1.3 Exercises

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===Exercise 1.3:2===
===Exercise 1.3:2===
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|width="50%"| <math>f(x)=x^3-9x^2+30x-15</math>
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===Exercise 1.3:3===
===Exercise 1.3:3===
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|width="50%"| <math>f(x)=(x^2-x-1)e^x</math> when <math>-3\le x\le 3</math>
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===Exercise 1.3:4===
===Exercise 1.3:4===
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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?
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?
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===Exercise 1.3:5===
===Exercise 1.3:5===
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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?
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?
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===Exercise 1.3:6===
===Exercise 1.3:6===
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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?
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===Exercise 1.3:7===
===Exercise 1.3:7===
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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?
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?
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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
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