Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 23

Warm-up-exercises

Exercise

Determine the Riemann sum over ${}[-3,+4]$ of the staircase function

${}f(t)={\begin{cases}5,{\text{ if }}-3\leq t\leq -2\,,\\-3,{\text{ if }}-2<t\leq -1\,,\\{\frac {3}{7}},{\text{ if }}-1<t<-{\frac {1}{2}}\,,\\13,{\text{ if }}t=-{\frac {1}{2}}\,,\\\pi ,{\text{ if }}-{\frac {1}{2}}<t<e\,,\\0,{\text{ if }}e\leq t\leq 3\,,\\1,{\text{ if }}3<t\leq 4\,.\end{cases}}\,$

Exercise

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

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

Exercise

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

Exercise

Let

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

be a staircase function and let

$g\colon [c,d]\longrightarrow \mathbb {R}$

be a function. Prove that the composite ${}g\circ f$ is also a staircase function.

Exercise

Give an example of a continuous function

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

and a staircase function

$g\colon [c,d]\longrightarrow \mathbb {R}$

such that the composite ${}g\circ f$ is not a staircase function.

Exercise

Determine the definite integral

$\int _{0}^{1}tdt$

explicitly with upper and lower staircase functions.

Exercise

Determine the definite integral

$\int _{1}^{2}t^{3}dt$

explicitly with upper and lower staircase functions.

Exercise

Let ${}I=[a,b]$ be a compact interval and let

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

be a function. Consider a sequence of staircase functions ${}{\left(s_{n}\right)}_{n\in \mathbb {N} }$ such that ${}s_{n}\leq f$ and a sequence of staircase functions ${}{\left(t_{n}\right)}_{n\in \mathbb {N} }$ 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

${}\lim _{n\rightarrow \infty }\int _{a}^{b}s_{n}(x)dx=\int _{a}^{b}f(x)dx=\lim _{n\rightarrow \infty }\int _{a}^{b}t_{n}(x)dx\,.$

Exercise

Let ${}I=[a,b]$ be a compact interval. Prove that ${}f$ is Riemann-integrable if and only if there is a partition

${}a=a_{0}<a_{1}<\cdots <a_{n}=b\,$

such that the restrictions

${}f_{i}=f{|}_{[a_{i-1},a_{i}]}\,$

are Riemann-integrable.

Exercise

Let ${}I=[a,b]\subseteq \mathbb {R}$ be a compact interval and let ${}f,g\colon I\rightarrow \mathbb {R}$ be two Riemann-integrable functions. Prove the following statements.

1. If ${}m\leq f(t)\leq M$ for all ${}t\in I$, then

   ${}m(b-a)\leq \int _{a}^{b}f(t)dt=M(b-a)\,.$

2. If ${}f(t)\leq g(t)$ for all ${}t\in I$, then

   ${}\int _{a}^{b}f(t)dt\leq \int _{a}^{b}g(t)dt\,.$

3. We have

   ${}\int _{a}^{b}f(t)+g(t)dt=\int _{a}^{b}f(t)dt+\int _{a}^{b}g(t)dt\,.$

4. For ${}c\in \mathbb {R}$ we have ${}\int _{a}^{b}cf(t)dt=c\int _{a}^{b}f(t)dt$.

Exercise

Let ${}I=[a,b]$ be a compact interval and let ${}f\colon I\rightarrow \mathbb {R}$ be a Riemann-integrable function. Prove that

${}\vert {\int _{a}^{b}f(t)dt}\vert \leq \int _{a}^{b}\vert {f(t)}\vert dt\,.$

Exercise

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

Hand-in-exercises

Exercise

Let

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

be two staircase functions. Prove that ${}f+g$ is also a staircase function.

Exercise

Determine the definite integral

$\int _{a}^{b}t^{2}dt$

as a function of ${}a$ and ${}b$ explicitly with lower and upper staircase functions.

Exercise

Determine the definite integral

$\int _{-2}^{7}-t^{3}+3t^{2}-2t+5dt$

explicitly with upper and lower staircase functions.

Exercise

Prove that for the function

$]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},$

neither the lower nor the upper integral exist.

Exercise

Prove that for the function

$]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{\sqrt {x}}},$

the lower integral exists, but the upper integral does not exist.

Exercise

Let ${}I$ be a compact interval and let

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

be a monotone function. Prove that ${}f$ is Riemann-integrable.