Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 25/en/latex
\setcounter{section}{25}
\zwischenueberschrift{Warm-up-exercises}
\inputexercise
{}
{
Compute the
\definitionsverweis {definite integral}{}{}
\mathdisp {\int_{ 0 }^{ \sqrt{\pi} } x \sin x^2 \, d x} { . }
}
{} {}
In the following exercises, which involve the determination of antiderivative functions, consider an appropriate domain of definition.
\inputexercise
{}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {\tan x} { . }
}
{} {}
\inputexercise
{}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {x^n \cdot \ln x} { . }
}
{} {}
\inputexercise
{}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {e^{\sqrt{x} }} { . }
}
{} {}
\inputexercise
{}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {{ \frac{ x^3 }{ \sqrt[5]{ x^4+2} } }} { . }
}
{} {}
\inputexercise
{}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {{ \frac{ \sin^{ 2 } x }{ \cos^{ 2 } x } }} { . }
}
{} {}
\inputexercise
{}
{
Determine for which
\mathl{a \in \R}{} the function
\mathdisp {a \longmapsto \int_{ -1 }^{ 2 } at^2-a^2t \, d t} { }
has a maximum or a minimum.
}
{} {}
\inputexercise
{}
{
According to recent studies the student's attention skills during the day are described by the following function
\maabbeledisp {} {[8,18]} {\R
} {x} {f(x) = -x^2+25x-100
} {.}
Here $x$ is the time in hours and
\mathl{y=f(x)}{} is the attention 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?
}
{} {}
\inputexercise
{}
{
Let $I$ be a real interval and let
\maabbdisp {f} {I} {\R
} {}
be a continuous function with antiderivative $F$. Let $G$ be an antiderivative of $F$ and let
\mathl{b,c \in \R}{.} Determine an antiderivative of the function
\mathdisp {(bt+c) \cdot f(t)} { . }
}
{} {}
\inputexercise
{}
{
Let
\mathl{n \in \N_+}{.} Determine an antiderivative of the function
\maabbeledisp {} {\R_+} {\R_+
} {x} {x^{1/n}
} {,}
using the antiderivative of $x^n$ and Theorem 25.4.
}
{} {}
\inputexercise
{}
{
Determine an antiderivative of the natural logarithm function using the antiderivative of its inverse function.
}
{} {}
\inputexercise
{}
{
Let
\maabbdisp {f} {[a,b]} {[c,d]
} {}
be a bijective, continuous differentiable function. Prove the formula for the antiderivative of the inverse function by the integral
\mathdisp {\int_a^b f^{-1}(y) dy} { }
using the substitution
\mathl{y=f(x)}{} and then integration by parts.
}
{} {}
\inputexercise
{}
{
Compute by an appropriate substitution an antiderivative of
\mathdisp {\sqrt{3x^2+5x-4}} { . }
}
{} {}
\inputexercise
{}
{
Compute the definite integral of the function
\maabbeledisp {f} {\R} {\R
} {x} {f(x) = 2x^3 +3e^x - \sin x
} {,}
on
\mathl{[-1,0]}{.}
}
{} {}
\inputexercise
{}
{
Compute the definite integral of the function
\maabbeledisp {f} {\R_+} {\R
} {x} {f(x) = \sqrt{x} - { \frac{ 1 }{ \sqrt{x} } } + { \frac{ 1 }{ 2x+3 } } -e^{-x}
} {,}
on
\mathl{[1,4]}{.}
}
{} {}
\zwischenueberschrift{Hand-in-exercises}
\inputexercise
{4}
{
Compute the definite integral
\mathl{\int_{ 0 }^{ 8 } f ( t) \, d t}{,} where the function $f$ is
\mathdisp {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}} { }
}
{} {}
\inputexercise
{3}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {x^3 \cdot \cos x -x^2 \cdot \sin x} { . }
}
{} {}
\inputexercise
{2}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {\arcsin x} { . }
}
{} {}
\inputexercise
{4}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {\sin ( \ln x)} { . }
}
{} {}
\inputexercise
{5}
{
Determine an
\definitionsverweis {antiderivative}{}{}
of the
\definitionsverweis {function}{}{}
\mathdisp {e^x \cdot { \frac{ x^2+1 }{ (x+1)^2 } }} { . }
}
{} {}
\inputexercise
{5}
{
Let $I$ be a real interval and let
\maabbdisp {f} {I} {\R
} {}
be a continuous function with antiderivative $F$. Let $G$ be an antiderivative of $F$ and $H$ an antiderivative of $G$. Let
\mathl{a,b,c \in \R}{.} Determine an antiderivative of the function
\mathdisp {(at^2+bt+c) \cdot f(t)} { . }
}
{} {}
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