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Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 21

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Warm-up-exercises

Determine the derivatives of hyperbolic sine and hyperbolic cosine.


Determine the derivative of the function


Determine the derivative of the function


Determine the derivatives of the sine and the cosine function by using fact.


Determine the -th derivative of the sine function.


Determine the derivative of the function


Determine the derivative of the function


Determine for the derivative of the function


Determine the derivative of the function


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

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


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


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 .


Let

be two differentiable functions. Let . Suppose we have that

Prove that


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 .


Consider the function

Determine the zeros and the local (global) extrema of . Sketch up roughly the graph of the function.


Discuss the behavior of the function graph of

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


Prove that the function

is continuous and that it has infinitely many zeros.


Determine the limit of the sequence


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

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


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

  1. ,
  2. ,
  3. .




Hand-in-exercises

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


Determine the derivative of the function


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

Determine the extrema of the function


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