1.3 Exercises

From Förberedande kurs i matematik 2

(Difference between revisions)
Jump to: navigation, search
Current revision (13:24, 17 September 2008) (edit) (undo)
m (Robot: Automated text replacement (-{{:1.3 - Figur - Plåtränna}} +{{:1.3 - Figure - Channel}}))
 
(22 intermediate revisions not shown.)
Line 2: Line 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|[[1.3 Max- och minproblem|Teori]]}}
+
{{Not selected tab|[[1.3 Maximum and minimum problems|Theory]]}}
-
{{Mall:Vald flik|[[1.3 Övningar|Övningar]]}}
+
{{Selected tab|[[1.3 Exercises|Exercises]]}}
| style="border-bottom:1px solid #000" width="100%"|  
| style="border-bottom:1px solid #000" width="100%"|  
|}
|}
-
===Övning 1.3:1===
+
===Exercise 1.3:1===
<div class="ovning">
<div class="ovning">
-
Bestäm kritiska punkter, terasspunkter, lokala extrempunkter och globala extrempunkter för funktionerna som beskrivs i graferna nedan. Ange också de intervall där funktionen är strängt växande respektive strängt avtagande.
+
Determine the critical points, the inflexion points, the local extrema and global extrema. Give also the intervals where the function is strictly increasing and strictly decreasing.
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
-
|width="50%"|
+
|width="50%"|{{:1.3 - Figure - The graph in exercise 1.3:1a}}
|b)
|b)
-
|width="50%"|
+
|width="50%"|{{:1.3 - Figure - The graph in exercise 1.3:1b}}
|-
|-
|c)
|c)
-
|width="50%"|
+
|width="50%"|{{:1.3 - Figure - The graph in exercise 1.3:1c}}
|d)
|d)
-
|width="50%"|
+
|width="50%"|{{:1.3 - Figure - The graph in exercise 1.3:1d}}
|}
|}
-
</div>{{#NAVCONTENT:Svar|Svar 1.3:1|Lösning a|Lösning 1.3:1a|Lösning b|Lösning 1.3:1b|Lösning c|Lösning 1.3:1c|Lösning d|Lösning 1.3:1d}}
+
</div>{{#NAVCONTENT:Answer|Answer 1.3:1|Solution a|Solution 1.3:1a|Solution b|Solution 1.3:1b|Solution c|Solution 1.3:1c|Solution d|Solution 1.3:1d}}
-
===Övning 1.3:2===
+
===Exercise 1.3:2===
<div class="ovning">
<div class="ovning">
-
Bestäm kritiska punkter, terasspunkter, lokala extrempunkter och globala extrempunkter för funktionerna som beskrivs i graferna nedan. Ange också de intervall där funktionen är strängt växande respektive strängt avtagande.
+
Determine the local extrema and sketch the graph of
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
Line 37: Line 37:
|width="50%"| <math>f(x)=x^3-9x^2+30x-15</math>
|width="50%"| <math>f(x)=x^3-9x^2+30x-15</math>
|}
|}
-
</div>{{#NAVCONTENT:Svar|Svar 1.3:2|Lösning a|Lösning 1.3:2a|Lösning b|Lösning 1.3:2b|Lösning c|Lösning 1.3:2c|Lösning d|Lösning 1.3:2d}}
+
</div>{{#NAVCONTENT:Answer|Answer 1.3:2|Solution a|Solution 1.3:2a|Solution b|Solution 1.3:2b|Solution c|Solution 1.3:2c|Solution d|Solution 1.3:2d}}
-
===Övning 1.3:3===
+
===Exercise 1.3:3===
<div class="ovning">
<div class="ovning">
-
Bestäm kritiska punkter, terasspunkter, lokala extrempunkter och globala extrempunkter för funktionerna som beskrivs i graferna nedan. Ange också de intervall där funktionen är strängt växande respektive strängt avtagande.
+
Determine the local extrema and sketch the graph of
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
Line 54: Line 54:
|-
|-
|e)
|e)
-
|width="50%"| <math>f(x)=(x^2-x-1)e^x</math> då <math>-3\le x\le 3</math>
+
|width="50%"| <math>f(x)=(x^2-x-1)e^x</math> when <math>-3\le x\le 3</math>
|}
|}
-
</div>{{#NAVCONTENT:Svar|Svar 1.3:3|Lösning a|Lösning 1.3:3a|Lösning b|Lösning 1.3:3b|Lösning c|Lösning 1.3:3c|Lösning d|Lösning 1.3:3d|Lösning e|Lösning 1.3:3e}}
+
</div>{{#NAVCONTENT:Answer|Answer 1.3:3|Solution a|Solution 1.3:3a|Solution b|Solution 1.3:3b|Solution c|Solution 1.3:3c|Solution d|Solution 1.3:3d|Solution e|Solution 1.3:3e}}
-
===Övning 1.3:4===
+
===Exercise 1.3:4===
<div class="ovning">
<div class="ovning">
-
Var på kurvan <math>y=1-x^2</math> i första kvadranten ska punkten <math>P</math> väljas för att rektangeln i figuren till höger ska ha maximal area?
+
{| width="100%"
-
Bild
+
| width="95%" |
 +
Where, in the first quadrant, on the curve <math>y=1-x^2</math> should the point <math>P</math> be chosen so that the rectangle in the figure to the right has maximum area?
 +
| width="5%" |
 +
||{{:1.3 - Figure - The parabola y = 1 - x² with a rectangle}}
 +
|}
 +
</div>{{#NAVCONTENT:Answer|Answer 1.3:4|Solution|Solution 1.3:4}}
 +
 
 +
===Exercise 1.3:5===
 +
<div class="ovning">
 +
{| width="100%"
 +
| width="95%" |
 +
A 30&nbsp;cm wide sheet of metal is to be used to make a channel. The edges are bent upwards parallel with the sheet's long sides, as shown in the figure. How large should the angle <math>\alpha</math> be so that the channel holds as much water as possible?
 +
| width="5%" |
 +
||{{:1.3 - Figure - Channel}}
 +
|}
 +
</div>{{#NAVCONTENT:Answer|Answer 1.3:5|Solution|Solution 1.3:5}}
 +
 
 +
===Exercise 1.3:6===
 +
<div class="ovning">
 +
A metal cup is to be made which has the form of a vertical circular cylinder. What radius and height should the cup have if it is to have a prescribed volume <math>V</math> as well as being made of as little metal as possible?
 +
 
 +
</div>{{#NAVCONTENT:Answer|Answer 1.3:6|Solution|Solution 1.3:6}}
 +
 
 +
===Exercise 1.3:7===
 +
<div class="ovning">
 +
A circular sector is cut out from a circular disc and the two radial edge which result are bound together to produce a cornet. What should the angle of the removed circular sector be so that the cornet has maximum volume?
-
</div>{{#NAVCONTENT:Svar|Svar 1.3:4|Lösning |Lösning 1.3:4}}
+
</div>{{#NAVCONTENT:Answer|Answer 1.3:7|Solution|Solution 1.3:7}}

Current revision

       Theory          Exercises      

Exercise 1.3:1

Determine the critical points, the inflexion points, the local extrema and global extrema. Give also the intervals where the function is strictly increasing and strictly decreasing.

a)

[Image]

b)

[Image]

c)

[Image]

d)

[Image]

Answer
Answer 1.3:1
Solution a
Solution 1.3:1a
Solution b
Solution 1.3:1b
Solution c
Solution 1.3:1c
Solution d
Solution 1.3:1d

Exercise 1.3:2

Determine the local extrema and sketch the graph of

a) \displaystyle f(x)= x^2 -2x+1 b) \displaystyle f(x)=2+3x-x^2
c) \displaystyle f(x)= 2x^3+3x^2-12x+1 d) \displaystyle f(x)=x^3-9x^2+30x-15
Answer
Answer 1.3:2
Solution a
Solution 1.3:2a
Solution b
Solution 1.3:2b
Solution c
Solution 1.3:2c
Solution d
Solution 1.3:2d

Exercise 1.3:3

Determine the local extrema and sketch the graph of

a) \displaystyle f(x)=-x^4+8x^3-18x^2 b) \displaystyle f(x)=e^{-3x} +5x
c) \displaystyle f(x)= x\ln x -9 d) \displaystyle f(x)=\displaystyle\frac{1+x^2}{1+x^4}
e) \displaystyle f(x)=(x^2-x-1)e^x when \displaystyle -3\le x\le 3
Answer
Answer 1.3:3
Solution a
Solution 1.3:3a
Solution b
Solution 1.3:3b
Solution c
Solution 1.3:3c
Solution d
Solution 1.3:3d
Solution e
Solution 1.3:3e

Exercise 1.3:4

Where, in the first quadrant, on the curve \displaystyle y=1-x^2 should the point \displaystyle P be chosen so that the rectangle in the figure to the right has maximum area?

[Image]

Answer
Answer 1.3:4
Solution
Solution 1.3:4

Exercise 1.3:5

A 30 cm wide sheet of metal is to be used to make a channel. The edges are bent upwards parallel with the sheet's long sides, as shown in the figure. How large should the angle \displaystyle \alpha be so that the channel holds as much water as possible?

[Image]

Answer
Answer 1.3:5
Solution
Solution 1.3:5

Exercise 1.3:6

A metal cup is to be made which has the form of a vertical circular cylinder. What radius and height should the cup have if it is to have a prescribed volume \displaystyle V as well as being made of as little metal as possible?

Answer
Answer 1.3:6
Solution
Solution 1.3:6

Exercise 1.3:7

A circular sector is cut out from a circular disc and the two radial edge which result are bound together to produce a cornet. What should the angle of the removed circular sector be so that the cornet has maximum volume?

Answer
Answer 1.3:7
Solution
Solution 1.3:7
Retrieved from "http://wiki.sommarmatte.se/wikis/2008/bridgecourse2-ImperialCollege/index.php/1.3_Exercises"