Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 27/en/latex

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\setcounter{section}{27}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Let \maabbdisp {f} {\R} {\R } {} be an \definitionsverweis {increasing function}{}{} and
\mathl{b \in \R}{.} Show that the sequence
\mathbed {f(n)} {}
{n \in \N} {}
{} {} {} {,} converges to $b$ if and only if


\mathdisp {\operatorname{lim}_{ x \rightarrow + \infty } \, f(x) =b} { }
holds, i.e. if the limit of the function for
\mathl{x \rightarrow +\infty}{} is $b$.

}
{} {}




\inputexercise
{}
{

Let $I$ be an interval,
\mathl{r}{} a boundary point of $I$ and \maabbdisp {f} {I} {\R } {} a continuous function. Prove that the existence of the improper integral
\mathdisp {\int_{ a }^{ r } f ( t) \, d t} { }
does not depend on the choice of the starting point
\mathl{a \in I}{.}

}
{} {}




\inputexercise
{}
{

Let $I={]r,s[}$ be a bounded open interval and \maabbdisp {f} {I} {\R } {} a continuous function, which can be extended continuously to
\mathl{[r,s]}{.} Prove that the improper integral
\mathdisp {\int_{ r }^{ s } f ( t) \, d t} { }
exists.

}
{} {}




\inputexercise
{}
{

Formulate and prove computation rules for improper integrals (analogous to Lemma 23.5).

}
{} {}




\inputexercise
{}
{

Decide whether the improper integral
\mathdisp {\int_{ 1 }^{ \infty } { \frac{ x^2-3x+5 }{ x^4+2x^3+5x+8 } } \, d x} { }
exists.

}
{} {}




\inputexercise
{}
{

Determine the improper integral
\mathdisp {\int_{ 0 }^{ \infty } e^{-t} \, d t} { . }

}
{} {}




\inputexercise
{}
{

Let
\mathl{I=[a,b]}{} be a bounded interval and let \maabbdisp {f} {]a,b[} {\R } {} be a continuous function. Let
\mathl{{ \left( x_n \right) }_{n \in \N }}{} be a decreasing sequence in $I$ with limit $a$ and let
\mathl{{ \left( y_n \right) }_{n \in \N }}{} be an increasing sequence in $I$ with limit $b$. Assume that the improper integral
\mathl{\int_{ a }^{ b } f ( t) \, d t}{} exists. Prove that the sequence
\mathdisp {w_n = \int_{ x_n }^{ y_n } f ( t) \, d t} { }
converges to this improper integral.

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{2}
{

Compute the energy that would be necessary to move the Earth, starting from the current position relative to the Sun, infinitely far away from the Sun.

}
{} {}




\inputexercise
{3}
{

Decide whether the improper integral
\mathdisp {\int_{ 0 }^{ \infty } { \frac{ 1 }{ (x+1) \sqrt{x} } } \, d x} { }
exists and compute it in case of existence.

}
{} {}




\inputexercise
{5}
{

Give an example of a not bounded, continuous function \maabbdisp {f} {\R_{\geq 0}} {\R_{\geq 0} } {,} such that the improper integral $\int_{ 0 }^{ \infty } f ( t) \, d t$ exists.

}
{} {}




\inputexercise
{2}
{

Decide whether the improper integral
\mathdisp {\int_{ -1 }^{ 1 } { \frac{ 1 }{ \sqrt{1-t^2} } } \, d t} { }
exists and compute it in case of existence.

}
{} {}




\inputexercise
{4}
{

Decide whether the improper integral
\mathdisp {\int_{ 1 }^{ \infty } { \frac{ x^3-3x+5 }{ x^4+2x^3+5x+8 } } \, d x} { }
exists.

}
{} {}




\inputexercise
{8}
{

Decide whether the improper integral
\mathdisp {\int_{ 0 }^{ \infty } { \frac{ \sin x }{ x } } \, d x} { }
exists.

}
{} {(Do not try to find an antiderivative for the integrand.)}



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