Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 15/refcontrol

Exercises

Exercise Create referencenumber

Prove that the function

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

is differentiable but not twice differentiable.

Exercise Create referencenumber

Let ${}P\in \mathbb {R} [X]$ be a polynomial, ${}a\in \mathbb {R}$ and ${}n\in \mathbb {N}$. Prove that ${}P$ is a multiple of ${}(X-a)^{n}$ if and only if ${}a$ is a zero of all the derivatives ${}P,P^{\prime },P^{\prime \prime },\ldots ,P^{(n-1)}$.

Exercise Create referencenumber

Consider the function

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

defined by

${}f(x)={\begin{cases}x-\lfloor x\rfloor ,{\text{ if }}\lfloor x\rfloor {\text{ is even}},\\\lfloor x\rfloor -x+1,{\text{ if }}\lfloor x\rfloor {\text{ is odd}}\,.\end{cases}}\,$

Examine ${}f$ in terms of continuity, differentiability and extremes.

Exercise Create referencenumber

Does there exist a real number, which, in its fourth power, reduced by the double of its third power, equals the negative of the square root of ${}42$?

Exercise Create referencenumber

Determine local and global extrema of the function

$f\colon [-2,5]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=2x^{3}-5x^{2}+4x-1.$

Exercise Create referencenumber

Determine local and global extrema of the function

$f\colon [-4,4]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=3x^{3}-7x^{2}+6x-3.$

Exercise Create referencenumber

Consider the function

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

Find the point ${}a\in [-3,3]$ such that the tangent of the function at ${}a$ is parallel to the secant between ${}-3$ and ${}3$.

Exercise Create referencenumber

The city ${}S=(0,0)$ shall be connected by rails with the two cities ${}T=(a,b)$ and ${}U=(a,-b)$ with ${}a\geq 0$, ${}b>0$. The rails shall run along the ${}x$-axis until it ramifies into the two directions. Determine the ramification point, with as few rails as possible.

Exercise Create referencenumber

Next to a rectilinear river we want to fence a rectangular area of ${}1000m^{2}$, one side of the area is the river itself. For the other three sides, we need a fence. Which is the minimal length of the fence we need?

Exercise Create referencenumber

We consider the function

${}F(x)=x^{4}-x^{3}\,.$

1. Determine the zeroes of this function.
2. Determine on which intervals the function is positive or negative is.
3. Determine the extrema of this function.

Exercise Create referencenumber

Let

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

be differentiable functions.MDLD/differentiable functions (R) Let ${}a\in \mathbb {R}$ be a point, and suppose that

$f(a)=g(a){\text{ and }}f'(x)=g'(x){\text{ for all }}x.$

Show that

$f(x)=g(x){\text{ for all }}x{\text{ gilt}}.$

Exercise Create referencenumber

Let

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

be two differentiable functions. Let ${}a\in \mathbb {R}$. Suppose we have that

$f(a)\geq g(a){\text{ and }}f'(x)\geq g'(x){\text{ for all }}x\geq a.$

Prove that

$f(x)\geq g(x){\text{ for all }}x\geq a.$

Exercise Create referencenumber

Let

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

be a continuously differentiable function,MDLD/continuously differentiable function (R) and suppose that its graph intersects the diagonal in at least two points ${}P\neq Q$. Show that the graph of the derivative ${}f'$ has an intersection point with the line given by ${}y=1$.

===Exercise Exercise 15.14

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Prove that a real polynomial function

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

of degree ${}d\geq 1$ has at most ${}d-1$ extrema, and moreover the real numbers can be divided into at most ${}d$ sections, where ${}f$ is strictly increasing or strictly decreasing.

===Exercise Exercise 15.15

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Let ${}I$ be a real interval,MDLD/real interval

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

a twice continuously differentiableMDLD/continuously differentiable (R) function,MDLD/function and let ${}a\in I$ be an inner point of the interval. Suppose that ${}f'(a)=0$. Show the following statements.

1. If ${}f^{\prime \prime }(a)>0$ holds, then ${}f$ has an isolated local minimumMDLD/isolated local minimum (R) in ${}a$.
2. If ${}f^{\prime \prime }(a)<0$ holds, then ${}f$ has an isolated local maximumMDLD/isolated local maximum (R) in ${}a$.

Exercise Create referencenumber

Let ${}D\subseteq \mathbb {R}$ and

$F\colon D\longrightarrow \mathbb {R}$

be a rational function. Prove that ${}F$ is a polynomial if and only if there is a higher derivative such that ${}F^{(n)}=0$.

===Exercise Exercise 15.17

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Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be an ${}n$-fold continuously differentiableMDLD/continuously differentiable (R) function with the property that its ${}n$-th derivative is everywhere positive. Show that ${}f$ has at most ${}n$ zeroes.

Exercise Create referencenumber

Discuss the following properties of the rational function

$f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {2x-3}{5x^{2}-3x+4}},$

domain, zeros, growth behavior, (localMDLD/local (R extremum)) extrema. Sketch the graph of the function.

Exercise Create referencenumber

Consider

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

a) Prove that the function ${}f$ has in the real interval ${}[0,1]$ exactly one zero.

b) Compute the first decimal digit in the decimal system of this zero point.

c) Find a rational number ${}q\in [0,1]$ such that ${}\vert {f(q)}\vert \leq {\frac {1}{10}}$.

Exercise Create referencenumber

Show that the function

$f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto e^{-{\frac {1}{x}}}\cdot \ln x$

is bounded from below.

Exercise Create referencenumber

Let ${}f\colon I\rightarrow \mathbb {R}$ be a continuously differentiableMDLD/continuously differentiable (R) function (defined on an open interval), and let ${}a\in I$ be a point with ${}f'(a)\neq 0$. Show that there exist open intervals ${}J\subseteq I$ with ${}a\in J$ and ${}J'\subseteq \mathbb {R}$, such that the restricted function ${}f\colon J\rightarrow J'$ is bijective.MDLD/bijective

Exercise Create referencenumber

Prove the mean value theorem out of the second mean value theorem.

Exercise Create referencenumber

Determine the limit of

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

at the point ${}x=1$, and specifically

a) by polynomial division.

b) by the rule of l'Hospital.

Exercise Create referencenumber

Determine the limit

$\operatorname {lim} _{x\rightarrow 2}\,{\frac {3x^{2}-5x-2}{x^{3}-4x^{2}+x+6}}$

by polynomial long division.

Exercise Create referencenumber

Determine the limit of the rational function

${\frac {x^{3}-2x^{2}+x+4}{x^{2}+x}}$

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

Exercise Create referencenumber

Determine the limitMDLD/limit (real function)

$\operatorname {lim} _{x\rightarrow 1}\,{\frac {\sqrt {1-x}}{\sqrt[{3}]{1-x^{2}}}}.$

Hand-in-exercises

Exercise (5 marks) Create referencenumber

From a sheet of paper with side lengths of ${}20$ cm and ${}30$ cm we want to realize a box (without cover) with the greatest possible volume. We do it in this way. We remove from each corner a square of the same size, then we lift up the sides and we glue them. Which box height do we need to realize the maximum volume?

Exercise (4 marks) Create referencenumber

Discuss the following properties of the rational function

$f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {3x^{2}-2x+1}{x-4}},$

domain, zeros, growth behavior, (localMDLD/local (R extremum)) extrema. Sketch the graph of the function.

Exercise (5 marks) Create referencenumber

Prove that a non-constant rational function of the shape

${}f(x)={\frac {ax+b}{cx+d}}\,$

(with ${}a,b,c,d\in \mathbb {R}$, ${}a,c\neq 0$,) has no local extrema.

Exercise (4 marks) Create referencenumber

Let

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

be a polynomial function of degree ${}d\geq 1$. Let ${}m$ be the number of local maxima of ${}f$ and ${}n$ the number of local minima of ${}f$. Prove that if ${}d$ is odd then ${}m=n$ and that if ${}d$ is even then

${}\vert {m-n}\vert =1\,.$

Exercise (3 marks) Create referencenumber

Determine the limit of the rational function

${\frac {x^{4}+2x^{3}-3x^{2}-4x+4}{2x^{3}-x^{2}-4x+3}}$

at the point ${}a=1$.

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