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

Warm-up-exercises

Exercise

Find a zero for the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{2}+x-1,$

in the interval ${}[0,1]$ using the interval bisection method with a maximum error of ${}1/100$.

Exercise

Let

$f\colon [0,1]\longrightarrow [0,1[$

be a continuous function. Show that ${}f$ is not surjective.

Exercise

Give an example of a bounded interval ${}I\subseteq \mathbb {R}$ and a continuous function

$f\colon I\longrightarrow \mathbb {R}$

such that the image of ${}f$ is bounded, but the function admits no maximum.

Exercise

Let

$f\colon [a,b]\longrightarrow \mathbb {R}$

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

${\tilde {f}}\colon \mathbb {R} \longrightarrow \mathbb {R}$

of ${}f$.

Exercise

Let

$f\colon I\longrightarrow \mathbb {R}$

be a continuous function defined over a real interval. The function has at points ${}x_{1},x_{2}\in I$, ${}x_{1}<x_{2}$, local maxima. Prove that the function has between ${}x_{1}$ and ${}x_{2}$ has at least one local minimum.

Exercise

Determine directly, for which ${}n\in \mathbb {N}$ the power function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

has an extremum at the point zero.

Exercise *

Show that the Intermediate value theorem for continuous functions from ${}\mathbb {Q}$ to ${}\mathbb {Q}$ does not hold.

Exercise

Determine the limit of the sequence

$x_{n}={\sqrt {\frac {7n^{2}-4}{3n^{2}-5n+2}}},\,n\in \mathbb {N} .$

Hand-in-exercises

Exercise (2 marks)

Determine the minimum of the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}+3x-5.$

Exercise (5 marks)

Find for the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{3}-3x+1,$

a zero in the interval ${}[0,1]$ using the interval bisection method, with a maximum error of ${}1/200$.

Exercise (2 marks)

Determine the limit of the sequence

$x_{n}={\sqrt[{3}]{\frac {27n^{3}+13n^{2}+n}{8n^{3}-7n+10}}},\,n\in \mathbb {N} .$

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

Exercise (4 marks)

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

be a continuous function. Show that one can write

${}f=g+h\,,$

with a continuous even function ${}g$ and a continuous odd function ${}h$.

The following task uses the notion of fixed point.

Exercise (4 marks)

Let

$f\colon [a,b]\longrightarrow [a,b]$

be a continuous function from the interval ${}[a,b]$ into itself. Prove that ${}f$ has a fixed point.