Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 20/en

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Warm-up-exercises

Exercise

Prove that the function

is differentiable but not twice differentiable.


Exercise

Consider the function

defined by

Examine in terms of continuity, differentiability and extremes.


Exercise

Determine local and global extrema of the function


Exercise

Determine local and global extrema of the function


Exercise

Consider the function

Find the point such that the tangent of the function at is parallel to the secant between and .


Exercise

Prove that a real polynomial function

of degree has at most extrema, and moreover the real numbers can be divided into at most sections, where is strictly increasing or strictly decreasing.


Exercise

Determine the limit

by polynomial division (see Example 20.11).


Exercise

Determine the limit of the rational function

at point .


Exercise

Next to a rectilinear river we want to fence a rectangular area of , one side of the area is the river itself. For the other three sides, we need a fence. Which is the minimal length of the fence we need?


Exercise

Discuss the following properties of the rational function

domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.


Exercise

Consider

a) Prove that the function has in the real interval exactly one zero.

b) Compute the first decimal digit in the decimal system of this zero point.

c) Find a rational number such that .


Exercise

Determine the limit of

at point

, and specifically

a) by polynomial division.

b) by the rule of l'Hospital.


Exercise

Let be a polynomial and . Prove that is a multiple of if and only if is a zero of all the derivatives .




Hand-in-exercises

Exercise (5 points)

From a sheet of paper with side lengths of cm and cm we want to realize a box (without cover) with the greatest possible volume. We do it in this way. We remove from each corner a square of the same size, then we lift up the sides and we glue them. Which box height do we need to realize the maximum volume?


Exercise (4 points)

Discuss the following properties of the rational function

domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.


Exercise (5 points)

Prove that a non-costant rational function of the shape

(with ), has no local extrema.


Exercise (3 points)

Determine the limit of the rational function

at point .


Exercise (5 points)

Let and

be a rational function. Prove that is a polynomial if and only if there is a higher derivative such that.



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