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

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Find a zero for the function \maabbeledisp {f} {\R} {\R } {x} {f(x) = x^2+x-1 } {,} in the interval $[0,1]$ using the interval bisection method with a maximum error of $1/100$.

}
{} {}




\inputexercise
{}
{

Let \maabbdisp {f} {[0,1]} {[0,1[ } {} be a continuous function. Prove that $f$ is not surjective.

}
{} {}




\inputexercise
{}
{

Give an example of a bounded interval $I\subseteq \R$ and a continuous function \maabbdisp {f} {I} {\R } {} such that the image of $f$ is bounded, but the function admits no maximum.

}
{} {}




\inputexercise
{}
{

Let \maabbdisp {f} {[a,b]} {\R } {} be a continuous function. Show that there exists a continuous extension \maabbdisp {\tilde{f}} {\R} {\R } {} of $f$.

}
{} {}




\inputexercise
{}
{

Let \maabbdisp {f} {I} {\R } {} be a continuous function defined over a real interval. The function has at points
\mathbed {x_1,x_2 \in I} {}
{x_1 < x_2} {}
{} {} {} {,} local maxima. Prove that the function has between \mathkor {} {x_1} {and} {x_2} {} has at least one local minimum.

}
{} {}




\inputexercise
{}
{

Determine directly, for which $n \in \N$ the power function \maabbeledisp {} {\R} {\R } {x} {x^n } {,} has an extremum at the point zero.

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

Determine the limit of the sequence
\mathdisp {x_n = \sqrt{ { \frac{ 7n^2-4 }{ 3n^2-5n+2 } } }, \, n \in \N} { . }

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{2}
{

Determine the minimum of the function \maabbeledisp {f} {\R} {\R } {x} {x^2+3x-5 } {.}

}
{} {}




\inputexercise
{4}
{

Find for the function \maabbeledisp {f} {\R} {\R } {x} {f(x) = x^3 -3x+1 } {,} a zero in the interval $[0,1]$ using the interval bisection method, with a maximum error of $1/200$.

}
{} {}




\inputexercise
{2}
{

Determine the limit of the sequence
\mathdisp {x_n = \sqrt[3]{ { \frac{ 27n^3+13n^2+n }{ 8n^3-7n+10 } } }, \, n \in \N} { . }

}
{} {}

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


A \definitionsverweis {function}{}{} \maabbdisp {f} {\R} {\R } {} is called \definitionswort {even}{,} if for all
\mathl{x \in \R}{} the equality
\mathdisp {f(x)=f(-x)} { }
holds.

A function \maabbdisp {f} {\R} {\R } {} is called \definitionswort {odd}{,} if for all
\mathl{x \in \R}{} the equality
\mathdisp {f(x)=-f(-x)} { }
holds.





\inputexercise
{4}
{

Let \maabbdisp {f} {\R} {\R } {} be a \definitionsverweis {continuous function}{}{.} Show that one can write
\mathdisp {f=g+h} { }
with a continuous \definitionsverweis {even function}{}{} $g$ and a continuous \definitionsverweis {odd function}{}{} $h$.

}
{} {} The following task uses the notion of \stichwort {fixed point} {.}


Let $M$ be a set and let \maabbdisp {f} {M} {M } {} be a function. An element
\mathl{x \in M}{} such that
\mathl{f(x)=x}{} is called a \definitionswort {fixed point}{} of the function.





\inputexercise
{4}
{

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

}
{} {}



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