Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 23/en/latex

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\setcounter{section}{23}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Determine the Riemann sum (Treppenintegral) over
\mathl{[-3,+4]}{} of the staircase function
\mathdisp {f(t) = \begin{cases} 5 , \text{ if } -3 \leq t \leq -2 \, , \\ -3 , \text{ if } -2 < t \leq -1 \, , \\ \frac{3}{7} , \text{ if } -1 < t < -\frac{1}{2} \, , \\ 13 , \text{ if } t = - \frac{1}{2} \, , \\ \pi , \text{ if } - \frac{1}{2} < t < e \, , \\ 0 , \text{ if } e \leq t \leq 3 \, , \\ 1 , \text{ if } 3 < t \leq 4 \, , \end{cases}} { . }

}
{} {}




\inputexercise
{}
{

a) Subdivide the interval
\mathl{[-4,5]}{} in six subintervals of equal length.

b) Determine the Riemann sum of the staircase function on
\mathl{[-4,5]}{,} which takes alternately the values
\mathl{2}{} and $-1$ on the subdivision constructed in a).

}
{} {}




\inputexercise
{}
{

Give an example of a function \maabb {f} {[a,b]} {\R } {} which assumes only finitely many values, but is not a staircase function.

}
{} {}




\inputexercise
{}
{

Let \maabbdisp {f} {[a,b]} { [c,d] } {} be a staircase function and let \maabbdisp {g} {[c,d]} {\R } {} be a function. Prove that the composite
\mathl{g \circ f}{} is also a staircase function.

}
{} {}




\inputexercise
{}
{

Give an example of a continuous function \maabbdisp {f} {[a,b]} {[c,d] } {} and a staircase function \maabbdisp {g} {[c,d]} {\R } {} such that the composite
\mathl{g \circ f}{} is not a staircase function.

}
{} {}




\inputexercise
{}
{

Determine the definite integral
\mathdisp {\int_{ 0 }^{ 1 } t \, d t} { }
explicitly with upper and lower staircase functions.

}
{} {}




\inputexercise
{}
{

Determine the definite integral
\mathdisp {\int_{ 1 }^{ 2 } t^3 \, d t} { }
explicitly with upper and lower staircase functions.

}
{} {}




\inputexercise
{}
{

Let $I =[a,b]$ be a compact interval and let \maabbdisp {f} {I} {\R } {} be a function. Consider a sequence of staircase functions
\mathbed {{ \left( s_n \right) }_{ n \in \N }} {such that}
{s_n \leq f} {}
{} {} {} {} and a sequence of staircase functions
\mathbed {{ \left( t_n \right) }_{ n \in \N }} {such that}
{t_n \geq f} {}
{} {} {} {.} Assume that the two Riemann sums corresponding to the sequences converge and that their limits coincide. Prove that $f$ is Riemann-integrable and that
\mavergleichskettedisp
{\vergleichskette
{ \lim_{n \rightarrow \infty} \int_{ a }^{ b } s_n ( x) \, d x }
{ =} { \int_{ a }^{ b } f ( x) \, d x }
{ =} { \lim_{n \rightarrow \infty} \int_{ a }^{ b } t_n ( x) \, d x }
{ } { }
{ } { }
} {}{}{.}

}
{} {}




\inputexercise
{}
{ Kompaktes Intervall/Reelle Funktion/Riemann integrierbar auf Unterteilung/Fakt/Beweis/Aufgabe/en }
{} {}




\inputexercise
{}
{

Let
\mathl{I =[a,b] \subseteq \R}{} be a compact interval and let \maabb {f,g} {I} {\R } {} be two Riemann-integrable functions. Prove the following statements. \aufzaehlungvier{If
\mathl{m \leq f(x) \leq M}{} for all
\mathl{x \in I}{,} then
\mathl{m(b-a) \leq \int_{ a }^{ b } f ( t) \, d t \leq M(b-a)}{.} }{If
\mathl{f(x) \leq g(x)}{} for all
\mathl{x \in I}{,} then
\mathl{\int_{ a }^{ b } f ( t) \, d t \leq \int_{ a }^{ b } g ( t) \, d t}{.} }{We have
\mathl{\int_{ a }^{ b } f(t)+g(t) \, d t = \int_{ a }^{ b } f ( t) \, d t + \int_{ a }^{ b } g ( t) \, d t}{.} }{For
\mathl{c \in \R}{} we have
\mathl{\int_{ a }^{ b } (cf)(t) \, d t = c \int_{ a }^{ b } f ( t) \, d t}{.} }

}
{} {}




\inputexercise
{}
{

Let
\mathl{I=[a,b]}{} be a compact interval and let \maabb {f} {I} {\R } {} be a Riemann-integrable function. Prove that
\mathdisp {\betrag { \int_{ a }^{ b } f ( t) \, d t } \leq \int_{ a }^{ b } \betrag { f(t) } \, d t} { . }

}
{} {}




\inputexercise
{}
{

Let
\mathl{I=[a,b]}{} be a \definitionsverweis {compact interval}{}{} and let \maabb {f,g} {I} {\R } {} be two Riemann-integrable functions. Prove that $fg$ is also Riemann-integrable.

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{2}
{

Let \maabbdisp {f,g} {[a,b]} {\R } {} be two staircase functions. Prove that
\mathl{f+g}{} is also a staircase function.

}
{} {}




\inputexercise
{3}
{

Determine the definite integral
\mathdisp {\int_{ a }^{ b } t^2 \, d t} { }
as a function of \mathkor {} {a} {and} {b} {} explicitly with lower and upper staircase functions

}
{} {}




\inputexercise
{4}
{

Determine the definite integral
\mathdisp {\int_{ -2 }^{ 7 } -t^3+3t^2-2t+5 \, d t} { }
explicitly with upper and lower staircase functions.

}
{} {}




\inputexercise
{3}
{

Prove that for the function \maabbeledisp {} {]0,1]} {\R } {x} { \frac{1}{x} } {,} neither the lower nor the upper integral exist.

}
{} {}




\inputexercise
{6}
{

Prove that for the function \maabbeledisp {} {]0,1]} {\R } {x} { \frac{1}{ \sqrt{x} } } {,} the lower integral exists, but the upper integral does not exist.

}
{} {}




\inputexercise
{5}
{

Let $I$ be a compact interval and let \maabbdisp {f} {I} {\R } {} be a monotone function. Prove that $f$ is Riemann-integrable.

}
{} {}



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