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

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\zwischenueberschrift{Warm-up-exercises}

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Prove that the function \maabbeledisp {} {\R} {\R } {x} {x \betrag { x } } {,} is differentiable but not twice differentiable.

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\inputexercise
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Consider the function \maabbdisp {f} {\R} {\R } {,} defined by
\mathdisp {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.

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\inputexercise
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Determine local and global extrema of the function \maabbeledisp {f} {[-2,5]} {\R } {x} { f ( x ) = 2x^3-5x^2+4x-1 } {.}

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\inputexercise
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Determine local and global extrema of the function \maabbeledisp {f} {[-4,4]} {\R } {x} { f ( x ) = 3x^3-7x^2+6x-3 } {.}

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\inputexercise
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Consider the function \maabbeledisp {f} {\R} {\R } {x} {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 \mathkor {} {-3} {and} {3} {.}

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\inputexercise
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Prove that a real polynomial function \maabbdisp {f} {\R} {\R } {} of degree $d \geq 1$ has at most
\mathl{d-1}{} extrema, and moreover the real numbers can be divided into at most $d$ sections, where $f$ is strictly increasing or strictly decreasing.

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\inputexercise
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Determine the limit
\mathdisp {\operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 3x^2-5x-2}{x^3-4x^2+x+6}} { }
by polynomial division \zusatzklammer {see Example 20.11} {} {.}

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\inputexercise
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Determine the limit of the rational function
\mathdisp {\frac{ x^3-2x^2+x+4}{ x^2+x }} { }
at point $a=-1$.

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\inputexercise
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Next to a rectilinear river we want to fence a rectangular area of $1000 m^2$, 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?

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\inputexercise
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Discuss the following properties of the rational function \maabbeledisp {f} {D} {\R } {x} {f(x) = \frac{ 2x-3 }{ 5x^2-3x+4 } } {,} domain, zeros, growth behavior, \zusatzklammer {\definitionsverweis {local}{}{}} {} {} extrema. Sketch the graph of the function.

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\inputexercise
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Consider
\mathdisp {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 $\betrag { f(q) } \leq \frac{1}{10}$.

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\inputexercise
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Determine the limit of
\mathdisp {\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.

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

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\zwischenueberschrift{Hand-in-exercises}

\inputexercise
{5}
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From a sheet of paper with side lengths of $20$ cm and
\mathl{30}{} cm we want to realize a box \zusatzklammer {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?

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\inputexercise
{4}
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Discuss the following properties of the rational function \maabbeledisp {f} {D} {\R } {x} {f(x) = \frac{ 3x^2-2x+1 }{ x-4 } } {,} domain, zeros, growth behavior, \zusatzklammer {\definitionsverweis {local}{}{}} {} {} extrema. Sketch the graph of the function.

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\inputexercise
{5}
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Prove that a non-costant rational function of the shape
\mathdisp {f(x) = \frac{ax+b}{cx+d}} { }
\zusatzklammer {with $a,b,c,d \in \R, a,c \neq 0$} {} {,} has no local extrema.

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\inputexercise
{3}
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Determine the limit of the rational function
\mathdisp {\frac{ x^4+2x^3-3x^2-4x+4}{ 2x^3-x^2-4x+3 }} { }
at point $a=1$.

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\inputexercise
{5}
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Let
\mathl{D \subseteq \R}{} and \maabbdisp {F} {D} {\R } {} be a rational function. Prove that $F$ is a polynomial if and only if there is a higher derivative such that$F^{(n)} =0$.

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