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}}\,

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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).

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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.

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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.

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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.

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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.

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Determine the definite integral

\int _{0}^{1}tdt

explicitly with upper and lower staircase functions.

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Determine the definite integral

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

explicitly with upper and lower staircase functions.

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Determine the definite integral

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

explicitly with upper and lower staircase functions.

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Show (without using primitive functions)

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

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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

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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\,.

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Let ${}I$ be a bounded intervalMDLD/bounded interval (R) and let ${}f\colon I\rightarrow \mathbb {R}$ denote a continuous function MDLD/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 integral MDLD/lower integral) exists and coincides with the supremum first mentioned.

===Exercise Exercise 18.14

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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

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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 ${}\int _{a}^{b}cf(t)dt=c\int _{a}^{b}f(t)dt$ .

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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

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

===Exercise Exercise 18.18

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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.