# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 20/en

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Warm-up-exercises

### Exercise

Prove that the function

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

is differentiable but not twice differentiable.

### Exercise

Consider the function

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

defined by

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

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

### Exercise

Determine local and global extrema of the function

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

### Exercise

Determine local and global extrema of the function

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

### Exercise

Consider the function

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

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

### Exercise

Prove that a real polynomial function

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

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

### Exercise

Determine the limit

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

by polynomial division (see Example 20.11).

### Exercise

Determine the limit of the rational function

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

at point ${\displaystyle {}a=-1}$.

### Exercise

Next to a rectilinear river we want to fence a rectangular area of ${\displaystyle {}1000m^{2}}$, 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

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

domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.

### Exercise

Consider

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

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

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

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

### Exercise

Determine the limit of

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

${\displaystyle {}x=1}$, and specifically

a) by polynomial division.

b) by the rule of l'Hospital.

### Exercise

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

Hand-in-exercises

### Exercise (5 points)

From a sheet of paper with side lengths of ${\displaystyle {}20}$ cm and ${\displaystyle {}30}$ 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 points)

Discuss the following properties of the rational function

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

domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.

### Exercise (5 points)

Prove that a non-costant rational function of the shape

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

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

### Exercise (3 points)

Determine the limit of the rational function

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

at point ${\displaystyle {}a=1}$.

### Exercise (5 points)

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

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

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