1.1 Övningar
Aus Förberedande kurs i matematik 2
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| - | {{ | + | {{Ej vald flik|[[1.1 Inledning till derivata|Theory]]}} |
| - | {{ | + | {{Vald flik|[[1.1 Övningar|exercises]]}} |
| style="border-bottom:1px solid #000" width="100%"| | | style="border-bottom:1px solid #000" width="100%"| | ||
|} | |} | ||
| - | === | + | ===exercise 1.1:1=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | {| width="100%" | |
| + | | width="95%" | | ||
| + | The graph for <math>f(x)</math> is shown in the figure. | ||
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
| - | |a) | + | | valign="top" |a) |
| - | |width="100%"| | + | | width="100%" | What are the signs of <math>f^{\,\prime}(-5)</math> and <math>f^{\,\prime}(1)</math>? |
|- | |- | ||
| - | |b) | + | | valign="top" |b) |
| - | |width="100%"| | + | |width="100%"| For what values of <math>x</math> is <math>f^{\,\prime}(x)=0</math>? |
|- | |- | ||
| - | |c) | + | | valign="top" |c) |
| - | |width="100%"| | + | |width="100%"| In which interval(s) is <math>f^{\,\prime}(x)</math> negative? |
| + | |} | ||
| + | (Each square in the grid of the figure has width and height 1.) | ||
| + | | width="5%" | | ||
| + | ||{{:1.1 - Figur - Grafen till f(x) i övning 1.1:1}} | ||
|} | |} | ||
</div>{{#NAVCONTENT:Svar|Svar 1.1:1|Lösning a|Lösning 1.1:1a|Lösning b|Lösning 1.1:1b|Lösning c|Lösning 1.1:1c}} | </div>{{#NAVCONTENT:Svar|Svar 1.1:1|Lösning a|Lösning 1.1:1a|Lösning b|Lösning 1.1:1b|Lösning c|Lösning 1.1:1c}} | ||
| - | === | + | ===exercise 1.1:2=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Determine the derivative <math>f^{\,\prime}(x)</math> when | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 42: | Zeile 48: | ||
</div>{{#NAVCONTENT:Svar|Svar 1.1:2|Lösning a|Lösning 1.1:2a|Lösning b|Lösning 1.1:2b|Lösning c|Lösning 1.1:2c|Lösning d|Lösning 1.1:2d|Lösning e|Lösning 1.1:2e|Lösning f|Lösning 1.1:2f}} | </div>{{#NAVCONTENT:Svar|Svar 1.1:2|Lösning a|Lösning 1.1:2a|Lösning b|Lösning 1.1:2b|Lösning c|Lösning 1.1:2c|Lösning d|Lösning 1.1:2d|Lösning e|Lösning 1.1:2e|Lösning f|Lösning 1.1:2f}} | ||
| - | === | + | ===exercise 1.1:3=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | 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? | |
</div>{{#NAVCONTENT:Svar|Svar 1.1:3|Lösning |Lösning 1.1:3}} | </div>{{#NAVCONTENT:Svar|Svar 1.1:3|Lösning |Lösning 1.1:3}} | ||
| - | === | + | ===exercise 1.1:4=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Determine the equation for the tangent and normal to the curve <math>y=x^2</math> at the point <math>(1,1)</math>. | |
</div>{{#NAVCONTENT:Svar|Svar 1.1:4|Lösning |Lösning 1.1:4}} | </div>{{#NAVCONTENT:Svar|Svar 1.1:4|Lösning |Lösning 1.1:4}} | ||
| - | === | + | ===exercise 1.1:5=== |
| - | <div | + | <div exercise ="ovning"> |
| - | + | 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>. | |
</div>{{#NAVCONTENT:Svar|Svar 1.1:5|Lösning |Lösning 1.1:5}} | </div>{{#NAVCONTENT:Svar|Svar 1.1:5|Lösning |Lösning 1.1:5}} | ||
Aktuelle Version
| Theory | exercises |
exercise 1.1:1
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The graph for \displaystyle f(x) is shown in the figure.
(Each square in the grid of the figure has width and height 1.) |
|
![[Image]](/wikis/2008/bridgecourse2-TU-Berlin/img_auth.php/metapost/6/a/2/6a260dcf937e32a9315ffb6cf330d510.png)
Laden...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) |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Lösning f
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?
Svar
Lösning
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).
Svar
Lösning
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).
Svar
Lösning
