# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 18/refcontrol

    Exercises

### Exercise Create referencenumber

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 Create referencenumber

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 Create referencenumber

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 Create referencenumber

Let
    $f,g\colon [a,b]\longrightarrow \mathbb {R}$
be two staircase functions.MDLD/staircase functions Show that also
* $f+g$,
* $f\cdot g$,
* ${\max {\left(f,g\right)}}$,
* ${\min {\left(f,g\right)}}$,
are staircase function.

### Exercise Create referencenumber

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 Create referencenumber

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 Create referencenumber

Determine the definite integral
    $\int _{0}^{1}tdt$
explicitly with upper and lower staircase functions.

### Exercise Create referencenumber

Determine the definite integral
    $\int _{1}^{2}t^{3}dt$
explicitly with upper and lower staircase functions.

### Exercise Create referencenumber

Determine the definite integral
    $\int _{3}^{5}7t^{3}+4tdt$
explicitly with upper and lower staircase functions.

### Exercise Create referencenumber

Show (without using primitive functions)
    $\int _{0}^{1}e^{x}dx=e-1\,.$

### Exercise Create referencenumber

We consider the function
    $[1,2]\longrightarrow \mathbb {R} ,t\longmapsto g(t)={\frac {1}{t}}.$
* Determine the area of the maximal lower staircase function for $g$ for an interval partitions of the form $1\leq x\leq 2$ in dependence on $x$.
* Determine $x$ between $1$ and $2$, for which the area of the lower maximal staircase function for $g$ for an interval partition given by $1\leq x\leq 2$ is maximal. What is the value of the corresponding area?
In the situation of the preceding exercise, a natural question is how the best staircase function looks like if we allow a finer subdivision, say with two points in between. We will deal with this question in the second semester, see exercise *****.
===Exercise Exercise 18.12
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 Create referencenumber

Let $I$ be a bounded intervalMDLD/bounded interval (R) and let $f\colon I\rightarrow \mathbb {R}$ denote a continuous functionMDLD/continuous function (R) which is bounded from below. Suppose that the
supremumMDLD/supremum (R) over all staircase integralsMDLD/staircase integrals for the equidistant lower staircase functions exists. Show that then the supremum for all staircase integrals for lower staircase functions (that is, the lower integralMDLD/lower integral) exists and coincides with the supremum first mentioned.
===Exercise Exercise 18.14
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 Exercise 18.14
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.
* If $m\leq f(t)\leq M$ for all $t\in I$, then
    $m(b-a)\leq \int _{a}^{b}f(t)dt=M(b-a)\,.$
* If $f(t)\leq g(t)$ for all $t\in I$, then
    $\int _{a}^{b}f(t)dt\leq \int _{a}^{b}g(t)dt\,.$
* We have
    $\int _{a}^{b}f(t)+g(t)dt=\int _{a}^{b}f(t)dt+\int _{a}^{b}g(t)dt\,.$
* For $c\in \mathbb {R}$ we have $\int _{a}^{b}cf(t)dt=c\int _{a}^{b}f(t)dt$.

### Exercise Create referencenumber

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 Exercise 18.17
Let $I=[a,b]$ denote a compact intervalMDLD/compact interval (R) and let $f,g\colon I\rightarrow \mathbb {R}$ denote Riemann-integrableMDLD/Riemann-integrable (compact) functions.MDLD/functions Show that also ${\max {\left(f,g\right)}}$ is Riemann-integrable.
===Exercise Exercise 18.18
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.
    The Christmas exercise for the whole family
===Exercise Exercise 18.19
Which construction principle is behind the sequence
    $1,\,11,\,21,\,1211,\,111221,\,312211,\,...?$
    Hand-in-exercises

### Exercise (2 marks) Create referencenumber

Let
    $f,g\colon [a,b]\longrightarrow \mathbb {R}$
be two staircase functions. Prove that $f+g$ is also a staircase function.

### Exercise (4 marks) Create referencenumber

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 (4 marks) Create referencenumber

Determine the definite integral
    $\int _{-2}^{7}-t^{3}+3t^{2}-2t+5dt$
explicitly with upper and lower staircase functions.

### Exercise (3 marks) Create referencenumber

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

### Exercise (6 marks) Create referencenumber

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. Hint: Use Exercise 9.7 .

### Exercise (5 marks) Create referencenumber

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

### Exercise (4 marks) Create referencenumber

We consider the mapping
    $f\colon \mathbb {N} \longrightarrow \mathbb {N} ,$
which is described in Exercise 18.19
(the natural numbers are given as finite sequences in the decimal system).
* Is $f$ increasing?
* Is $f$ surjective?
* Is $f$ injective?
* Does $f$ have a fixed point?MDLD/fixed point