Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 28/en/latex
\setcounter{section}{28}
\zwischenueberschrift{Warm-up-exercises}
\inputexercise
{}
{
Let
\mathl{x \in \R}{} and consider the function
\maabbeledisp {f} {\R_+} {\R
} {t} { f(t) = t^x e^{-t}
} {.}
Determine the extrema of this function.
}
{} {}
\inputexercise
{}
{
Prove that for the factorial function the relationship
\mathdisp {\operatorname{Fak} \, { \left({ \frac{ 2k-1 }{ 2 } }\right) } = { \frac{ \prod_{i = 1}^{k} (2i-1) }{ 2^k } } \cdot \sqrt{\pi}} { }
holds.
}
{} {}
\inputexercise
{}
{
a) Prove that for
\mathl{x \geq 1}{} the estimate
\mavergleichskettedisp
{\vergleichskette
{ \int_{ 1 }^{ \infty } t^x e^{-t} \, d t
}
{ \leq} {1
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds.
b)
Prove that the function
\mathl{H(x)}{} defined by
\mathdisp {H(x) = \int_{ 1 }^{ \infty } t^x e^{-t} \, d t} { }
for
\mathl{x \geq 1}{} is increasing.
c) Prove that
\mathl{10! \geq e^{11} +1}{.}
d) Prove that for the factorial function for
\mathl{x \geq 10}{} the estimate
\mathdisp {\operatorname{Fak} \, (x) \geq e^x} { }
holds.
}
{} {}
\inputexercise
{}
{
Solve the initial value problem
\mathdisp {y'= \sin t \text{ with } y(\pi) = 7} { . }
}
{} {}
\inputexercise
{}
{
Solve the initial value problem
\mathdisp {y'= 3t^2-4t+7 \text{ with } y(2) = 5} { . }
}
{} {}
\inputexercise
{}
{
Find all the solutions for the ordinary differential equation
\mathdisp {y'=y} { . }
}
{} {}
\inputexercise
{}
{
Make clear and mathematically clear to yourself that in a location-independent differential equation (i.e. $f(t,y)$ does not depend on $y$) the difference between two solutions \mathkor {} {y_1} {and} {y_2} {}
does not depend on time, that is
\mathl{y_1(t)- y_2(t)}{} is constant.
Show with an example that this may not happen in a time-independent differential equation.
}
{} {}
\zwischenueberschrift{Hand-in-exercises}
\inputexercise
{2}
{
Prove that for the factorial function the relationship
\mathdisp {\operatorname{Fak} \, (x) = \int_{ 0 }^{ 1 } (- \ln t)^x \, d t} { }
holds.
}
{} {}
\inputexercise
{3}
{
Solve the initial value problem
\mathdisp {y'= 3t^3-2t+5 \text{ with } y(3) = 4} { . }
}
{} {}
\inputexercise
{3}
{
Find a solution for the ordinary differential equation
\mathdisp {y'=t + y} { . }
}
{} {}
\inputexercise
{4}
{
Solve the initial value problem
\mathdisp {y'= { \frac{ t^3 }{ t^2+1 } } \text{ with } y(1) = 2} { . }
}
{} {}
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