through the point \displaystyle (1,1).

Answer
Answer 1.1:5
Solution
Solution 1.1:5
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