Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 24

Warm-up-exercises

Exercise

Compute the definite integral

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

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 ${}g\colon \mathbb {R} \rightarrow \mathbb {R}$ 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 ${}f(c)=g(c)$.

Hand-in-exercises

Exercise (2 marks)

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

Exercise (3 marks)

Compute the definite integral

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

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 (4 marks)

We consider the function

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

with

${}f(t)={\begin{cases}0{\text{ for }}t=0,\\\sin {\frac {1}{t}}{\text{ for }}t\neq 0\,.\end{cases}}\,$

Show, with reference to the function

${}g(x)=x^{2}\cos {\frac {1}{x}}\,,$

that ${}f$ has an antiderivative.

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

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