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

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

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 $f:[a,b]\to ℝ$ which assumes only finitely many values, but is not a staircase function.

Exercise

Let

$f:[a,b]⟶[c,d]$

be a staircase function and let

$g:[c,d]⟶ℝ$

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

Exercise

Give an example of a continuous function

$f:[a,b]⟶[c,d]$

and a staircase function

$g:[c,d]⟶ℝ$

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

$f:I⟶ℝ$

be a function. Consider a sequence of staircase functions ${\left(s_n\right)}_{n\in \mathbb{N}}$ such that $s_n\le f$ and a sequence of staircase functions ${\left(t_n\right)}_{n\in \mathbb{N}}$ such that $t_n\ge f$. Assume that the two Riemann sums corresponding to the sequences converge and that their limits coincide. Prove that $f$ is Riemann-integrable and that

$\underset{n\to \mathrm{\infty }}{lim}{\int }_a^b{s}_n(x)dx={\int }_a^{b}f(x)dx=\underset{n\to \mathrm{\infty }}{lim}{\int }_a^b{t}_n(x)dx.$

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 $I=[a,b]\subseteq ℝ$ be a compact interval and let $f,g:I\to ℝ$ be two Riemann-integrable functions. Prove the following statements.

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

   $m(b-a)\le {\int }_a^{b}f(t)dt=M(b-a).$

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

   ${\int }_a^{b}f(t)dt\le {\int }_a^{b}g(t)dt.$

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 $c\in ℝ$ 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 $f:I\to ℝ$ be a Riemann-integrable function. Prove that

$|{\int }_a^{b}f(t)dt|\le {\int }_a^b|f(t)|dt.$

Exercise

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

Hand-in-exercises

Exercise

Let

$f,g:[a,b]⟶ℝ$

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

$]0,1]⟶ℝ,x⟼\frac{1}{x},$

neither the lower nor the upper integral exist.

Exercise

Prove that for the function

$]0,1]⟶ℝ,x⟼\frac{1}{\sqrt{x}},$

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

Exercise

Let $I$ be a compact interval and let

$f:I⟶ℝ$

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