1.3 Übungen
Aus Online Mathematik Brückenkurs 2
(Unterschied zwischen Versionen)
(Translated links into English) |
|||
| Zeile 3: | Zeile 3: | ||
| 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]]}} | {{Ej vald flik|[[1.3 Max- och minproblem|Theory]]}} | ||
| - | {{Vald flik|[[1.3 Övningar| | + | {{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. | 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. | ||
| Zeile 21: | Zeile 21: | ||
|width="50%"|{{:1.3 - Figur - Grafen till övning 1.3:1d}} | |width="50%"|{{:1.3 - Figur - Grafen till övning 1.3:1d}} | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 1.3:1|Solution a|Lösning 1.3:1a|Solution b|Lösning 1.3:1b|Solution c|Lösning 1.3:1c|Solution d|Lösning 1.3:1d}} |
| - | === | + | ===Exercise 1.3:2=== |
<div class="ovning"> | <div class="ovning"> | ||
Determine the local extrema and sketch the graph of | Determine the local extrema and sketch the graph of | ||
| Zeile 37: | Zeile 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: | + | </div>{{#NAVCONTENT:Answer|Svar 1.3:2|Solution a|Lösning 1.3:2a|Solution b|Lösning 1.3:2b|Solution c|Lösning 1.3:2c|Solution d|Lösning 1.3:2d}} |
| - | === | + | ===Exercise 1.3:3=== |
<div class="ovning"> | <div class="ovning"> | ||
Determine the local extrema and sketch the graph of | Determine the local extrema and sketch the graph of | ||
| Zeile 54: | Zeile 54: | ||
|- | |- | ||
|e) | |e) | ||
| - | |width="50%"| <math>f(x)=(x^2-x-1)e^x</math> | + | |width="50%"| <math>f(x)=(x^2-x-1)e^x</math> when <math>-3\le x\le 3</math> |
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 1.3:3|Solution a|Lösning 1.3:3a|Solution b|Lösning 1.3:3b|Solution c|Lösning 1.3:3c|Solution d|Lösning 1.3:3d|Solution e|Lösning 1.3:3e}} |
| - | === | + | ===Exercise 1.3:4=== |
<div class="ovning"> | <div class="ovning"> | ||
{| width="100%" | {| width="100%" | ||
| Zeile 66: | Zeile 66: | ||
||{{:1.3 - Figur - Parabeln y = 1 - x² med rektangel}} | ||{{:1.3 - Figur - Parabeln y = 1 - x² med rektangel}} | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 1.3:4|Solution|Lösning 1.3:4}} |
| - | === | + | ===Exercise 1.3:5=== |
<div class="ovning"> | <div class="ovning"> | ||
{| width="100%" | {| width="100%" | ||
| Zeile 76: | Zeile 76: | ||
||{{:1.3 - Figur - Plåtränna}} | ||{{:1.3 - Figur - Plåtränna}} | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT: | + | </div>{{#NAVCONTENT:Answer|Svar 1.3:5|Solution|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? | 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: | + | </div>{{#NAVCONTENT:Answer|Svar 1.3:6|Solution|Lösning 1.3:6}} |
| - | === | + | ===Exercise 1.3:7=== |
<div class="ovning"> | <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 | + | 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: | + | </div>{{#NAVCONTENT:Answer|Svar 1.3:7|Solution|Lösning 1.3:7}} |
Version vom 07:08, 3. Sep. 2008
|
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) | 1.3 - Figur - Grafen till övning 1.3:1a | b) | 1.3 - Figur - Grafen till övning 1.3:1b |
| c) | 1.3 - Figur - Grafen till övning 1.3:1c | d) | 1.3 - Figur - Grafen till övning 1.3:1d |
Answer
Laden...
Solution a
Solution b
Solution c
Solution 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 |
Answer
Solution a
Solution b
Solution c
Solution 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 when \displaystyle -3\le x\le 3 |
Answer
Solution a
Solution b
Solution c
Solution d
Solution 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? | 1.3 - Figur - Parabeln y = 1 - x² med rektangel |
Answer
Solution
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? | 1.3 - Figur - Plåtränna |
Answer
Solution
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
Solution
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
Solution
