1.1 Exercises

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{{Mall:Ej vald flik|[[1.1 Inledning till derivata|Teori]]}}
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{{Not selected tab|[[1.1 Introduction to derivatives|Theory]]}}
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{{Mall:Vald flik|[[1.1 Övningar|Övningar]]}}
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{{Selected tab|[[1.1 Exercises|Exercises]]}}
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===Exercise 1.1:1===
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<div class="ovning">
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{| width="100%"
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The graph for <math>f(x)</math> is shown in the figure.
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{| width="100%" cellspacing="10px"
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| valign="top" |a)
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| width="100%" | What are the signs of <math>f^{\,\prime}(-4)</math> and <math>f^{\,\prime}(1)</math>?
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|-
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| valign="top" |b)
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|width="100%"| For what values of <math>x</math> is <math>f^{\,\prime}(x)=0</math>?
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|-
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| valign="top" |c)
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|width="100%"| In which interval(s) is <math>f^{\,\prime}(x)</math> negative?
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(Each square in the grid of the figure has width and height 1.)
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| width="5%" |
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||{{:1.1 - Figure - The graph of f(x) in exercise 1.1:1}}
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</div>{{#NAVCONTENT:Answer|Answer 1.1:1|Solution a|Solution 1.1:1a|Solution b|Solution 1.1:1b|Solution c|Solution 1.1:1c}}
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===Exercise 1.1:2===
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<div class="ovning">
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Determine the derivative <math>f^{\,\prime}(x)</math> when
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{| width="100%" cellspacing="10px"
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|a)
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|width="33%"| <math>f(x) = x^2 -3x +1</math>
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|b)
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|width="33%"| <math>f(x)=\cos x -\sin x</math>
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|c)
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|width="33%"| <math>f(x)= e^x-\ln x</math>
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|-
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|d)
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|width="33%"| <math>f(x)=\sqrt{x}</math>
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|e)
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|width="33%"| <math>f(x) = (x^2-1)^2</math>
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|f)
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|width="33%"| <math>f(x)= \cos (x+\pi/3)</math>
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|}
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</div>{{#NAVCONTENT: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}}
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===Exercise 1.1:3===
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<div class="ovning">
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A small ball, that is released from a height of <math>h=10</math>m above the ground at time <math>t=0</math>, is at a height <math>h(t)=10-\displaystyle\frac{9{,}82}{2}\,t^2</math> at time <math>t</math> (measured in seconds) What is the speed of the ball when it hits the grounds?
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</div>{{#NAVCONTENT:Answer|Answer 1.1:3|Solution |Solution 1.1:3}}
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===Exercise 1.1:4===
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<div class="ovning">
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Determine the equation for the tangent and normal to the curve <math>y=x^2</math> at the point <math>(1,1)</math>.
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</div>{{#NAVCONTENT:Answer|Answer 1.1:4|Solution |Solution 1.1:4}}
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===Exercise 1.1:5===
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<div exercise ="ovning">
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Determine all the points on the curve <math>y=-x^2</math> which have a tangent that goes through the point <math>(1,1)</math>.
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</div>{{#NAVCONTENT:Answer|Answer 1.1:5|Solution |Solution 1.1:5}}

Current revision

       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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