# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 21

    Warm-up-exercises

### Exercise

Determine the derivatives of hyperbolic sine and hyperbolic cosine.

### Exercise

Determine the derivative of the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}\cdot \exp {\left(x^{3}-4x\right)}.$

### Exercise

Determine the derivative of the function
    $\ln \colon \mathbb {R} _{+}\longrightarrow \mathbb {R}$

### Exercise

Determine the derivatives of the sine and the cosine function by using fact.

### Exercise

Determine the $1034871$-th derivative of the sine function.

### Exercise

Determine the derivative of the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin {\left(\cos x\right)}.$

### Exercise

Determine the derivative of the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto (\sin x)(\cos x).$

### Exercise

Determine for $n\in \mathbb {N}$ the derivative of the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto (\sin x)^{n}.$

### Exercise

Determine the derivative of the function
    $D\longrightarrow \mathbb {R} ,x\longmapsto \tan x={\frac {\sin x}{\cos x}}.$

### Exercise

Prove that the real sine function induces a bijective, strictly increasing function
    $[-\pi /2,\pi /2]\longrightarrow [-1,1],$
and that the real cosine function induces a bijective, strictly decreasing function
    $[0,\pi ]\longrightarrow [-1,1].$

### Exercise

Determine the derivatives of arc-sine and arc-cosine functions.

### Exercise

We consider the function
    $f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto f(x)=1+\ln x-{\frac {1}{x}}.$
a) Prove that $f$ gives a continuous bijection between $\mathbb {R} _{+}$ and $\mathbb {R}$.
b) Determine the inverse image $u$ of $0$ under $f$, then compute $f'(u)$ and $(f^{-1})'(0)$. Draw a rough sketch for the inverse function $f^{-1}$.

### Exercise

Let
    $f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$
be two differentiable functions. Let $a\in \mathbb {R}$. Suppose we have that
    $f(a)\geq g(a){\text{ and }}f'(x)\geq g'(x){\text{ for all }}x\geq a.$
Prove that
    $f(x)\geq g(x){\text{ for all }}x\geq a.$

### Exercise

We consider the function
    $f\colon \mathbb {R} \setminus \{0\}\longrightarrow \mathbb {R} ,x\longmapsto f(x)=e^{-{\frac {1}{x}}}.$
a) Investigate the monotony behavior of this function.
b) Prove that this function is injective.
c) Determine the image of $f$.
d) Determine the inverse function on the image for this function.
e) Sketch the graph of the function $f$.

### Exercise

Consider the function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=(2x+3)e^{-x^{2}}.$
Determine the zeros and the local (global) extrema of $f$. Sketch up roughly the graph of the function.

### Exercise

Discuss the behavior of the function graph of
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=e^{-2x}-2e^{-x}.$
Determine especially the monotonicity behavior, the extrema of $f$, $\operatorname {lim} _{x\rightarrow \infty }\,f(x)$ and also for the derivative $f'$.

### Exercise

Prove that the function
    $f(x)={\begin{cases}x\sin {\frac {1}{x}}{\text{ for }}x\in0,1]},\\0{\text{ for }}x=0,\end{cases}}$
is continuous and that it has infinitely many zeros.

### Exercise

Determine the limit of the sequence
    ${\frac {\sin n}{n}},\,n\in \mathbb {N} _{+}.$

### Exercise

Determine for the following functions if the function limit exists and, in case, what value it takes.
* $\operatorname {lim} _{x\rightarrow 0}\,{\frac {\sin x}{x}}$,
* $\operatorname {lim} _{x\rightarrow 0}\,{\frac {(\sin x)^{2}}{x}}$,
* $\operatorname {lim} _{x\rightarrow 0}\,{\frac {\sin x}{x^{2}}}$,
* $\operatorname {lim} _{x\rightarrow 1}\,{\frac {x-1}{\ln x}}$.

### Exercise

Determine for the following functions, if the limit function for $x\in \mathbb {R} \setminus \{0\}$, $x\rightarrow 0$, exists, and, in case, what value it takes.
* $\sin {\frac {1}{x}}$,
* $x\cdot \sin {\frac {1}{x}}$,
* ${\frac {1}{x}}\cdot \sin {\frac {1}{x}}$.
    Hand-in-exercises

### Exercise

Determine the linear functions that are tangent to the exponential function.

### Exercise

Determine the derivative of the function
    $\mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto x^{x}.$
The following task should be solved without reference to the second derivative.

### Exercise

Determine the extrema of the function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin x+\cos x.$

### Exercise

Let
    $f\colon \mathbb {R} \longrightarrow \mathbb {R}$
be a polynomial function of degree $d\geq 1$. Let $m$ be the number of local maxima of $f$ and $n$ the number of local minima of $f$. Prove that if $d$ is odd then $m=n$ and that if $d$ is even then
    $\vert {m-n}\vert =1\,.$