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

Exercises

Exercise

Prove that the function

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

is differentiable but not twice differentiable.

Exercise

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

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

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

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

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

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

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

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

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

Let

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

be differentiable functions. 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

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

Let

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

be a continuously differentiable function, 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

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

Let ${}I$ be a real interval,

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

a twice continuously differentiable 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 minimum in ${}a$.
2. If ${}f^{\prime \prime }(a)<0$ holds, then ${}f$ has an isolated local maximum in ${}a$.

Exercise

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

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

Exercise

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, (local) extrema. Sketch the graph of the function.

Exercise

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

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

Let ${}f\colon I\rightarrow \mathbb {R}$ be a continuously differentiable 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.

Exercise

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

Exercise

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

Determine the limit

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

by polynomial long division.

Exercise

Determine the limit of the rational function

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

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

Exercise

Determine the limit

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

Hand-in-exercises

Exercise (5 marks)

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)

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, (local) extrema. Sketch the graph of the function.

Exercise (5 marks)

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)

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)

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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