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

$ℝ⟶ℝ,x⟼\sin x\cos x,$

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 $f:ℝ\to ℝ$ be a function. Compare the polynomial interpolation for $n+1$ given point and the Taylor-polynomials of degree $n$ in a point.

Exercise

Let $f:ℝ\to ℝ$ 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 $f:ℝ\to ℝ$ 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 $f,g:ℝ\to ℝ$ be polynomials of degree $n$, let $a_1,\dots ,a_k\in ℝ$ be points and $n_1,\dots ,n_k\ge 1$ 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 $(n_j-1)$-th derivative. Show $f=g$.

Exercise

Let $f(x):=\frac{x^2-x+5}{x^2+3}$. Determine a polynomial $h$ of degree $\le 3$, 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 $\le 3$, 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 $a\in ℝ$.

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 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

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

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

$g(y)=\sum _{k=0}^{\mathrm{\infty }}{b}_k{y}^k$

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

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

Exercise (5 marks)

Let $f(x):=\frac{x^2+2x+1}{x^2+5}$. Determine a polynomial $h$ of degree $\le 3$, 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

$f:[0,2\pi ]⟶ℝ,x⟼f(x)=\sin x\cos x,$

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 $A_n\le A_{n+1}$.

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