Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 17/refcontrol

Exercises

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Determine all the Taylor polynomials of the function

${}f(x)=x^{4}-2x^{3}+2x^{2}-3x+5\,$

at the point ${}a=3$.

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Write the polynomial

${}f(z)=z^{3}+3z^{2}-7z-4\,$

in the new variable ${}z-2$, using two different ways, namely

a) directly by inserting,

b) via the Taylor-polynomial in the point ${}2$.

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Determine the Taylor polynomialMDLD/Taylor polynomial (R) of order ${}4$ for the function ${}f(x)={\frac {x}{x^{2}+1}}$ in the point ${}a=3$.

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Determine the Taylor series of the function

${}f(x)={\frac {1}{x}}\,$

at point ${}a=2$ up to order ${}4$ (Give also the Taylor polynomial of degree ${}4$ at point ${}2$, where the coefficients must be stated in the most simple form).

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Determine the Taylor polynomialMDLD/Taylor polynomial (R) of degree ${}3$ of the rational functionMDLD/rational function (R)

${}f(x)={\frac {3x^{2}-2x+5}{x-2}}\,$

in the point ${}0$.

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Determine the Taylor polynomial of degree ${}4$ of the function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin x\cos x,$

at the zero point.

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We consider the function

${}f(x)={\frac {1}{\sin x}}\,$

over the real numbers.

a) Determine the range of ${}f$.

b) Sketch ${}f$ for ${}x$ between ${}-2\pi$ and ${}2\pi$.

c) Determine the first three derivatives of ${}f$.

d) Determine the Taylor-polynomial of order ${}3$ of ${}f$ in the point ${}{\frac {\pi }{2}}$.

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Determine the Taylor polynomial of degree ${}3$ of the function

${}f(x)=x\cdot \sin x\,$

at point

${}a={\frac {\pi }{2}}\,.$

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Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a function. Compare the polynomial interpolationMDLD/polynomial interpolation for ${}n+1$ given point and the Taylor-polynomialsMDLD/Taylor-polynomials (R) of degree ${}n$ in a point.

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Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be an ${}n$-fold differentiableMDLD/differentiable (R) function in the point ${}a$. Show that the ${}n$-th Taylor polynomialMDLD/Taylor polynomial for ${}f$ in the point ${}a$, written in the shifted variable ${}x-a$, equals the ${}n$-th Taylor polynomial of the function ${}g(x)=f(x+a)$ in the zero point.

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Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a function. Is it possible to get the ${}n$-th Taylor polynomialMDLD/Taylor polynomial (R) of ${}f$ in the point ${}b$ from the ${}n$-th Taylor polynomial of ${}f$ in the point ${}a$.

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Let ${}f,g\colon \mathbb {R} \rightarrow \mathbb {R}$ be polynomials of degree ${}n$, let ${}a_{1},\ldots ,a_{k}\in \mathbb {R}$ be points and ${}n_{1},\ldots ,n_{k}\geq 1$ natural numbers fulfilling

${}\sum _{j=1}^{k}n_{j}>n\,.$

Suppose that the derivatives of ${}f$ and ${}g$ coincide in den points ${}a_{j}$ up to the ${}{\left(n_{j}-1\right)}$-th derivative. Show ${}f=g$.

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Let ${}f(x):={\frac {x^{2}-x+5}{x^{2}+3}}$. Determine a polynomial ${}h$ of degree ${}\leq 3$, with the property that its linear approximation at the points ${}x=0$ and ${}x=1$ coincide with those of ${}f$.

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Let ${}f(x)=\sin x$. Determine polynomials ${}P,Q,R$ of degree ${}\leq 3$, fulfilling the following conditions.

(a) ${}P$ coincides with ${}f$ at the points ${}-\pi ,0,\pi$.

(b) ${}Q$ coincides with ${}f$ in ${}0$ and in ${}\pi$ up to the first derivative.

(c) ${}R$} coincides with ${}f$ in ${}\pi /2$ up to the third derivative.

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Determine the Taylor seriesMDLD/Taylor series (R) of the der exponential functionMDLD/exponential function (R) for an arbitrary point ${}a\in \mathbb {R}$.

===Exercise Exercise 17.16

change===

Let ${}p\in \mathbb {R} [Y]$ be a polynomial and

$g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto g(x)=p{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}.$

Prove that the derivative ${}g'(x)$ has also the shape

${}g'(x)=q{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}\,,$

where ${}q$ is a polynomial.

===Exercise Exercise 17.17

change===

We consider the function

$f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto f(x)=e^{-{\frac {1}{x}}}.$

Prove that for all ${}n\in \mathbb {N}$ the ${}n$-th derivative ${}f^{(n)}$ satisfies the following property

${}\operatorname {lim} _{x\rightarrow 0}\,f^{(n)}(x)=0\,.$

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Determine the Taylor polynomialMDLD/Taylor polynomial (R) of the third order of the function ${}{\frac {1}{x^{2}+1}}$ in the zero point, using the power series approach described in remark *****.

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Let

${}f(x)=-3x+x^{3}\,.$

Because of

${}f'(x)=-3+3x^{2}\,,$

this function is on the open interval ${}]-1,1[$ strictly decreasing and therefore injective (with the image interval ${}]-2,2[$). Also, ${}f(0)=0$. Let

${}g(y)=\sum _{k=0}^{\infty }b_{k}y^{k}\,$

be the inverse function, which we want to understand as a power series. Determine from the condition

${}(g(f(x))=x\,,$

the coefficients ${}b_{0},b_{1},b_{2},b_{3},b_{4}$.

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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 an remark.

Hand-in-exercises

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Find the Taylor polynomials in ${}0$ up to degree ${}4$ of the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin {\left(\cos x\right)}+x^{3}\exp {\left(x^{2}\right)}.$

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Let ${}f(x):={\frac {x^{2}+2x+1}{x^{2}+5}}$. Determine a polynomial ${}h$ of degree ${}\leq 3$, which in the two points ${}x=0$ and ${}x=-1$ has the same linear approximation as ${}f$.

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Discuss the behavior of the function

$f\colon [0,2\pi ]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin x\cos x,$

concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.

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Discuss the behavior of the function

$f\colon [-{\frac {\pi }{2}},{\frac {\pi }{2}}]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin ^{3}x-{\frac {1}{4}}\sin x,$

concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.

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

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

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