# 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 ${\displaystyle {}30}$ km and in the following three hours, she also makes ${\displaystyle {}30}$ km. What is her average velocity?

### Exercise

Prove the mean value theorem for differential calculus for differentiable functions

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

and a compact interval ${\displaystyle {}[a,b]\subset \mathbb {R} }$, 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

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

### Exercise

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

### Exercise

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

### Exercise

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

### Exercise

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

### Exercise

Determine a primitive function for

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

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

### Exercise

Compute the definite integral

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

### Exercise

Compute the definite integral

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

### Exercise

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

### Exercise

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

### Exercise

Compute the definite integral of the function

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

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

### Exercise

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

### Exercise

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

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

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

### Exercise

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

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

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 ${\displaystyle {}[6,22]}$ (in hours) is given by the function

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

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

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

Here, ${\displaystyle {}x}$ is the time in hours and ${\displaystyle {}y=f(x)}$ 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 ${\displaystyle {}g\colon \mathbb {R} \rightarrow \mathbb {R} }$ be a differentiable function and let ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ be a continuous function. Prove that the function

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

is differentiable and determine its derivative.

### Exercise

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

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

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

### Exercise

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

be a Riemann-integrable function. Prove that the series
${\displaystyle \sum _{n=1}^{\infty }\int _{0}^{a_{n}}f(x)dx}$
is absolutely convergent.

### Exercise

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

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

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

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

### Exercise

Prove that the equation

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

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

### Exercise

Let

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

be two continuous functions such that

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

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

### Exercise

Let

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

be a continuous function with

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

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

Hand-in-exercises

### Exercise (3 marks)

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

${\displaystyle {}f(t)={\begin{cases}t+1,{\text{ if }}0\leq t\leq 2\,,\\t^{2}-6t+11,{\text{ if }}2

### Exercise (3 marks)

Compute the definite integral

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

### Exercise (2 marks)

Determine the area below the graph[1] of the sine function between ${\displaystyle {}0}$ and ${\displaystyle {}\pi }$.

### Exercise (3 marks)

Determine an antiderivative for the function

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

### Exercise (4 marks)

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

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

### Exercise (3 marks)

Let

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

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

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

Footnotes
1. We mean the area between the graph and the ${\displaystyle {}x}$-axis.