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 power series.

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

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

is continuous and that it has infinitely many zeros.

Exercise

Determine the limit of the sequence

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

Exercise

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

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

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.

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

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

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