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

$ℝ⟶ℝ,x⟼\sin x\cos x,$

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

Exercise

Let $p\in ℝ[Y]$ be a polynomial and

$g:{ℝ}_{+}⟶ℝ,x⟼g(x)=p\left(\frac{1}{x}\right)e^{-\frac{1}{x}}.$

Prove that the derivative $g^{\prime }(x)$ has also the shape

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

where $q$ is a polynomial.

Exercise

We consider the function

$f:{ℝ}_{+}⟶ℝ,x⟼f(x)=e^{-\frac{1}{x}}.$

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

${\lim }_{x\to 0}{f}^{(n)}(x)=0.$

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

$f:ℝ⟶ℝ,x⟼f(x),$

be a differentiable function with the property

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

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

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

$f:ℝ⟶ℝ,x⟼\sin (\cos x)+x^3\exp \left(x^2\right).$

Exercise (4 marks)

Discuss the behavior of the function

$f:[0,2\pi ]⟶ℝ,x⟼f(x)=\sin x\cos x,$

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

Exercise (4 marks)

Discuss the behavior of the function

$f:[-\frac{\pi }{2},\frac{\pi }{2}]⟶ℝ,x⟼f(x)={\sin }^{3}x-\frac{1}{4}\sin x,$

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