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

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

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

be two staircase functions. Show that also

1. ${}f+g$,
2. ${}f\cdot g$,
3. ${}{\max {\left(f,g\right)}}$,
4. ${}{\min {\left(f,g\right)}}$,

are 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

Determine the definite integral

$\int _{3}^{5}7t^{3}+4tdt$

explicitly with upper and lower staircase functions.

Exercise

Show (without using primitive functions)

${}\int _{0}^{1}e^{x}dx=e-1\,.$

Exercise

We consider the function

$[1,2]\longrightarrow \mathbb {R} ,t\longmapsto g(t)={\frac {1}{t}}.$

1. 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 ${}x$.
2. 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

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 ${}I$ be a bounded interval and let ${}f\colon I\rightarrow \mathbb {R}$ denote a continuous function which is bounded from below. Suppose that the supremum over all 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 integral) exists and coincides with the supremum first mentioned.

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 ${}I=[a,b]$ denote a compact interval and let ${}f,g\colon I\rightarrow \mathbb {R}$ denote Riemann-integrable functions. Show that also ${}{\max {\left(f,g\right)}}$ is Riemann-integrable.

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.

The Christmas exercise for the whole family

Exercise

Which construction principle is behind the sequence

$1,\,11,\,21,\,1211,\,111221,\,312211,\,...?$

(Some people claim that this exercise is for primary school children very easy and for mathematicians quite hard.)

Hand-in-exercises

Exercise (2 marks)

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)

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)

Determine the definite integral

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

explicitly with upper and lower staircase functions.

Exercise (3 marks)

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)

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)

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)

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).

1. Is ${}f$ increasing?
2. Is ${}f$ surjective?
3. Is ${}f$ injective?
4. Does ${}f$ have a fixed point?

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