Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 11/refcontrol

Exercises

Exercise Create referencenumber

Give an example of a continuous functionMDLD/continuous function (R)

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

which takes exactly two values​​.

Exercise Create referencenumber

Let

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

be a continuous function which takes only finitely many values. Prove that ${}f$ is constant.

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Does there exist a real number such that its third power, reduced by the fourfold of its second power, equals the square root of ${}42$?

Exercise Create referencenumber

Find a zero for the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{2}+x-1,$

in the interval ${}[0,1]$ using the interval bisection method with a maximum error of ${}1/100$.

Exercise Create referencenumber

We consider the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{3}+4x^{2}-x+3.$

Determine, starting with the interval ${}[-5,-4]$ and using the bisection method, an interval of length ${}1/8$ which contains a zero of ${}f$.

Exercise Create referencenumber

We consider the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{3}-4x+2.$

Determine, starting with the interval ${}[1,2]$ and using the bisection method, an interval of length ${}1/8$ which contains a zero of ${}f$.

Exercise Create referencenumber

We consider the mappingMDLD/mapping ${}f\colon \mathbb {R} \setminus {\{0,1\}}\rightarrow \mathbb {R}$ given by

${}f(x)={\frac {1}{x^{3}}}+{\frac {1}{(x-1)^{3}}}\,.$

Show, using the intermediate value theorem, that ${}f$ obtains every value ${}c\neq 0$ at least in two points.

Exercise Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous functionMDLD/continuous function (R) and let ${}x$ be "close“ to a zero of ${}f$. Is then ${}f(x)$ close to ${}0$?

Exercise Create referencenumber

Fridolin says:

"Something is wrong about the Intermediate value theorem. For the continuous function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},$

we have ${}f(-1)=-1$ and ${}f(1)=1$. Due to the Intermediate value theorem, there must be a zero between ${}-1$ and ${}1$, hence a number ${}x\in [-1,1]$ with ${}f(x)=0$. However, we always have ${}{\frac {1}{x}}\neq 0$.“

Where is the mistake in this argument?

Exercise Create referencenumber

Let ${}z\in \mathbb {R}$ be a real number. Show that the following properties are equivalent.

1. There exist a polynomial ${}P\in \mathbb {R} [X]$, ${}P\neq 0$, with integer coefficients and with ${}P(z)=0$.
2. There exists a polynomial ${}Q\in \mathbb {Q} [X]$, ${}Q\neq 0$, wit ${}Q(z)=0$.
3. There exists a normed polynomial ${}R\in \mathbb {Q} [X]$ with ${}R(z)=0$.

Exercise Create referencenumber

Let

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

be continuous functionsMDLD/continuous functions (R) with ${}f(a)\geq g(a)$ and ${}f(b)\leq g(b)$. Show that there is a point ${}c\in [a,b]$ with ${}f(c)=g(c)$.

The next exercises use following terms.

Let ${}M$ be a set and let

$f\colon M\longrightarrow M$

be a mapping. An element ${}x\in M$ such that ${}f(x)=x$ is called a fixed point

of ${}f$.

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Determine the fixed pointsMDLD/fixed points of the mappingMDLD/mapping

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}.$

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Let ${}P\in \mathbb {R} [X]$ be a polynomial of degree ${}d\geq 1$, ${}P\neq X$. Show that ${}P$ has at most ${}d$ fixed points.MDLD/fixed points

Exercise Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous function,MDLD/continuous function (R) and suppose that there exist ${}x,y\in \mathbb {R}$ with

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

and

${}f(y)\geq y\,.$

Show that ${}f$ has a fixed point.MDLD/fixed point

Exercise Create referencenumber

Show that the image of a closed interval under a continuous function is not necessarily closed.

Exercise Create referencenumber

Show that the image of an open interval under a continuous function is not necessarily open.

Exercise Create referencenumber

Show that the image of a bounded interval under a continuous function is not necessarily bounded.

Exercise Create referencenumber

Let ${}I$ be a real interval and let

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

denote a continuous injective function. Show that ${}f$ is strictly increasingMDLD/strictly increasing (R) or strictly decreasing.

Exercise Create referencenumber

Show that the function defined by

${}f(x)={\frac {x}{\vert {x}\vert +1}}\,$

is a continuous, strictly increasing, bijective function

$f\colon \mathbb {R} \longrightarrow {]{-1},1[}$

and that its inverse function is also continuous.

Exercise Create referencenumber

1. Sketch the graphs of the functions

   $f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto x-1,$

   and

   $g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},$

2. Determine the intersection points of these graphs.

Exercise Create referencenumber

Show that for every real number ${}a\in \mathbb {R}$, there exists a continuous functionMDLD/continuous function (R)

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

such that ${}a$ is the only zero of ${}f$.

Exercise Create referencenumber

Show that for every real number ${}x\in \mathbb {R}$, there exists a continuous functionMDLD/continuous function (R)

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

such that ${}x$ is the only zero of ${}f$ and such that for every rational number ${}q$, also ${}f(q)$ is rational.

Exercise Create referencenumber

Show that for every real number ${}x\in \mathbb {R}$, there exists a strictly increasingMDLD/strictly increasing (R) continuous functionMDLD/continuous function (R)

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

such that ${}x$ is the only zero of ${}f$ and such that for every rational number ${}q$, also ${}f(q)$ is rational.

Exercise Create referencenumber

Let

$f\colon [0,1]\longrightarrow [0,1[$

be a continuous function.MDLD/continuous function (R) Show that ${}f$ is not surjective.MDLD/surjective

Exercise Create referencenumber

Give an example of a bounded interval ${}I\subseteq \mathbb {R}$ and a continuous function

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

such that the image of ${}f$ is bounded, but the function admits no maximum.

Exercise Create referencenumber

Let

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

be a continuous function defined over a real interval. The function has at points ${}x_{1},x_{2}\in I$, ${}x_{1}<x_{2}$, local maxima. Prove that the function has between ${}x_{1}$ and ${}x_{2}$ has at least one local minimum.

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Determine directly, for which ${}n\in \mathbb {N}$ the power function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

has an extremum at the point zero.

Hand-in-exercises

Exercise (5 marks) Create referencenumber

Find for the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{3}-3x+1,$

a zero in the interval ${}[0,1]$ using the interval bisection method, with a maximum error of ${}1/200$.

Exercise (3 marks) Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ denote a continuous functionMDLD/continuous function (R) having the property that the imageMDLD/image of ${}f$ is unbounded in both directions. Show that ${}f$ is surjective.MDLD/surjective

Exercise (4 marks) Create referencenumber

Show that a real polynomial of odd degree has at least one real zero.

Exercise (5 marks) Create referencenumber

Write a computer-program (in pseudocode) which for a polynomial ${}dX^{3}+cX^{2}+bX+a$ of degree ${}3$ computes a zero within an accuracy of a given number ${}e>0$ berechnet.

• The computer has as many memory units as needed, which can contain nonnegative real numbers.

• It can write the content of a memory unit into another memory unit.

• It can halve the content of a memory unit and write the result into another memory unit.

• It can add the content of two memory units and write the result into another memory unit.

• It can multiply the content of two memory units and write the result into another memory unit.

• It can compare the content of memory units and can, depending on the outcome, switch to a certain program line.

• It can print contents of memory units and it can print given texts.

• There is a stop command.

The initial configuration is

$(a,b,c,d,e,1,0,0,\ldots )$

with ${}a,b,c\geq 0$ and ${}d,e>0$ (hence, the coefficients of the polynomial, the accuracy ${}e$ and ${}1$ are in the first memory units). The program shall print a sentence telling the bounds of an interval for a zero with the wished-for accuracy and stop.
Caution: The main difficulty is here that the polynomials do not have any zero on ${}\mathbb {R} _{+}$ due to our condition. Hence we have to find a zero in the negative real numbers. However, the memory units do not accept negative numbers. Therefore we have to emulate/simulate negative numbers by nonnegative numbers.

Exercise (4 marks) Create referencenumber

Let

$f\colon [a,b]\longrightarrow [a,b]$

be a continuous function from the interval ${}[a,b]$ into itself. Prove that ${}f$ has a fixed point.

Exercise (2 marks) Create referencenumber

Determine the limit of the sequence

$x_{n}={\sqrt[{3}]{\frac {27n^{3}+13n^{2}+n}{8n^{3}-7n+10}}},\,n\in \mathbb {N} .$

Exercise (2 marks) Create referencenumber

Determine the minimum of the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}+3x-5.$

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