Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 19/en

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Warm-up-exercises

The following task should be solved both directly and through the derivation rules.

Exercise

Determine the derivative of the function

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

for all ${\displaystyle {}n\in \mathbb {N} }$.

Exercise

Determine the derivative of the function

${\displaystyle f\colon \mathbb {R} \setminus \{0\}\longrightarrow \mathbb {R} ,\,x\longmapsto f(x)=x^{n}}$

for all ${\displaystyle {}n\in \mathbb {Z} }$.

Exercise

Determine the derivative of the function

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

for all ${\displaystyle {}n\in \mathbb {N} _{+}}$.

Exercise

Determine directly

(without the use of derivation rules) the derivative of the function
${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto f(x)=x^{3}+2x^{2}-5x+3,}$
at any point ${\displaystyle {}a\in \mathbb {R} }$.

Exercise

Prove that the real absolute value

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto \vert {x}\vert ,}$

is not differentiable at the point zero.

Exercise

Determine the derivative of the function

${\displaystyle f\colon \mathbb {R} \setminus \{0\}\longrightarrow \mathbb {R} ,\,x\longmapsto f(x)={\frac {x^{2}+1}{x^{3}}}.}$

Exercise

Prove that the derivative of a rational function is also a rational function.

Exercise

Consider ${\displaystyle {}f(x)=x^{3}+4x^{2}-1}$ and ${\displaystyle {}g(y)=y^{2}-y+2}$. Determine the derivative of the composite function ${\displaystyle {}h(x)=g(f(x))}$ directly and by the chain rule (Theorem 19.8).

Exercise

Prove that a polynomial ${\displaystyle {}P\in \mathbb {R} [X]}$ has degree ${\displaystyle {}\leq d}$ (or it is ${\displaystyle {}P=0}$), if and only if the ${\displaystyle {}(d+1)}$-th derivative of ${\displaystyle {}P}$ is the zero poynomial.

Exercise

Let

${\displaystyle f,g\colon \mathbb {R} \longrightarrow \mathbb {R} }$
be two differentiable functions and consider
${\displaystyle h(x)=(g(f(x)))^{2}f(g(x)).}$

a) Determine the derivative ${\displaystyle {}h'}$ from the derivatives of ${\displaystyle {}f}$ and ${\displaystyle {}g}$.

b) Let now
${\displaystyle f(x)=x^{2}-1{\text{ and }}g(x)=x+2.}$
Compute ${\displaystyle {}h'(x)}$ in two ways, one directly from ${\displaystyle {}h(x)}$ and the other by the formula of part ${\displaystyle {}a)}$.

For the “linear approximation” of differentiable maps we need the definition of affine-linear maps.

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be vector spaces over ${\displaystyle {}K}$. A map

${\displaystyle \alpha \colon V\longrightarrow W,\,v\longmapsto \alpha (v)=\varphi (v)+w,}$

where ${\displaystyle {}\varphi }$ is a linear map and ${\displaystyle {}w\in W}$ is a vector, is called affine-linear.

Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Prove that given two vectors ${\displaystyle {}u,v\in W}$ there exists exactly one affine-linear map

${\displaystyle \alpha \colon K\longrightarrow W}$

sucht that ${\displaystyle {}\alpha (0)=u}$ and ${\displaystyle {}\alpha (1)=v}$.

Exercise

Determine the affine-linear map

${\displaystyle \alpha \colon \mathbb {R} \longrightarrow \mathbb {R} ^{3}}$

such that ${\displaystyle {}\alpha (0)=(2,3,4)}$ and ${\displaystyle {}\alpha (1)=(5,-2,-1)}$.

Hand-in-exercises

Exercise (3 points)

Determine the derivative of the function

${\displaystyle f\colon D\longrightarrow \mathbb {R} ,\,x\longmapsto f(x)={\frac {x^{2}+x-1}{x^{3}-x+2}},}$

where ${\displaystyle {}D}$ is the set where the denominator does not vanish.

Exercise (4 points)

Determine the tangents to the graph of the function ${\displaystyle {}f(x)=x^{3}-x^{2}-x+1}$, which are parallel to ${\displaystyle {}y=x}$.

Exercise (4 points)

Let ${\displaystyle {}f(x)={\frac {x^{2}+5x-2}{x+1}}}$ and ${\displaystyle {}g(y)={\frac {y-2}{y^{2}+3}}}$. Determine the derivative of the composite ${\displaystyle {}h(x)=g(f(x))}$ directly and by the chain rule (Theorem 19.8).

Exercise (2 points)

Determine the affine-linear map

${\displaystyle \alpha \colon \mathbb {R} \longrightarrow \mathbb {R} ,}$

whose graph passes through the two points ${\displaystyle {}(-2,3)}$ and ${\displaystyle {}(5,-7)}$.

Exercise (3 points)

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ be a subset and let

${\displaystyle f_{i}\colon D\longrightarrow \mathbb {R} ,\,i=1,\ldots ,n,}$
be differentiable functions. Prove the formula
${\displaystyle (f_{1}{\cdots }f_{n})'=\sum _{i=1}^{n}f_{1}{\cdots }f_{i-1}f_{i}'f_{i+1}{\cdots }f_{n}.}$