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

Exercises

Exercise

Let

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

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

Exercise

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

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

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

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

We consider the 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

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

Exercise

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

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

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

Exercise

Let

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

be continuous functions 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

.

Exercise

Determine the fixed points of the mapping

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

Exercise

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.

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

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

f\colon I\longrightarrow \mathbb {R}

denote a continuous injective function. Show that ${}f$ is strictly increasing or strictly decreasing.

Exercise

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

Sketch the graphs of the functions
$f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto x-1,$

and

$g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},$
Determine the intersection points of these graphs.

Exercise

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

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

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

Exercise

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

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

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

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

Let

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

be a continuous function. Show that ${}f$ is not surjective.

Exercise

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

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.

Exercise

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)

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)

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

Exercise (4 marks)

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

Exercise (5 marks)

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.