# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 22

Warm-up-exercises

### 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

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

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

### Exercise

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

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

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

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

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 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 function

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

at point

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

### Exercise

Let

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x),}$

be a differentiable function with the property

${\displaystyle f'=f{\text{ and }}f(0)=1.}$

Prove that ${\displaystyle {}f(x)=\exp x}$ for all ${\displaystyle {}a\in \mathbb {R} }$.

### 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

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

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

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

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

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 ${\displaystyle {}A_{n}\leq A_{n+1}}$.