Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 22/en/latex

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Determine the Taylor polynomial of degree $4$ of the function \maabbeledisp {} {\R} {\R } {x} { \sin x \cos x } {,} at the zero point.

}
{} {}




\inputexercise
{}
{

Determine all the Taylor polynomials of the function
\mathdisp {f(x) = x^4-2x^3+2x^2-3x+5} { }
at the point $a=3$.

}
{} {}




\inputexercise
{}
{

Let $\sum _{ n= 0}^\infty c_n (x-a)^{ n }$ be a convergent power series. Determine the derivative $f^{(k)}(a)$.

}
{} {}




\inputexercise
{}
{

Let $p \in \R[Y]$ be a polynomial and \maabbeledisp {g} { \R_+} {\R } {x} { g(x) = p(\frac{1}{x}) e^{- \frac{1}{x} } } {.} Prove that the derivative $g'(x)$ has also the shape
\mathdisp {g'(x)=q(\frac{1}{x}) e^{- \frac{1}{x} }} { , }
where $q$ is a polynomial.

}
{} {}




\inputexercise
{}
{

We consider the function \maabbeledisp {f} {\R_+} {\R } {x} {f(x) = e^{- \frac{1}{x} } } {.} Prove that for all $n \in \N$ the $n$-th derivative
\mathl{f^{(n)}}{} satisfies the following property
\mathdisp {\operatorname{lim}_{ x \in \R_+ , \, x \rightarrow 0 } \, f^{(n)}(x) = 0} { . }

}
{} {}




\inputexercise
{}
{

Determine the Taylor series of the function $f(x)=\frac{1}{x}$ at point $a=2$ up to order $4$ \zusatzklammer {Give also the Taylor polynomial of degree $4$ at point $2$, where the coefficients must be stated in the most simple form} {} {.}

}
{} {}




\inputexercise
{}
{

Determine the Taylor polynomial of degree $3$ of the function
\mathdisp {f(x) =x \cdot \sin x} { }
at point
\mathl{a= { \frac{ \pi }{ 2 } }}{.}

}
{} {}




\inputexercise
{}
{

Let \maabbeledisp {f} {\R} {\R } {x} {f(x) } {,} be a differentiable function with the property
\mathdisp {f'=f \text{ and } f(0)=1} { . }
Prove that $f(x)= \exp x$ for all $x \in \R$.

}
{} {}




\inputexercise
{}
{

Determine the Taylor polynomial up to fourth order of the inverse of the sine function at the point $0$ with the power series approach described in Remark 22.8.

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{4}
{

Find the Taylor polynomials in $0$ up to degree $4$ of the function \maabbeledisp {f} {\R} {\R } {x} { \sin \left( \cos x \right) + x^3 \exp \left( x^2 \right) } {.}

}
{} {}




\inputexercise
{4}
{

Discuss the behavior of the function \maabbeledisp {f} {[0,2 \pi]} {\R } {x} {f(x) = \sin x \cos x } {,} concerning zeros, growth behavior, \zusatzklammer {local} {} {} extrema. Sketch the graph of the function.

}
{} {}




\inputexercise
{4}
{

Discuss the behavior of the function \maabbeledisp {f} {[- { \frac{ \pi }{ 2 } } ,{ \frac{ \pi }{ 2 } }]} {\R } {x} {f(x) = \sin^{ 3 } x - { \frac{ 1 }{ 4 } } \sin x } {,} concerning zeros, growth behavior, \zusatzklammer {local} {} {} extrema. Sketch the graph of the function.

}
{} {}




\inputexercise
{4}
{

Determine the Taylor polynomial up to fourth order of the natural logarithm at point $1$ with the power series approach described in Remark 22.8 from the power series of the exponential function.

}
{} {}




\inputexercise
{8}
{

For
\mathl{n \geq 3}{} let
\mathl{A_n}{} be the area of ​​a circle inscribed in the unit regular $n$-gon. Prove that
\mathl{A_n \leq A_{n+1}}{.}

}
{} {}



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