1.1 Övningar

Aus Förberedande kurs i matematik 2

(Unterschied zwischen Versionen)
Wechseln zu: Navigation, Suche
(Ny sida: __NOTOC__ {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | style="border-bottom:1px solid #000" width="5px" |   {{Mall:Ej vald flik|[[1.1 Inledning till derivat...)
Aktuelle Version (12:33, 7. Aug. 2008) (bearbeiten) (rückgängig)
K
 
(Der Versionsvergleich bezieht 13 dazwischen liegende Versionen mit ein.)
Zeile 2: Zeile 2:
{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
| style="border-bottom:1px solid #000" width="5px" |  
| style="border-bottom:1px solid #000" width="5px" |  
-
{{Mall:Ej vald flik|[[1.1 Inledning till derivata|Teori]]}}
+
{{Ej vald flik|[[1.1 Inledning till derivata|Theory]]}}
-
{{Mall:Vald flik|[[1.1 Övningar|Övningar]]}}
+
{{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">
 +
{| width="100%"
 +
| width="95%" |
 +
The graph for <math>f(x)</math> is shown in the figure.
 +
{| width="100%" cellspacing="10px"
 +
| valign="top" |a)
 +
| width="100%" | What are the signs of <math>f^{\,\prime}(-5)</math> and <math>f^{\,\prime}(1)</math>?
 +
|-
 +
| valign="top" |b)
 +
|width="100%"| For what values of <math>x</math> is <math>f^{\,\prime}(x)=0</math>?
 +
|-
 +
| valign="top" |c)
 +
|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}}
 +
 +
===exercise 1.1:2===
 +
<div class="ovning">
 +
Determine the derivative <math>f^{\,\prime}(x)</math> when
 +
{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="33%"| <math>f(x) = x^2 -3x +1</math>
 +
|b)
 +
|width="33%"| <math>f(x)=\cos x -\sin x</math>
 +
|c)
 +
|width="33%"| <math>f(x)= e^x-\ln x</math>
 +
|-
 +
|d)
 +
|width="33%"| <math>f(x)=\sqrt{x}</math>
 +
|e)
 +
|width="33%"| <math>f(x) = (x^2-1)^2</math>
 +
|f)
 +
|width="33%"| <math>f(x)= \cos (x+\pi/3)</math>
 +
|}
 +
</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">
 +
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}}
 +
 +
===exercise 1.1:4===
 +
<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}}
 +
 +
===exercise 1.1:5===
 +
<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}}

Aktuelle Version

       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}(-5) 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]

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

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
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

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
Svar 1.1:3
Lösning
Lösning 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).

Svar
Svar 1.1:4
Lösning
Lösning 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).

Svar
Svar 1.1:5
Lösning
Lösning 1.1:5
Von „http://wiki.sommarmatte.se/wikis/2008/bridgecourse2-TU-Berlin/index.php/1.1_%C3%96vningar“