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



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