Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 16

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

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x),$

be a differentiable function with the property

$f'=f{\text{ and }}f(0)=1.$

Prove that ${}f(x)=\exp x$ for all ${}a\in \mathbb {R}$.

Exercise

Determine the derivatives of the sine and the cosine function by using Theorem 16.1 .

Exercise

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

Exercise

Determine the derivative of the function

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

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

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

Exercise

Let ${}\sum _{n=0}^{\infty }c_{n}(x-a)^{n}$ be a convergent power series. Determine the derivatives ${}f^{(k)}(a)$.

Exercise

Show that the function

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

is strictly increasing.

Exercise

Determine the local and the global extrema of the function

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

Exercise

Show that the sine function and the cosine function have the following values.

a)

${}\sin {\frac {\pi }{4}}=\cos {\frac {\pi }{4}}={\frac {1}{\sqrt {2}}}\,.$

b)

${}\cos {\frac {\pi }{3}}={\frac {1}{2}}\,.$

c)

${}\sin {\frac {\pi }{3}}={\frac {\sqrt {3}}{2}}\,.$

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

Show that the real tangent function induces a bijective, strictly increasing function

$]-\pi /2,\pi /2[\longrightarrow \mathbb {R}$

and that the real cotangent function induces a bijective strictly decreasing function

$[0,\pi ]\longrightarrow \mathbb {R} .$

Exercise

Let

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

be a periodic function and

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

any function. a) Prove that the composite function ${}g\circ f$ is also periodic. b) Prove that the composite function ${}f\circ g$ does not need to be periodic.

Exercise

Let

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

be a continuous periodic function. Prove that ${}f$ is bounded.

Exercise

Let

$f_{1},f_{2}\colon \mathbb {R} \longrightarrow \mathbb {R}$

be periodic functions with periods respectively ${}L_{1}$ and ${}L_{2}$. The quotient ${}L_{1}/L_{2}$ is a rational number. Prove that ${}f_{1}+f_{2}$ is also a periodic function.

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

Determine the derivative of the function

$\mathbb {R} _{+}\longrightarrow \mathbb {R} _{+},x\longmapsto f(x)=\pi ^{x}+x^{e}.$

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

Sketch the function

$g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto \sin {\frac {1}{x}}.$

Exercise

Show that the function

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

defined by

${}f(x)={\begin{cases}x\cdot \sin {\frac {1}{x}}{\text{ for }}x\neq 0\,,\\0{\text{ else}}\,,\end{cases}}\,$

is continuous. Is it possible to sketch the graph of this function?

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. ${}{\frac {1}{x}}\cdot \sin {\frac {1}{x}}$.

Exercise

For an initial value ${}x_{0}\in [0,{\frac {\pi }{2}}]$, we consider the sequence defined by the recursive relation

${}x_{n+1}:=\sin x_{n}\,.$

Decide whether ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ converges and, if applicable, determine its limit.

Exercise

Show that the sequence

${}x_{n}:=\sin n\,$

does not converge.

Hand-in-exercises

Exercise (3 marks)

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

Exercise (2 marks)

Determine the derivative of the function

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

The following exercise shall be solved without using the second derivative.

Exercise (4 marks)

Determine the extrema of the function

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

Exercise (5 marks)

We want to determine ${}\pi /2$ approximately as the smallest zero of cosine with the help of the cosine series

${}\cos x=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)!}}\,$

and the interval bisection method of the intermediate value theorem (in the sense of Method 11.3 ). Here, we encounter the problem that we can not compute the cosine exactly, as it involves infinitely many summands. Therefore we apply the following idea: as the ${}n$-th approximation ${}y_{n}$ for ${}\pi /2$, we use the lower bound of the ${}n$-th interval coming from the interval bisection (with the initial interval ${}[1,2]$) for the zero of the truncated cosine series ${}\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}$ (so we are using finer nested intervals of better approximations of the cosine function).

Design a computer program (pseudocode) that computes the values ${}y_{n}$ and prints them, under the following conditions.

• The computer has as many memory units as needed. They can store rational numbers.
• The natural numbers are in some data base
(it is not necessary to generate them).
• The computer can write the content of a memory unit into another memory unit.
• The computer can do arithmetic operations with rational numbers
(addition, subtraction, multiplication, division by a number ${}\neq 0$) and store the result in a memory unit.
• The computer can compare the content of two memory units and can jump, depending on the outcome, to program lines.
• The computer can print the content of a memory unit and stored texts.

Exercise (2 marks)

Determine the function limit ${}\operatorname {lim} _{x\rightarrow 1}\,{\frac {\ln x}{x-1}}$.

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