Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 16/en

Aus Wikiversity
Zur Navigation springen Zur Suche springen



Warm-up-exercises

Exercise

Find a zero for the function

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


Exercise

Let

be a continuous function. Prove that is not surjective.


Exercise

Give an example of a bounded interval and a continuous function

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


Exercise

Let

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

of .


Exercise

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.


Exercise

Determine directly, for which the power function

has an extremum at the point zero.


Exercise

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


Exercise

Determine the limit of the sequence




Hand-in-exercises

Exercise (2 points)

Determine the minimum of the function


Exercise (4 points)

Find for the function

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


Exercise (2 points)

Determine the limit of the sequence


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


A function

is called even, if for all the equality
holds. A function
is called odd, if for all the equality
holds.


Exercise (4 points)

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.


Let be a set and let

be a function. An element such that is called a fixed point of the function.


Exercise (4 points)

Let

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




<< | Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I | >>

PDF-Version dieses Arbeitsblattes (PDF englisch)

Zur Vorlesung (PDF)