Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 19

Warm-up-exercises

Exercise

Determine the derivative of the functions

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

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

Exercise

Determine the derivative of the function

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

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

Exercise

Determine the derivative of the function

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

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

Exercise

Determine directly (without the use of derivation rules) the derivative of the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{3}+2x^{2}-5x+3,$

at any point ${}a\in \mathbb {R}$.

Exercise

Prove that the real absolute value

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

is not differentiable at the point zero.

Exercise

Determine the derivative of the function

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

Exercise

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

Exercise

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

Exercise

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

Exercise

Let

$f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$

be two differentiable functions and consider

${}h(x)=(g(f(x)))^{2}f(g(x))\,.$

a) Determine the derivative ${}h'$ from the derivatives of ${}f$ and ${}g$. b) Let now

$f(x)=x^{2}-1{\text{ and }}g(x)=x+2.$

Compute ${}h'(x)$ in two ways, one directly from ${}h(x)$ and the other by the formula of part ${}a)$.

Exercise

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

$\alpha \colon K\longrightarrow W$

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

Exercise

Determine the affine-linear map

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

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

Hand-in-exercises

Exercise (3 marks)

Determine the derivative of the function

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

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

Exercise (4 marks)

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

Exercise (7 (2+2+3) marks)

Let

${}f(x)={\frac {x^{2}+5x-2}{x+1}}\,$

and

${}g(y)={\frac {y-2}{y^{2}+3}}\,.$

Determine the derivative of the composite

${}h(x)=g(f(x))\,$

directly and by the chain rule.

Exercise (2 marks)

Determine the affine-linear map

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

whose graph passes through the two points

${}(-2,3)$ and ${}(5,-7)$.

Exercise (3 marks)

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

$f_{i}\colon D\longrightarrow \mathbb {R} ,\,i=1,\ldots ,n,$

be differentiable functions. Prove the formula

${}{\left(f_{1}\cdots f_{n}\right)}'=\sum _{i=1}^{n}f_{1}\cdots f_{i-1}f_{i}'f_{i+1}\cdots f_{n}\,.$