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

Exercises

Exercise

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

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).

Exercise

Determine the second derivative of the function

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

Exercise

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?

Exercise

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

Exercise

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

Exercise

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

Exercise

Determine a primitive function for

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

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

Exercise

Compute the definite integral

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

Exercise

Compute the definite integral

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

Exercise

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

Exercise

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 ${}[0,a]$.

Exercise

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 ${}[-1,0]$.

Exercise

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 ${}4$.

Exercise

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 ${}[1,2]$.

Exercise

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.

Exercise

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.

Exercise

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?

Exercise

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

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.

Exercise

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.

Exercise

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

Prove that the equation

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

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

Exercise

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

Let

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

be a continuous 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)

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)

Compute the definite integral

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

Exercise (2 marks)

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

Exercise (3 marks)

Determine an antiderivative for the function

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

Exercise (4 marks)

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)

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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