# 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 $f\colon [0,1]\rightarrow \mathbb {R}$
    $\sum _{n=1}^{\infty }\int _{0}^{a_{n}}f(x)dx$

### 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
* ↑ We mean the area between the graph and the $x$-axis.