Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 18/en/latex

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\setcounter{section}{18}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Prove the following properties of the hyperbolic sine and the hyperbolic cosine \aufzaehlungdrei{
\mathdisp {\cosh x + \sinh x = e^x} { . }
}{
\mathdisp {\cosh x - \sinh x = e^{-x }} { . }
}{
\mathdisp {( \cosh x )^2 - ( \sinh x )^2 = 1} { . }
}

}
{} {}




\inputexercise
{}
{

Show that the hyperbolic sine on $\R$ is strictly increasing.

}
{} {}




\inputexercise
{}
{

Prove that the hyperbolic tangent satisfies the following estimate
\mathdisp {-1 \leq \tanh x \leq 1 {{\text{ for all }}} x \in \R} { . }

}
{} {}




\inputexercise
{}
{

Prove by elementary geometric considerations the Sine theorem, i.e. the statement that in a triangle the equalities
\mathdisp {{ \frac{ \sin \alpha }{ a } } = { \frac{ \sin \beta }{ b } } = { \frac{ \sin \gamma }{ c } }} { }
hold, where
\mathl{a,b,c}{} are the side lengths of the edges and
\mathl{\alpha, \beta, \gamma}{} are respectively the opposite angles.

}
{} {}




\inputexercise
{}
{

Compute the determinants of plane and spatial rotations.

}
{} {}




\inputexercise
{}
{

Prove the addition theorems for sine and cosine using the rotation matrices.

}
{} {}




\inputexercise
{}
{

We look at a clock with minute and second hands, both moving continuously. Determine a formula which calculates the angular position of the second hand from the angular position of the minute hand \zusatzklammer {each starting from the 12-clock-position measured in the clockwise direction} {} {.}

}
{} {}




\inputexercise
{}
{

Prove that the series
\mathdisp {\sum_{n=1}^\infty { \frac{ \sin n }{ n^2 } }} { }
converges.

}
{} {}




\inputexercise
{}
{

Determine the coefficients up to $z^6$ in the series product $\sum _{ n= 0}^\infty c_n z^{ n }$ of the sine series and the cosine series.

}
{} {}

The next tasks require the definition of a \stichwort {periodic function} {.}


A function \maabbdisp {f} {\R} {\R } {} is called \definitionswort {periodic}{} with \definitionswort {period}{}
\mathl{L>0}{,} if for all
\mathl{x \in \R}{} the equality
\mathdisp {f(x)=f(x+L)} { }
holds.





\inputexercise
{}
{

Let \maabbdisp {f} {\R} {\R } {} be a periodic function and \maabbdisp {g} {\R} {\R } {} any function.

a) Prove that the composite function
\mathl{g \circ f}{} is also periodic.

b) Prove that the composite function
\mathl{f \circ g}{} does not need to be periodic.

}
{} {}




\inputexercise
{}
{

Let \maabbdisp {f} {\R} {\R } {} be a continuous periodic function. Prove that $f$ is bounded.

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{3}
{

Prove that in the power series
\mathl{\sum_{n= 0}^\infty c_nx^n}{} of the hyperbolic cosine the coefficients $c_n$ are $0$ if $n$ is odd.

}
{} {}




\inputexercise
{6}
{

Prove that the hyperbolic cosine is strictly decreasing on $\R_{\leq 0}$ and strictly increasing on $\R_{\geq 0}$.

}
{} {}




\inputexercise
{4}
{

Let \maabbdisp {\varphi} {\R^3} {\R^3 } {} be the space rotation by $45$ degree around the $z$-axis counterclockwise. How does the matrix describing $\varphi$ with respect to the basis
\mathdisp {\begin{pmatrix} 1 \\2\\ 4 \end{pmatrix} ,\, \begin{pmatrix} 3 \\3\\ -1 \end{pmatrix} ,\, \begin{pmatrix} 5 \\0\\ 7 \end{pmatrix}} { }
look like?

}
{} {}




\inputexercise
{5}
{

Prove the addition theorem
\mathdisp {\sin (x+y) = \sin x \cdot \cos y + \cos x \cdot \sin y} { }
for the sine using the defining power series.

}
{} {}




\inputexercise
{4}
{

Let \maabbdisp {f_1,f_2} {\R} {\R } {} be periodic functions with periods respectively \mathkor {} {L_1} {and} {L_2} {.} The quotient
\mathl{L_1/L_2}{} is a rational number. Prove that
\mathl{f_1+f_2}{} is also a periodic function.

}
{} {}




\inputexercise
{5}
{

Consider $n$ complex numbers
\mathl{z_1,z_2 , \ldots , z_n}{} lying in the disc $B$ with center $(0,0)$ and radius $1$, that is in
\mathl{B={ \left\{ z \in {\mathbb C} \mid \betrag { z } \leq 1 \right\} }}{.} Prove that there exists a point
\mathl{w \in B}{} such that
\mathdisp {\sum_{i =1}^n \betrag { z_i-w } \geq n} { . }

}
{} {}



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