Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 24/en/latex

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Compute the \definitionsverweis {definite integral}{}{}
\mathdisp {\int_{ 2 }^{ 5 } \frac{x^2+3x-6}{x-1} \, d x} { . }

}
{} {}




\inputexercise
{}
{

Determine the second derivative of the function
\mathdisp {F(x)= \int_{ 0 }^{ x } \sqrt{t^5-t^3+2t} \, d t} { . }

}
{} {}




\inputexercise
{}
{

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
\mathl{v(t)}{} and the distance
\mathl{s(t)}{} as a function of time $t$. After which time the object has traveled $100$ meters?

}
{} {}




\inputexercise
{}
{

Let \maabb {g} {\R} {\R } {} be a differentiable function and let \maabb {f} {\R} {\R } {} be a continuous function. Prove that the function
\mathdisp {h(x)= \int_{ 0 }^{ g(x) } f(t) \, d t} { }
is differentiable and determine its derivative.

}
{} {}




\inputexercise
{}
{

Let \maabb {f} {[0,1]} {\R } {} be a continuous function. Consider the following sequence
\mathdisp {a_n := \int_{ \frac{1}{n+1} }^{ \frac{1}{n} } f(t) \, d t} { . }
Determine whether this sequence converges and, in case, determine its limit.

}
{} {}




\inputexercise
{}
{

Let
\mathl{\sum_{n=1}^{\infty} a_n}{} be a convergent series with
\mathl{a_n \in [0,1]}{} for all
\mathl{n \in \N}{} and let \maabb {f} {[0,1]} {\R } {} be a Riemann-integrable function. Prove that the series
\mathdisp {\sum_{n=1}^{\infty} \int_{0}^{a_n} f(x) dx} { }
is absolutely convergent.

}
{} {}




\inputexercise
{}
{

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
\mathdisp {\int_{a}^{b} f(x)dx >0} { . }

}
{} {}




\inputexercise
{}
{

Prove that the equation
\mathdisp {\int_{0}^{x} e^{t^2} dt =1} { }
has exactly one solution $x \in [0,1]$.

}
{} {}




\inputexercise
{}
{

Let \maabbdisp {f,g} {[a,b]} {\R } {} be two continuous functions such that
\mathdisp {\int_{a}^{b} f(x) dx =\int_{a}^{b} g(x) dx} { . }
Prove that there exists
\mathl{c \in [a,b]}{} such that
\mathl{f(c)=g(c)}{.}

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{2}
{

Determine the area below\zusatzfussnote {Here we mean the area between the graph and the $x$-axis} {.} {} the graph of the sine function between \mathkor {} {0} {and} {\pi} {.}

}
{} {}




\inputexercise
{3}
{

Compute the \definitionsverweis {definite integral}{}{}
\mathdisp {\int_{ 1 }^{ 7 } \frac{x^3-2x^2-x+5}{x+1} \, d x} { . }

}
{} {}




\inputexercise
{3}
{

Determine an \definitionsverweis {antiderivative}{}{} for the \definitionsverweis {function}{}{}
\mathdisp {\frac{1}{\sqrt{x} + \sqrt{x+1} }} { . }

}
{} {}




\inputexercise
{4}
{

Compute the area of ​​the surface, which is enclosed by the graphs of the two functions \mathkor {} {f} {and} {g} {} such that
\mathdisp {f(x)=x^2 \text{ and } g(x)=-2x^2+3x+4} { . }

}
{} {}




\inputexercise
{4}
{

We consider the function \maabbeledisp {f} {\R} {\R } {t} {f(t) } {,} with
\mathdisp {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
\mathl{g(x)=x^2 \cos \frac{1}{x}}{,} that $f$ has an antiderivative.

}
{} {}




\inputexercise
{3}
{

Let \maabbdisp {f,g} {[a,b]} {\R } {} be two continuous functions and let
\mathl{g(t) \geq 0}{} for all
\mathl{t \in [a,b]}{.} Prove that there exists
\mathl{s \in [a,b]}{} such that
\mathdisp {\int_{ a }^{ b } f(t)g(t) \, d t =f(s) \int_{ a }^{ b } g(t) \, d t} { . }

}
{} {}




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