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

Exercises

### Exercise

Determine all the Taylor polynomials of the function

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

at the point ${\displaystyle {}a=3}$.

### Exercise

Write the polynomial

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

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

a) directly by inserting,

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

### Exercise

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

### Exercise

Determine the Taylor series of the function

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

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

### Exercise

Determine the Taylor polynomial of degree ${\displaystyle {}3}$ of the rational function

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

in the point ${\displaystyle {}0}$.

### Exercise

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

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

at the zero point.

### Exercise

We consider the function

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

over the real numbers.

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

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

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

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

### Exercise

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

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

at point

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

### Exercise

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

### Exercise

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

### Exercise

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

### Exercise

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

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

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

### Exercise

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

### Exercise

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

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

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

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

### Exercise

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

### Exercise

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

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

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

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

where ${\displaystyle {}q}$ is a polynomial.

### Exercise

We consider the function

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

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

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

### Exercise

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

### Exercise

Let

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

Because of

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

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

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

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

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

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

### Exercise

Determine the Taylor polynomial up to fourth order of the inverse of the sine function at the point ${\displaystyle {}0}$ with the power series approach described in an remark.

Hand-in-exercises

### Exercise (4 marks)

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

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

### Exercise (5 marks)

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

### Exercise (4 marks)

Discuss the behavior of the function

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

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

### Exercise (4 marks)

Discuss the behavior of the function

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

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 ${\displaystyle {}1}$ with the power series approach described in remark from the power series of the exponential function.

### Exercise (6 marks)

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