# 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

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}\cdot \exp {\left(x^{3}-4x\right)}.}$

### Exercise

Determine the derivative of the function

${\displaystyle \ln \colon \mathbb {R} _{+}\longrightarrow \mathbb {R} }$

### Exercise

Determine the derivatives of the sine and the cosine function by using power series.

### Exercise

Determine the ${\displaystyle {}1034871}$-th derivative of the sine function.

### Exercise

Determine the derivative of the function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin {\left(\cos x\right)}.}$

### Exercise

Determine the derivative of the function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto (\sin x)(\cos x).}$

### Exercise

Determine for ${\displaystyle {}n\in \mathbb {N} }$ the derivative of the function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto (\sin x)^{n}.}$

### Exercise

Determine the derivative of the function

${\displaystyle D\longrightarrow \mathbb {R} ,x\longmapsto \tan x={\frac {\sin x}{\cos x}}.}$

### Exercise

Prove that the real sine function induces a bijective, strictly increasing function

${\displaystyle [-\pi /2,\pi /2]\longrightarrow [-1,1],}$

and that the real cosine function induces a bijective, strictly decreasing function

${\displaystyle [0,\pi ]\longrightarrow [-1,1].}$

### Exercise

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

### Exercise

We consider the function

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

a) Prove that ${\displaystyle {}f}$ gives a continuous bijection between ${\displaystyle {}\mathbb {R} _{+}}$ and ${\displaystyle {}\mathbb {R} }$.

b) Determine the inverse image ${\displaystyle {}u}$ of ${\displaystyle {}0}$ under ${\displaystyle {}f}$, then compute ${\displaystyle {}f'(u)}$ and ${\displaystyle {}(f^{-1})'(0)}$. Draw a rough sketch for the inverse function ${\displaystyle {}f^{-1}}$.

### Exercise

Let

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

be two differentiable functions. Let ${\displaystyle {}a\in \mathbb {R} }$. Suppose we have that

${\displaystyle f(a)\geq g(a){\text{ and }}f'(x)\geq g'(x){\text{ for all }}x\geq a.}$

Prove that

${\displaystyle f(x)\geq g(x){\text{ for all }}x\geq a.}$

### Exercise

We consider the function

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

a) Investigate the monotony behavior of this function.

b) Prove that this function is injective.

c) Determine the image of ${\displaystyle {}f}$.

d) Determine the inverse function on the image for this function.

e) Sketch the graph of the function ${\displaystyle {}f}$.

### Exercise

Consider the function

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

Determine the zeros and the local (global) extrema of ${\displaystyle {}f}$. Sketch up roughly the graph of the function.

### Exercise

Discuss the behavior of the function graph of

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=e^{-2x}-2e^{-x}.}$

Determine especially the monotonicity behavior, the extrema of ${\displaystyle {}f}$, ${\displaystyle {}\operatorname {lim} _{x\rightarrow \infty }\,f(x)}$ and also for the derivative ${\displaystyle {}f'}$.

### Exercise

Prove that the function

${\displaystyle f(x)={\begin{cases}x\sin {\frac {1}{x}}{\text{ for }}x\in {]0,1]},\\0{\text{ for }}x=0,\end{cases}}}$

is continuous and that it has infinitely many zeros.

### Exercise

Determine the limit of the sequence

${\displaystyle {\frac {\sin n}{n}},\,n\in \mathbb {N} _{+}.}$

### Exercise

Determine for the following functions if the function limit exists and, in case, what value it takes.

1. ${\displaystyle {}\operatorname {lim} _{x\rightarrow 0}\,{\frac {\sin x}{x}}}$,
2. ${\displaystyle {}\operatorname {lim} _{x\rightarrow 0}\,{\frac {(\sin x)^{2}}{x}}}$,
3. ${\displaystyle {}\operatorname {lim} _{x\rightarrow 0}\,{\frac {\sin x}{x^{2}}}}$,
4. ${\displaystyle {}\operatorname {lim} _{x\rightarrow 1}\,{\frac {x-1}{\ln x}}}$.

### Exercise

Determine for the following functions, if the limit function for ${\displaystyle {}x\in \mathbb {R} \setminus \{0\}}$, ${\displaystyle {}x\rightarrow 0}$, exists, and, in case, what value it takes.

1. ${\displaystyle {}\sin {\frac {1}{x}}}$,
2. ${\displaystyle {}x\cdot \sin {\frac {1}{x}}}$,
3. ${\displaystyle {}{\frac {1}{x}}\cdot \sin {\frac {1}{x}}}$.

Hand-in-exercises

### Exercise

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


### Exercise

Determine the derivative of the function

${\displaystyle \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto x^{x}.}$

The following task should be solved without reference to the second derivative.

### Exercise

Determine the extrema of the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin x+\cos x.}$

### Exercise

Let

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$

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

${\displaystyle {}\vert {m-n}\vert =1\,.}$