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

Exercises

Exercise

Give an example of a continuous function

$f:\mathbb{Q}⟶ℝ,$

which takes exactly two values​​.

Exercise

Let

$f:ℝ⟶ℝ$

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

$f:ℝ⟶ℝ,x⟼f(x)=x^2+x-1,$

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

Exercise

We consider the function

$f:ℝ⟶ℝ,x⟼x^3+4x^2-x+3.$

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

$f:ℝ⟶ℝ,x⟼x^3-4x+2.$

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 $f:ℝ\setminus \{0,1\}\to ℝ$ given by

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

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

Exercise

Let $f:ℝ\to ℝ$ 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

$f:ℝ⟶ℝ,x⟼\frac{1}{x},$

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 $\frac{1}{x}\ne 0$.“

Where is the mistake in this argument?

Exercise

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

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

Exercise

Let

$f,g:[a,b]⟶ℝ$

be continuous functions with $f(a)\ge g(a)$ and $f(b)\le g(b)$. 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

$f:M⟶M$

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

$f:ℝ⟶ℝ,x⟼x^2.$

Exercise

Let $P\in ℝ[X]$ be a polynomial of degree $d\ge 1$, $P\ne X$. Show that $P$ has at most $d$ fixed points.

Exercise

Let $f:ℝ\to ℝ$ be a continuous function, and suppose that there exist $x,y\in ℝ$ with

$f(x)\le x$

and

$f(y)\ge y.$

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

$f:I⟶ℝ$

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

Exercise

Show that the function defined by

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

is a continuous, strictly increasing, bijective function

$f:ℝ⟶]-1,1[$

and that its inverse function is also continuous.

Exercise

1. Sketch the graphs of the functions

   $f:{ℝ}_{+}⟶ℝ,x⟼x-1,$

   and

   $g:{ℝ}_{+}⟶ℝ,x⟼\frac{1}{x},$

2. Determine the intersection points of these graphs.

Exercise

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

$f:ℝ⟶ℝ$

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

Exercise

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

$f:ℝ⟶ℝ$

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 $x\in ℝ$, there exists a strictly increasing continuous function

$f:ℝ⟶ℝ$

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

Exercise

Let

$f:[0,1]⟶[0,1[$

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

Exercise

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

$f:I⟶ℝ$

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

Exercise

Let

$f:I⟶ℝ$

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

$ℝ⟶ℝ,x⟼x^n,$

has an extremum at the point zero.

Hand-in-exercises

Exercise (5 marks)

Find for the function

$f:ℝ⟶ℝ,x⟼f(x)=x^3-3x+1,$

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

Exercise (3 marks)

Let $f:ℝ\to ℝ$ 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 $d{X}^3+c{X}^2+bX+a$ 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

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

with $a,b,c\ge 0$ 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 ${ℝ}_{+}$ 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)

Let

$f:[a,b]⟶[a,b]$

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

Exercise (2 marks)

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)

Determine the minimum of the function

$f:ℝ⟶ℝ,x⟼x^2+3x-5.$

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