Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 10/refcontrol

Exercises

Exercise Create referencenumber

Show that a linear functionMDLD/linear function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax,$

is continuous.

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Let ${}D\subseteq \mathbb {R}$ be a subset, ${}f\colon D\rightarrow \mathbb {R}$ a function and ${}a\in D$ a point. Show that the following properties are equivalent.

1. ${}f$ is continuousMDLD/continuous (R) in ${}a$.
2. For every ${}n\in \mathbb {N} _{+}$, there exists some ${}m\in \mathbb {N} _{+}$ such that for

   ${}\vert {x-a}\vert \leq {\frac {1}{m}}\,$

   the estimate

   ${}\vert {f(x)-f(a)}\vert \leq {\frac {1}{n}}\,$

   holds.

3. For every ${}s\in \mathbb {N}$, there exists some ${}r\in \mathbb {N}$ such that for

   ${}\vert {x-a}\vert \leq {\frac {1}{10^{r}}}\,$

   the estimate

   ${}\vert {f(x)-f(a)}\vert \leq {\frac {1}{10^{s}}}\,$

   holds.

Exercise Create referencenumber

Prove that the function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,$

is continuous.

===Exercise Exercise 10.4

change===

Prove that the function

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto {\sqrt {x}},$

is continuous.

Exercise Create referencenumber

Farmer Ernst wants to make a square-shaped field for melons. The field shall be ${}100$ square meters in size, but he thinks that a size between ${}99$ and ${}101$ square meters is acceptable. What error is allowed for the side lengths in order to stick within this tolerance?

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Let

${}f(x)=2x^{3}-4x+5\,.$

Show that for all ${}x\in \mathbb {R}$, the following relation holds: If

${}\vert {x-3}\vert \leq {\frac {1}{800}}\,,$

then

${}\vert {f(x)-f(3)}\vert \leq {\frac {1}{10}}\,.$

Exercise Create referencenumber

For the function

${}f(x)=2x^{3}-4x^{2}+x-6\,$

and the point ${}a=1$, determine for ${}\epsilon ={\frac {1}{10}}$ an explicit ${}\delta >0$ such that

${}d(x,a)\leq \delta \,$

implies the estimate

${}d(f(x),f(a))\leq \epsilon \,.$

Exercise Create referencenumber

Let ${}T\subseteq \mathbb {R}$ be a subset and let

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

be a continuous function. Let ${}x\in T$ be a point such that ${}f(x)>0$. Prove that ${}f(y)>0$ for all ${}y$ in a non-empty open interval ${}]x-a,x+a[$.

Exercise Create referencenumber

Let ${}a\in \mathbb {R}$ and let

$f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$

be continuous functions with

${}f(a)>g(a)\,.$

Show that there exists some ${}\delta >0$ such that

${}f(x)>g(x)\,$

holds for all ${}x\in [a-\delta ,a+\delta ]$.

Exercise Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous function.MDLD/continuous function (R) Show the following statements.

1. The function ${}f$ is uniquely determined by its values on ${}\mathbb {Q}$.
2. The value ${}f(a)$ is determined by the values ${}f(x)$, ${}x\neq a$.
3. If for all ${}x<a$, the estimate

   ${}f(x)\leq c\,$

   holds, then also

   ${}f(a)\leq c\,$

   holds.

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Let ${}a<b<c$ be real numbers and let

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

and

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

be continuous functions such that ${}f(b)=g(b)$. Prove that the function

$h\colon [a,c]\longrightarrow \mathbb {R}$

such that

$h(t)=f(t){\text{ for }}t\leq b{\text{ and }}h(t)=g(t){\text{ for }}t>b$

is also continuous.

Exercise Create referencenumber

Let

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

be a continuous function. Show that there exists a continuous extension

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

of ${}f$.

Exercise Create referencenumber

Let ${}T\subseteq \mathbb {R}$ be a finite subset and let

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

be a function.MDLD/function Show that ${}f$ is continuous.MDLD/continuous (R)

Exercise Create referencenumber

Show that there exists a continuous functionMDLD/continuous function (R)

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

such that ${}f$ obtains on every interval of the form ${}[0,\delta ]$ with ${}\delta >0$ positive as well as negative values.

Is it possible to draw such a function? See also Exercise 16.25 .

Exercise Create referencenumber

Compute the limit of the sequence

${}x_{n}=5\left({\frac {2n+1}{n}}\right)^{3}-4\left({\frac {2n+1}{n}}\right)^{2}+2\left({\frac {2n+1}{n}}\right)-3\,$

for ${}n\rightarrow \infty$.

Exercise Create referencenumber

Prove that the function

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

defined by

${}f(x)={\begin{cases}x,\,{\text{ if }}x\in \mathbb {Q} \,,\\0,\,{\text{ otherwise}}\,,\end{cases}}\,$

is only at the zero point ${}0$ continuous.

Exercise Create referencenumber

Determine the limit of the sequence

$x_{n}={\sqrt {\frac {7n^{2}-4}{3n^{2}-5n+2}}},\,n\in \mathbb {N} .$

Exercise Create referencenumber

The sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ is recursively defined by ${}x_{0}=1$ and

${}x_{n+1}={\sqrt {x_{n}+1}}\,.$

Show that this sequence converges and determine its limit.

Exercise Create referencenumber

Prove directly the computing rules from Lemma 10.6 (without referring to the sequence crtierion).

Exercise Create referencenumber

Show that the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto {\frac {2x^{7}-3x\vert {6x^{3}-11}\vert }{\vert {3x-5}\vert +\vert {4x^{3}-5x+1}\vert }},$

is continuous.MDLD/continuous (R)

Exercise Create referencenumber

Give an example for a continuous functionMDLD/continuous function (R) ${}f\colon \mathbb {R} _{\geq 0}\rightarrow \mathbb {R} _{\geq 0}$ and an absolutely convergentMDLD/absolutely convergent (R) real seriesMDLD/real series ${}\sum _{k=0}^{\infty }a_{k}$ with ${}a_{k}\geq 0$ such that the series ${}\sum _{k=0}^{\infty }f(a_{k})$ does not converge.

Exercise Create referencenumber

Let ${}a\in \mathbb {R}$ and let ${}f,g,h\colon \mathbb {R} \rightarrow \mathbb {R}$ be functions. Suppose that ${}g$ and ${}h$ are continuousMDLD/continuous (R) in ${}a$, that ${}g(a)=f(a)=h(a)$ holds and suppose further that ${}g(x)\leq f(x)\leq h(x)$ holds for all ${}x\in \mathbb {R}$. Show that also ${}f$ is continuous in ${}a$.

Exercise Create referencenumber

Determine the limitMDLD/limit (Function R) of the rational functionMDLD/rational function (R)

${\frac {x-1}{x^{2}-1}}$

in the point ${}a=1$.

===Exercise Exercise 10.24

change===

Let ${}T\subseteq \mathbb {R}$ denote a subset and ${}a\in \mathbb {R}$ a point. Let

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

be a functionMDLD/function and ${}b\in \mathbb {R}$ a point. Show that the following statements are equivalent.

1. We have

   ${}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$

2. For every sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ in ${}T$ which convergesMDLD/converges (R) to ${}a$, also the image sequence ${}{\left(f(x_{n})\right)}_{n\in \mathbb {N} }$ converges to ${}b$.

Hint: This is proved similarly to the sequence criterion for continuity.

Exercise Create referencenumber

Let

${}T={\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\subseteq \mathbb {R} \,$

be the set of the unit fractions and let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote a real sequence. Let ${}b\in \mathbb {R}$ and ${}D=T\cup \{0\}$. Show that the following properties are equivalent.

1. The sequence convergesMDLD/converges (R) to ${}b$.
2. The function

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

   given by

   ${}f{\left({\frac {1}{n}}\right)}=x_{n}\,$

   has a limitMDLD/limit (real function) ${}\operatorname {lim} _{x\rightarrow 0}\,f(x)=b$.

3. The function

   ${\tilde {f}}\colon D\longrightarrow \mathbb {R}$

   given by

   ${}{\tilde {f}}{\left({\frac {1}{n}}\right)}=x_{n}\,$

   and ${}{\tilde {f}}(0)=b$ is continuous.

Hand-in-exercises

Exercise (3 marks) Create referencenumber

For the function

${}f(x)=x^{3}+5x^{2}-3x+2\,$

and the point ${}a=3$, determine for ${}\epsilon ={\frac {1}{100}}$ an explicite ${}\delta >0$ such that

${}d(x,a)\leq \delta \,$

implies the estimate

${}d(f(x),f(a))\leq \epsilon \,.$

Exercise (2 marks) Create referencenumber

We consider the function

${}f(x)={\begin{cases}1{\text{ for }}x\leq -1\,,\\x^{2}{\text{ for }}-1<x<2\,,\\-2x+7{\text{ for }}x\geq 2\,.\end{cases}}\,$

Determine the points ${}x\in \mathbb {R}$ where ${}f$ is continuous.MDLD/continuous (R)

Exercise (3 marks) Create referencenumber

Prove that the function ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ defined by

${}f(x)={\begin{cases}1,{\text{ if }}x\in \mathbb {Q} \,,\\0\,{\text{ otherwise}}\,,\end{cases}}\,$

is for no point ${}x\in \mathbb {R}$ continuous.

Exercise (3 marks) Create referencenumber

Compute the limit of the sequence

${}b_{n}=2a_{n}^{4}-6a_{n}^{3}+a_{n}^{2}-5a_{n}+3\,,$

where

${}a_{n}={\frac {3n^{3}-5n^{2}+7}{4n^{3}+2n-1}}\,.$

Exercise (4 marks) Create referencenumber

Determine the limitMDLD/limit (Function R) of the rational functionMDLD/rational function (R)

${\frac {2x^{3}+3x^{2}-1}{x^{3}-x^{2}+x+3}}$

in the point ${}a=-1$.

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