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 $g:ℝ\to ℝ$ be a differentiable function and let $f:ℝ\to ℝ$ 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 $f:[0,1]\to ℝ$ 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 $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ be a convergent series with $a_n\in [0,1]$ for all $n\in \mathbb{N}$ and let $f:[0,1]\to ℝ$

be a Riemann-integrable function. Prove that the series

$\sum _{n=1}^{\mathrm{\infty }}{\int }_0^{a_n}f(x)dx$

is absolutely convergent.

Exercise

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

$f(x)\ge 0$

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

$f,g:[a,b]⟶ℝ$

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

$f:ℝ⟶ℝ,t⟼f(t),$

with

$f(t)=\{\begin{array}{c}0\text{ for }t=0,\\ \sin \frac{1}{t}\text{ for }t\ne 0.\end{array}$

Show, with reference to the function

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

that $f$ has an antiderivative.

Exercise (3 marks)

Let

$f,g:[a,b]⟶ℝ$

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

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