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

Exercises

Exercise

Show that a linear function

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

is continuous.

Exercise

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 continuous 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

Prove that the function

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

is continuous.

Exercise

Prove that the function

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

is continuous.

Exercise

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?

Exercise

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

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

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

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

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous function. 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.

Exercise

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

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

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

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

be a function. Show that ${}f$ is continuous.

Exercise

Show that there exists a continuous function

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

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

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

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

Determine the limit of the sequence

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

Exercise

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

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

Exercise

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.

Exercise

Give an example for a continuous function ${}f\colon \mathbb {R} _{\geq 0}\rightarrow \mathbb {R} _{\geq 0}$ and an absolutely convergent 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

Let ${}a\in \mathbb {R}$ and let ${}f,g,h\colon \mathbb {R} \rightarrow \mathbb {R}$ be functions. Suppose that ${}g$ and ${}h$ are continuous 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

Determine the limit of the rational function

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

in the point ${}a=1$.

Exercise

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

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

be a 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 converges 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

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 converges to ${}b$.
2. The function

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

   given by

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

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

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)

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)

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.

Exercise (3 marks)

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)

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)

Determine the limit of the rational function

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

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

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