Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 21/en/latex

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\zwischenueberschrift{Warm-up-exercises}

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Determine the derivatives of hyperbolic sine and hyperbolic cosine.

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Determine the derivative of the function \maabbeledisp {} {\R} {\R } {x} { x^2 \cdot \exp \left( x^3-4x \right) } {.}

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Determine the derivative of the function \maabbdisp {\ln} {\R_+} { \R } {.}

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Determine the derivatives of the sine and the cosine function by using Theorem 21.1.

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Determine the $1034871$-th derivative of the sine function.

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Determine the derivative of the function \maabbeledisp {} {\R} {\R } {x} { \sin \left( \cos x \right) } {.}

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Determine the derivative of the function \maabbeledisp {} {\R} {\R } {x} { ( \sin x )( \cos x ) } {.}

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Determine for $n \in \N$ the derivative of the function \maabbeledisp {} {\R} {\R } {x} { (\sin x )^n } {.}

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Determine the derivative of the function \maabbeledisp {} {D} {\R } {x} { \tan x = \frac{ \sin x }{ \cos x } } {.}

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Prove that the real sine function induces a bijective, strictly increasing function \maabbdisp {} {[- \pi/2, \pi/2]} {[-1,1] } {,} and that the real cosine function induces a bijective, strictly decreasing function \maabbdisp {} {[0,\pi]} {[-1,1] } {.}

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Determine the derivatives of arc-sine and arc-cosine functions.

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We consider the function \maabbeledisp {f} {\R_+} {\R } {x} {f(x) = 1 + \ln x - \frac{1}{x} } {.}

a) Prove that $f$ gives a continuous bijection between \mathkor {} {\R_+} {and} {\R} {.}

b) Determine the inverse image $u$ of $0$ under $f$, then compute $f'(u)$ and $(f^{-1})'(0)$. Draw a rough sketch for the inverse function $f^{-1}$.

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Let \maabbdisp {f,g} {\R} {\R } {} be two differentiable functions. Let $a \in \R$. We have that
\mathdisp {f(a) \geq g(a) \text{ and } f'(x) \geq g'(x) \text{ for all } x \geq a} { . }
Prove that
\mathdisp {f(x) \geq g(x) \text{ for all } x \geq a} { . }

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We consider the function \maabbeledisp {f} {\R \setminus \{0\}} {\R } {x} {f(x) = e^{ - { \frac{ 1 }{ x } } } } {.}

a) Investigate the monotony behavior of this function.

b) Prove that this function is injective.

c) Determine the image of $f$.

d) Determine the inverse function on the image for this function.

e) Sketch the graph of the function $f$.

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Consider the function \maabbeledisp {f} {\R} {\R } {x} {f(x) = (2x+3)e^{-x^2} } {.} Determine the zeros and the local (global) extrema of $f$. Sketch up roughly the graph of the function.

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Discuss the behavior of the function graph of \maabbeledisp {f} {\R} {\R } {x} {f(x) = e^{-2x} -2e^{-x} } {.} Determine especially the monotony behavior, the extrema of $f$,
\mathl{\operatorname{lim}_{ x \rightarrow \infty } \, f(x)}{} and also for the derivative $f'$.

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Prove that the function
\mathdisp {f(x) = \begin{cases} x \sin \frac{1}{x} \text{ for } x \in {]0,1]}, \\ 0 \text{ for } x = 0,\end{cases}} { }
is continuous and that it has infinitely many zeros.

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Determine the limit of the sequence
\mathdisp {\frac{ \sin n }{n} , \, n \in \N_+} { . }

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Determine for the following functions if the function limit exists and, in case, what value it takes. \aufzaehlungvier{$\operatorname{lim}_{ x \rightarrow 0 } \, \frac{ \sin x }{x}$, }{$\operatorname{lim}_{ x \rightarrow 0 } \, \frac{ (\sin x)^2 }{x}$, }{$\operatorname{lim}_{ x \rightarrow 0 } \, \frac{ \sin x }{x^2}$, }{$\operatorname{lim}_{ x \rightarrow 1 } \, \frac{x-1}{ \ln x }$. }

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Determine for the following functions, if the limit function for
\mathbed {x \in \R \setminus \{0\}} {}
{x \rightarrow 0} {}
{} {} {} {,} exists, and, in case, what value it takes. \aufzaehlungdrei{$\sin \frac{1}{x}$, }{$x \cdot \sin \frac{1}{x}$, }{$\frac{1}{x} \cdot \sin \frac{1}{x}$. }

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\zwischenueberschrift{Hand-in-exercises}

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Determine the linear functions that are tangent to the exponential function.

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\inputexercise
{3}
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Determine the derivative of the function \maabbeledisp {} {\R_+ } {\R } {x} { x^x } {.}

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The following task should be solved without reference to the second derivative.

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Determine the extrema of the function \maabbeledisp {f} {\R} {\R } {x} {f(x) = \sin x + \cos x } {.}

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Let \maabbdisp {f} {\R} {\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 $\betrag { m-n }=1$.

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