Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 19/refcontrol

Exercises

Exercise Create referencenumber

Lucy Sonnenschein is riding on her bike for five hours. In the first two hours, she makes ${}30$ km and in the following three hours, she also makes ${}30$ km. What is her average velocity?

Exercise Create referencenumber

Prove the mean value theorem for differential calculus for differentiable functions

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

and a compact interval ${}[a,b]\subset \mathbb {R}$, using the mean value theorem of integral calculus (you do not have to show that the average velocity is obtained in the interior of the interval).

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Determine the second derivative of the function

${}F(x)=\int _{0}^{x}{\sqrt {t^{5}-t^{3}+2t}}dt\,.$

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An object is released at time ${}0$ and it falls freely without air resistance from a certain height down to the earth thanks to the (constant) gravity force. Determine the velocity ${}v(t)$ and the distance ${}s(t)$ as a function of time ${}t$. After which time the object has traveled ${}100$ meters?

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Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R} ,x\mapsto f(x)$, be a continuous function,MDLD/continuous function (R) and let ${}F(x)$ be a primitive functionMDLD/primitive function for ${}f(x)$. Show that ${}F(x-a)$ is a primitive function for ${}f(x-a)$.

Exercise Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R} ,x\mapsto f(x)$, be a continuous functionMDLD/continuous function (R) and let ${}F(x)$ be a primitive functionMDLD/primitive function for ${}f(x)$. Show that ${}-F(-x)$ is a primitive function for ${}f(-x)$.

Exercise Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R} ,x\mapsto f(x)$, be a continuous functionMDLD/continuous function (R) and let ${}F(x)$ be a primitive functionMDLD/primitive function for ${}f(x)$. Show that ${}F(x)+cx$ is a primitive function for ${}f(x)+c$.

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Determine a primitive functionMDLD/primitive function for

${}f(x)=4x^{2}-3x+2\,,$

whose value at ${}3$ equals ${}5$.

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Compute the definite integralMDLD/definite integral

$\int _{-1}^{4}3x^{2}-5x+6dx$

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Compute the definite integralMDLD/definite integral

$\int _{2}^{5}{\frac {x^{2}+3x-6}{x-1}}dx$

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Compute the area of the surface, which is enclosed by the graphs of ${}f(x)=x^{2}$ and of ${}g(x)={\sqrt {x}}$.

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Let ${}a$ be the minimal positive number fulfilling ${}\sin a=\cos a$. Compute the area of the surface, which is enclosed by the graph of the cosine function and the graph of the sine function above ${}[0,a]$.

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Compute the definite integral of the function

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

on ${}[-1,0]$.

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Determine the average value of the square root ${}{\sqrt {x}}$ for ${}x\in [1,4]$. Compare this value with the square root of the arithmetic mean of ${}1$ and ${}4$ and with the arithmetic mean of the square root of ${}1$ and of the square root of ${}4$.

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Show that for every ${}n\in \mathbb {N} _{+}$ the estimate

${}{\frac {1}{n+1}}+{\frac {1}{n+2}}+\cdots +{\frac {1}{2n}}\leq \ln 2\,$

holds. Hint: Consider the function ${}f(x)={\frac {1}{x}}$ on the interval ${}[1,2]$.

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Determine for which ${}a\in \mathbb {R}$ the function

$a\longmapsto \int _{-1}^{2}at^{2}-a^{2}tdt$

has a maximum or a minimum.

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A person wants to sun bath for an hour. The intensity of the sun in the time interval ${}[6,22]$ (in hours) is given by the function

$f\colon [6,22]\longrightarrow \mathbb {R} ,t\longmapsto f(t)=-t^{3}+27t^{2}-120t.$

Determine the starting point for the sun bath in order to get the maximal amount of sun.

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According to recent studies, the student's attention skills during the day are described by the following function

$[8,18]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=-x^{2}+25x-100.$

Here, ${}x$ is the time in hours and ${}y=f(x)$ is the attention,n measured in micro-credit points per second. When should one start a one and a half hour lecture, such that the total attention skills are optimal? How many micro-credit points will be added during this lecture?

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Let ${}g\colon \mathbb {R} \rightarrow \mathbb {R}$ be a differentiable function and let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous function. Prove that the function

${}h(x)=\int _{0}^{g(x)}f(t)dt\,$

is differentiable and determine its derivative.

Exercise Create referencenumber

Let ${}f\colon [0,1]\rightarrow \mathbb {R}$ be a continuous function. Consider the following sequence

${}a_{n}:=\int _{\frac {1}{n+1}}^{\frac {1}{n}}f(t)dt\,.$

Determine whether this sequence converges and, in case, determine its limit.

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Let ${}\sum _{n=1}^{\infty }a_{n}$ be a convergent series with ${}a_{n}\in [0,1]$ for all ${}n\in \mathbb {N}$ and let ${}f\colon [0,1]\rightarrow \mathbb {R}$

be a Riemann-integrable function. Prove that the series

$\sum _{n=1}^{\infty }\int _{0}^{a_{n}}f(x)dx$

is absolutely convergent.

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Let ${}f$ be a Riemann-integrable function on ${}[a,b]$ with

${}f(x)\geq 0\,$

for all ${}x\in [a,b]$. Show that if ${}f$ is continuous at a point ${}c\in [a,b]$ with ${}f(c)>0$, then

${}\int _{a}^{b}f(x)dx>0\,.$

Exercise Create referencenumber

Prove that the equation

${}\int _{0}^{x}e^{t^{2}}dt=1\,$

has exactly one solution ${}x\in [0,1]$.

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Let

$f,g\colon [a,b]\longrightarrow \mathbb {R}$

be two continuous functions such that

${}\int _{a}^{b}f(x)dx=\int _{a}^{b}g(x)dx\,.$

Prove that there exists ${}c\in [a,b]$ such that ${}f(c)=g(c)$.

Exercise Create referencenumber

Let

$f\colon [a,b]\longrightarrow \mathbb {R}$

be a continuousMDLD/continuous (R) function with

${}\int _{a}^{b}f(x)g(x)dx=0\,$

for every continuous function ${}g\colon [a,b]\rightarrow \mathbb {R}$. Show ${}f=0$.

Hand-in-exercises

Exercise (3 marks) Create referencenumber

Compute the definite integral ${}\int _{0}^{8}f(t)dt$, where the function ${}f$ is

${}f(t)={\begin{cases}t+1,{\text{ if }}0\leq t\leq 2\,,\\t^{2}-6t+11,{\text{ if }}2<t\leq 5\,,\\6,{\text{ if }}5<t\leq 6\,,\\-2t+18\,,{\text{ if }}6<t\leq 8\,.\end{cases}}\,$

Exercise (3 marks) Create referencenumber

Compute the definite integralMDLD/definite integral

$\int _{1}^{7}{\frac {x^{3}-2x^{2}-x+5}{x+1}}dx$

Exercise (2 marks) Create referencenumber

Determine the area below the graph of the sine function between ${}0$ and ${}\pi$.

Exercise (3 marks) Create referencenumber

Determine an antiderivativeMDLD/antiderivative (R) for the functionMDLD/function

${\frac {1}{{\sqrt {x}}+{\sqrt {x+1}}}}.$

Exercise (4 marks) Create referencenumber

Compute the area of ​​the surface, which is enclosed by the graphs of the two functions ${}f$ and ${}g$ such that

$f(x)=x^{2}{\text{ and }}g(x)=-2x^{2}+3x+4.$

Exercise (3 marks) Create referencenumber

Let

$f,g\colon [a,b]\longrightarrow \mathbb {R}$

be two continuous functions and let ${}g(t)\geq 0$ for all ${}t\in [a,b]$. Prove that there exists ${}s\in [a,b]$ such that

${}\int _{a}^{b}f(t)g(t)dt=f(s)\int _{a}^{b}g(t)dt\,.$

Footnotes

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