Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 20

Warm-up-exercises

Exercise

Prove that the function

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

is differentiable but not twice differentiable.

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

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

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

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

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

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

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

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 ${}P,P^{\prime },P^{\prime \prime },\ldots ,P^{(n-1)}$.

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 (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 ${}a=1$.

Exercise (5 marks)

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 ${}F^{(n)}=0$.