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