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

Warm-up-exercises

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

Determine all the Taylor polynomials of the function

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

at the point ${}a=3$.

Exercise

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

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 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 function

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

at point

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

Exercise

Let

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

be a differentiable function with the property

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

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

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