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

    Exercises

### Exercise

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

### Exercise

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 $2$.

### Exercise

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

### Exercise

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).

### Exercise

Determine the Taylor polynomial of degree $3$ of the rational function
    $f(x)={\frac {3x^{2}-2x+5}{x-2}}\,$
in the point $0$.

### Exercise

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.

### Exercise

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 ${\frac {\pi }{2}}$.

### Exercise

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

### Exercise

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

### Exercise

Let $f\colon \mathbb {R} \rightarrow \mathbb {R}$ be an $n$-fold differentiable function in the point $a$. Show that the $n$-th 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.

### Exercise

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

### Exercise

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 $f=g$.

### Exercise

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 $f$.

### Exercise

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.

### Exercise

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

### Exercise

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

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

### Exercise

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

### Exercise

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

### Exercise

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

### Exercise (4 marks)

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

### Exercise (5 marks)

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 $f$.

### Exercise (4 marks)

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.

### Exercise (4 marks)

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.

### Exercise (4 marks)

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.

### Exercise (6 marks)

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