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

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 division (see Example 20.11).

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 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 points)

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?