Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 16



Warm-up-exercises

Find a zero for the function

 

in the interval   using the interval bisection method with a maximum error of  .


Let

 

be a continuous function. Show that   is not surjective.


Give an example of a bounded interval   and a continuous function

 

such that the image of   is bounded, but the function admits no maximum.


Let

 

be a continuous function. Show that there exists a continuous extension

 

of  .


Let

 

be a continuous function defined over a real interval. The function has at points  ,  , local maxima. Prove that the function has between   and   has at least one local minimum.


Determine directly, for which   the power function

 

has an extremum at the point zero.


Show that the Intermediate value theorem for continuous functions from   to   does not hold.


Determine the limit of the sequence

 




Hand-in-exercises

Exercise (2 marks)

Determine the minimum of the function

 


Exercise (5 marks)

Find for the function

 

a zero in the interval   using the interval bisection method, with a maximum error of  .


Exercise (2 marks)

Determine the limit of the sequence

 


The next task uses the notion of an even and an odd function.

Exercise (4 marks)

Let

 

be a continuous function. Show that one can write

 

with a continuous even function   and a continuous odd function  .


The following task uses the notion of fixed point.

Exercise (4 marks)

Let

 

be a continuous function from the interval   into itself. Prove that   has a fixed point.