1.3 Övningar
Aus Förberedande kurs i matematik 2
(Unterschied zwischen Versionen)
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{| 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" | | ||
| - | {{ | + | {{Ej vald flik|[[1.3 Max- och minproblem|Theory]]}} |
| - | {{ | + | {{Vald flik|[[1.3 Övningar|exercises]]}} |
| style="border-bottom:1px solid #000" width="100%"| | | style="border-bottom:1px solid #000" width="100%"| | ||
|} | |} | ||
| - | === | + | ===exercise 1.3:1=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | 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 - Figur - Grafen till övning 1.3:1a}} |
|b) | |b) | ||
| - | |width="50%"| | + | |width="50%"|{{:1.3 - Figur - Grafen till övning 1.3:1b}} |
|- | |- | ||
|c) | |c) | ||
| - | |width="50%"| | + | |width="50%"|{{:1.3 - Figur - Grafen till övning 1.3:1c}} |
|d) | |d) | ||
| - | |width="50%"| | + | |width="50%"|{{:1.3 - Figur - Grafen till övning 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: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}} | ||
| - | === | + | ===exercise 1.3:2=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Determine the local extrema and sketch the graph of | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 39: | Zeile 39: | ||
</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: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}} | ||
| - | === | + | ===exercise 1.3:3=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Determine the local extrema and sketch the graph of | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 58: | Zeile 58: | ||
</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: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}} | ||
| - | === | + | ===exercise 1.3:4=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | {| width="100%" | |
| - | + | | 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 - Figur - Parabeln y = 1 - x² med rektangel}} | ||
| + | |} | ||
</div>{{#NAVCONTENT:Svar|Svar 1.3:4|Lösning |Lösning 1.3:4}} | </div>{{#NAVCONTENT:Svar|Svar 1.3:4|Lösning |Lösning 1.3:4}} | ||
| - | === | + | ===exercise 1.3:5=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | {| width="100%" | |
| - | + | | width="95%" | | |
| - | + | 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 <math>\alpha</math> be so that the channel holds as much water as possible? | |
| + | | width="5%" | | ||
| + | ||{{:1.3 - Figur - Plåtränna}} | ||
| + | |} | ||
</div>{{#NAVCONTENT:Svar|Svar 1.3:5|Lösning |Lösning 1.3:5}} | </div>{{#NAVCONTENT:Svar|Svar 1.3:5|Lösning |Lösning 1.3:5}} | ||
| - | === | + | ===exercise 1.3:6=== |
<div class="ovning"> | <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:Svar|Svar 1.3:6|Lösning |Lösning 1.3:6}} | </div>{{#NAVCONTENT:Svar|Svar 1.3:6|Lösning |Lösning 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 be angle of the removed circular sector so that the cornet has maximum volume? | ||
| + | |||
| + | </div>{{#NAVCONTENT:Svar|Svar 1.3:7|Lösning |Lösning 1.3:7}} | ||
Aktuelle Version
| 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) |
| b) |
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| c) |
| d) |
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![[Image]](/wikis/2008/bridgecourse2-TU-Berlin/img_auth.php/metapost/3/6/c/36c6d4d37e0b3fc74d0df35ee94ec850.png)
![[Image]](/wikis/2008/bridgecourse2-TU-Berlin/img_auth.php/metapost/e/c/0/ec0655f2f2d7e20cddf013b01fbd299a.png)
![[Image]](/wikis/2008/bridgecourse2-TU-Berlin/img_auth.php/metapost/3/f/a/3fadb27533bd453e1b86362d65f8e744.png)
![[Image]](/wikis/2008/bridgecourse2-TU-Berlin/img_auth.php/metapost/c/b/f/cbfbe26d9af3d32a2f6f7b11dfe5462e.png)
Svar
Laden...
Lösning a
Lösning b
Lösning c
Lösning d
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 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
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 då \displaystyle -3\le x\le 3 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
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]](/wikis/2008/bridgecourse2-TU-Berlin/img_auth.php/metapost/6/a/8/6a879a0e0075e3738edb770c4fb972bb.png)
Svar
Lösning
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]](/wikis/2008/bridgecourse2-TU-Berlin/img_auth.php/metapost/8/6/0/8607c1119dac485e3910f0df120625af.png)
Svar
Lösning
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?
Svar
Lösning
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 be angle of the removed circular sector so that the cornet has maximum volume?
Svar
Lösning
