Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 21/en

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Warm-up-exercises

Exercise

Determine the derivatives of hyperbolic sine and hyperbolic cosine.


Exercise

Determine the derivative of the function


Exercise

Determine the derivative of the function


Exercise

Determine the derivatives of the sine and the cosine function by using Theorem 21.1.


Exercise

Determine the -th derivative of the sine function.


Exercise

Determine the derivative of the function


Exercise

Determine the derivative of the function


Exercise

Determine for the derivative of the function


Exercise

Determine the derivative of the function


Exercise

Prove that the real sine function induces a bijective, strictly increasing function

and that the real cosine function induces a bijective, strictly decreasing function


Exercise

Determine the derivatives of arc-sine and arc-cosine functions.


Exercise

We consider the function

a) Prove that gives a continuous bijection between and .


b) Determine the inverse image of under , then compute and . Draw a rough sketch for the inverse function .


Exercise

Let

be two differentiable functions. Let . We have that

Prove that


Exercise

We consider the function

a) Investigate the monotony behavior of this function.

b) Prove that this function is injective.

c) Determine the image of .

d) Determine the inverse function on the image for this function.

e) Sketch the graph of the function .


Exercise

Consider the function

Determine the zeros and the local (global) extrema of .

Sketch up roughly the graph of the function.


Exercise

Discuss the behavior of the function graph of

Determine especially the monotony behavior, the extrema of , and also for the derivative .


Exercise

Prove that the function

is continuous and that it has infinitely many zeros.


Exercise

Determine the limit of the sequence


Exercise

Determine for the following functions if the function limit exists and, in case, what value it takes.

  1. ,
  2. ,
  3. ,
  4. .


Exercise

Determine for the following functions, if the limit function for , , exists, and, in case, what value it takes.

  1. ,
  2. ,
  3. .




Hand-in-exercises

Exercise (3 points)

Determine the linear functions that are tangent to the exponential function.


Exercise (3 points)

Determine the derivative of the function


The following task should be solved without reference to the second derivative.

Exercise (4 points)

Determine the extrema of the function


Exercise (6 points)

Let

be a polynomial function of degree . Let be the number of local maxima of and the number of local minima of . Prove that if is odd then and that if is even then .




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