Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19/latex
\setcounter{section}{19}
\zwischenueberschrift{Mean value theorem for integrals}
For a
Riemann-integrable
function
$f \colon [a,b] \rightarrow \R$,
one may consider
\mathdisp {{ \frac{ \int_{ a }^{ b } f ( t) \, d t }{ b-a } }} { }
as the mean height of the function, since this value, multiplied with the length \mathl{b-a}{} of the interval, yields the area below the graph of $f$. The \stichwort {Mean value theorem for definite integrals} {} claims that, for a continuous function, this \stichwort {mean value} {} is in fact obtained by the function somewhere.
\inputfaktbeweis
{Mean value theorem for definite integrals/Riemann/Fact}
{Theorem}
{}
{
\faktsituation {Suppose that $[a,b]$ is a
compact interval,
and let
\mathdisp {f \colon [a,b] \longrightarrow \R} { }
be a
continuous function.}
\faktfolgerung {Then there exists some
\mavergleichskette
{\vergleichskette
{c
}
{ \in }{ [a,b]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that
\mavergleichskettedisp
{\vergleichskette
{ \int_{ a }^{ b } f ( t) \, d t
}
{ =} { f(c)(b-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\faktzusatz {}
}
{
On the
compact interval,
the function $f$ is bounded from above and from below, let
\mathkor {} {m} {and} {M} {}
denote the
minimum
and the
maximum
of the function. Due to
Theorem 11.13
,
they are both obtained. Then, in particular,
\mavergleichskette
{\vergleichskette
{ m
}
{ \leq }{ f(x)
}
{ \leq }{ M
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mavergleichskette
{\vergleichskette
{ x
}
{ \in }{ [a,b]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and so
\mavergleichskettedisp
{\vergleichskette
{ m(b-a)
}
{ \leq} { \int_{ a }^{ b } f ( t) \, d t
}
{ \leq} { M(b-a)
}
{ } {
}
{ } {
}
}
{}{}{.}
Therefore,
\mavergleichskette
{\vergleichskette
{ \int_{ a }^{ b } f ( t) \, d t
}
{ = }{ d (b-a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with some
\mavergleichskette
{\vergleichskette
{d
}
{ \in }{ [m,M]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Due to
the Intermediate value theorem,
there exists a
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ [a,b]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that
\mavergleichskette
{\vergleichskette
{ f(c)
}
{ = }{ d
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
\zwischenueberschrift{The Fundamental theorem of calculus}
It is useful to allow bounds for an integral, where the lower bound is larger than the upper bound. For
\mavergleichskette
{\vergleichskette
{a
}
{ < }{b
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and an integrable function
$f \colon [a,b] \rightarrow \R$,
we define
\mavergleichskettedisp
{\vergleichskette
{ \int_{ b }^{ a } f ( t) \, d t
}
{ \defeq} { -\int_{ a }^{ b } f ( t) \, d t
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
\inputdefinition
{ }
{
Let $I$ denote a real interval, let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
Riemann-integrable function,
and let
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then the function
\mathdisp {I \longrightarrow \R
, x \longmapsto \int_{ a }^{ x } f ( t) \, d t} { , }
}
This function is also called the \stichwort {indefinite integral} {.}
The following statement is called \stichwort {Fundamental theorem of calculus} {.}
\inputfaktbeweis
{Fundamental theorem of calculus/Riemann/Fact}
{Theorem}
{}
{
\faktsituation {Let $I$ denote a real interval, and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
continuous function.
Let
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and let
\mavergleichskettedisp
{\vergleichskette
{ F(x)
}
{ \defeq} { \int_{ a }^{ x } f ( t) \, d t
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
denote the corresponding
integral function.}
\faktfolgerung {Then $F$ is
differentiable,
and the identity
\mavergleichskettedisp
{\vergleichskette
{ F'(x)
}
{ =} { f(x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds for all
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktzusatz {}
}
{
Let $x$ be fixed. The
difference quotient
is
\mavergleichskettedisp
{\vergleichskette
{\frac{F(x+h)-F(x) }{h}
}
{ =} { \frac{1}{h} { \left( \int_{ a }^{ x+h } f ( t) \, d t - \int_{ a }^{ x } f ( t) \, d t \right) }
}
{ =} { \frac{1}{h} \int_{ x }^{ x+h } f ( t) \, d t
}
{ } {
}
{ } {
}
}
{}{}{.}
We have to show that for \mathl{h \rightarrow 0}{,} the
limit
exists and equals \mathl{f(x)}{.} Because of
the Mean value theorem for definite integrals,
for every $h$, there exists a
\mavergleichskette
{\vergleichskette
{c_h
}
{ \in }{ [x,x+h]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with
\mavergleichskettedisp
{\vergleichskette
{ f(c_h) \cdot h
}
{ =} { \int_x^{x+h} f(t)dt
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and therefore
\mavergleichskettedisp
{\vergleichskette
{ f(c_h)
}
{ =} { { \frac{ \int_x^{x+h} f(t)dt }{ h } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For \mathl{h \rightarrow 0}{,} $c_h$ converges to $x$, and because of the continuity of $f$, also \mathl{f(c_h)}{} converges to \mathl{f(x)}{.}
\zwischenueberschrift{Primitive functions}
\inputdefinition
{ }
{
Let
\mavergleichskette
{\vergleichskette
{I
}
{ \subseteq }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote an
interval,
and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
function.
A function
\mathdisp {F \colon I \longrightarrow \R} { }
is called a \definitionswort {primitive function}{} for $f$, if $F$ is
differentiable
on $I$ and if
\mavergleichskette
{\vergleichskette
{ F'(x)
}
{ = }{ f(x)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds for all
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
}
A primitive function is also called an \stichwort {antiderivative} {.} The fundamental theorem of calculus might be rephrased, in connection with Theorem 18.17 , as an existence theorem for primitive functions.
\inputfaktbeweis
{Continuous Function/Primitive function exists/Fact}
{Corollary}
{}
{
\faktsituation {Let $I$ denote a real interval, and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
continuous function.}
\faktfolgerung {Then $f$ has a
primitive function.}
\faktzusatz {}
}
{
Let
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an arbitrary point. Due to
Theorem 18.17
,
there exists the function
\mavergleichskettedisp
{\vergleichskette
{ F(x)
}
{ =} { \int_{ a }^{ x } f ( t) \, d t
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and because of
the Fundamental theorem,
the identity
\mavergleichskette
{\vergleichskette
{ F'(x)
}
{ = }{ f(x)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds. This means that $F$ is a primitive function for $f$.
\inputfaktbeweis
{Interval/Primitive function/Constant difference/Fact}
{Lemma}
{}
{
\faktsituation {Let $I$ denote a real interval, and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
function.}
\faktvoraussetzung {Suppose that
\mathkor {} {F} {and} {G} {}
are
primitive functions
of $f$.}
\faktfolgerung {Then \mathl{F-G}{} is a
constant function.}
\faktzusatz {}
}
{
We have
\mavergleichskettedisp
{\vergleichskette
{ (F-G)'
}
{ =} { F'-G'
}
{ =} { f-f
}
{ =} { 0
}
{ } {
}
}
{}{}{.}
Therefore, due to
Corollary 15.6
,
the difference \mathl{F-G}{} is constant.
The following statement is also a version of the fundamental theorem, it is called the \stichwort {Newton-Leibniz-formula} {.}
\inputfaktbeweis
{Main theorem of calculus/Riemann/Newton-Leibniz-formula/Fact}
{Corollary}
{}
{
\faktsituation {Let $I$ denote a
real interval,
and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
continuous function.}
\faktvoraussetzung {Suppose that $F$ is a
primitive function
for $f$.}
\faktfolgerung {Then for
\mavergleichskette
{\vergleichskette
{a,b
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the identity
\mavergleichskettedisp
{\vergleichskette
{ \int_{ a }^{ b } f ( t) \, d t
}
{ =} { F(b)- F(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds.}
\faktzusatz {}
}
{
Due to
Theorem 18.17
,
the integral exists. With the
integral function
\mavergleichskettedisp
{\vergleichskette
{ G(x)
}
{ \defeq} { \int_{ a }^{ x } f ( t) \, d t
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
we have the relation
\mavergleichskettedisp
{\vergleichskette
{ \int_{ a }^{ b } f ( t) \, d t
}
{ =} { G(b)
}
{ =} { G(b) - G(a)
}
{ } {
}
{ } {
}
}
{}{}{.}
Because of
Theorem 19.3
,
the function $G$ is
differentiable
and
\mavergleichskettedisp
{\vergleichskette
{ G'(x)
}
{ =} { f(x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds. Hence $G$ is a primitive function for $f$. Due to
Lemma 19.6
,
we have
\mavergleichskette
{\vergleichskette
{F(x)
}
{ = }{ G(x)+c
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Therefore,
\mavergleichskettedisp
{\vergleichskette
{ \int_{ a }^{ b } f ( t) \, d t
}
{ =} { G(b) - G(a)
}
{ =} { F(b) - c - F(a) + c
}
{ =} { F(b) -F(a)
}
{ } {
}
}
{}{}{.}
Since a primitive function is only determined up to an additive constant, we sometimes write
\mavergleichskettedisp
{\vergleichskette
{ \int_{ }^{ } f ( t) \, d t
}
{ =} { F+c
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Here $c$ is called a \stichwort {constant of integration} {.} In certain situations, in particular in relation with \stichwort {differential equations} {,} this constant is determined by further conditions.
\inputnotation
{ }
{
Let $I$ denote a
real interval,
and
\mathdisp {F \colon I \longrightarrow \R} { }
a
primitive function
for a function
$f \colon I \rightarrow \R$.
Suppose that
\mavergleichskette
{\vergleichskette
{ a,b
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then one sets
\mavergleichskettedisp
{\vergleichskette
{ F | _{ a } ^{ b }
}
{ \defeq} { F(b) -F(a)
}
{ =} { \int_{ a }^{ b } f ( t) \, d t
}
{ } {
}
{ } {
}
}
}
This notation is basically used for computations, in particular, when we want to determine definite integrals.
Using known results about the derivatives of differentiable functions, we obtain a list of primitive functions for some important functions. In general however, it is difficult to find a primitive function.
The primitive function of \mathl{x^a}{,} where
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{\R_+
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathbed {a \in \R} {}
{a \neq -1} {}
{} {} {} {,}
is \mathl{{ \frac{ 1 }{ a+1 } }x^{a+1}}{.}
\inputbeispiel{}
{
Suppose that the distance between two masses
\zusatzklammer {thought of as mass points} {} {}
\mathkor {} {M} {and} {m} {}
is $R_0$. Because of gravitation, this system contains a certain potential energy. How is this potential energy changing, when we move these masses to a distance
\mavergleichskette
{\vergleichskette
{ R_1
}
{ \geq }{ R_0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{?}
The needed energy is force times path, where the force itself depends on the distance between the masses. Due to the gravitation law, the force, given the distance $r$ between the masses, equals
\mavergleichskettedisp
{\vergleichskette
{ F(r)
}
{ =} { \gamma { \frac{ Mm }{ r^2 } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
where $\gamma$ denotes the constant of gravitation. Therefore, the energy needed to increase the distance from $R_0$ to $R_1$, equals
\mavergleichskettedisp
{\vergleichskette
{E
}
{ =} { \int_{ R_0 }^{ R_1 } \gamma { \frac{ Mm }{ r^2 } } \, d r
}
{ =} { \gamma M m \int_{ R_0 }^{ R_1 } { \frac{ 1 }{ r^2 } } \, d r
}
{ =} { \gamma M m { \left( - { \frac{ 1 }{ r } } | _{ R_0 } ^{ R_1 }\right) }
}
{ =} { \gamma M m { \left( { \frac{ 1 }{ R_0 } } - { \frac{ 1 }{ R_1 } }\right) }
}
}
{}{}{.}
Hence it is possible to assign a value to the difference between the potential energies for the two distances
\mathkor {} {R_0} {and} {R_1} {,}
though it is not possible to assign an absolute value to the potential energy for a given distance.
}
The primitive function of the function \mathl{{ \frac{ 1 }{ x } }}{} is the natural logarithm.
The primitive function of the exponential function is the exponential function itself.
The primitive function of \mathl{\sin x}{} is \mathl{-\cos x}{,} the primitive function of \mathl{\cos x}{} is $\sin x$.
The primitive function of \mathl{{ \frac{ 1 }{ 1+x^2 } }}{} is \mathl{\arctan x}{,} due to Theorem 16.20 .
The primitive function of \mathl{{ \frac{ 1 }{ 1-x^2 } }}{}
\zusatzklammer {for
\mavergleichskette
{\vergleichskette
{ x
}
{ \in }{ {]{-1},1[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {}
is \mathl{{ \frac{ 1 }{ 2 } } \ln { \frac{ 1+x }{ 1-x } }}{,} because we have
\mavergleichskettealign
{\vergleichskettealign
{ { \left( { \frac{ 1 }{ 2 } } \cdot \ln { \frac{ 1+x }{ 1-x } }\right) }^\prime
}
{ =} { { \frac{ 1 }{ 2 } } \cdot { \frac{ 1-x }{ 1+x } } \cdot { \frac{ (1-x)+ (1+x) }{ (1-x)^2 } }
}
{ =} { { \frac{ 1 }{ 2 } } \cdot { \frac{ 2 }{ (1+x)(1-x) } }
}
{ =} { { \frac{ 1 }{ (1-x^2) } }
}
{ } {
}
}
{}
{}{.}
Caution! Integration rules are only applicable for functions, which are defined on the whole interval. In particular, the following is not true
\mavergleichskettedisp
{\vergleichskette
{ \int_{ -a }^{ a } { \frac{ dt }{ t^2 } } \, d t
}
{ =} { - { \frac{ 1 }{ x } } | _{ -a } ^{ a }
}
{ =} { - { \frac{ 1 }{ a } } - { \frac{ 1 }{ a } }
}
{ =} { - { \frac{ 2 }{ a } }
}
{ } {
}
}
{}{}{,}
since we integrate over a point where the function is not defined.
\inputbeispiel{}
{
We consider the function
\mathdisp {f \colon \R \longrightarrow \R
, t \longmapsto f(t)} { , }
given by
\mavergleichskettedisp
{\vergleichskette
{ f(t)
}
{ \defeq} { \begin{cases} 0 \text{ for } t = 0, \\ \frac{1}{t} \sin \frac{1}{t^2} \text{ for } t \neq 0 \, .\end{cases}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
This function is not
Riemann-integrable,
because it it neither bounded from above nor from below. Hence, there exist no
upper step functions
for $f$. However, $f$ still has a
primitive function.
To see this, we consider the function
\mavergleichskettedisp
{\vergleichskette
{ H(t)
}
{ \defeq} { \begin{cases} 0 \text{ for } t = 0, \\ \frac{ t^2}{2} \cos \frac{1}{t^2} \text{ for } t \neq 0 \, .\end{cases}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
This function is
differentiable.
For
\mavergleichskette
{\vergleichskette
{t
}
{ \neq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the derivative is
\mavergleichskettedisp
{\vergleichskette
{ H'(t)
}
{ =} {t \cos \frac{1}{t^2} + \frac{1}{t} \sin \frac{1}{t^2}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For
\mavergleichskette
{\vergleichskette
{t
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the
difference quotient
is
\mavergleichskettedisp
{\vergleichskette
{ \frac{ \frac{h^2}{2} \cos \frac{1}{h^2} }{h}
}
{ =} { \frac{h}{2} \cos \frac{1}{h^2}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For \mathl{h \mapsto 0}{,} the
limit
exists and equals $0$, so that $H$ is differentiable everywhere
\zusatzklammer {but not continuously differentiable} {} {.}
The first summand in $H'$ is
continuous,
and therefore, due to
Theorem 18.17
,
it has a primitive function $G$. Hence \mathl{H - G}{} is a primitive function for $f$. This follows for
\mavergleichskette
{\vergleichskette
{t
}
{ \neq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
from the explicit derivative and for
\mavergleichskette
{\vergleichskette
{t
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
from
\mavergleichskettedisp
{\vergleichskette
{ H'(0)-G'(0)
}
{ =} { 0-0
}
{ =} { 0
}
{ } {
}
{ } {
}
}
{}{}{.}
}
\zwischenueberschrift{Primitive functions for power series}
We recall that the derivative of a convergent power series is obtained by derivating the summands.
\inputfaktbeweisnichtvorgefuehrt
{Convergent power series/R/Primitive function/Fact}
{Lemma}
{}
{
\faktsituation {Let
\mavergleichskette
{\vergleichskette
{f
}
{ = }{ \sum_{n = 0}^\infty a_n x^n
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote a
power series
which
converges
on \mathl{]-r,r[}{.}}
\faktfolgerung {Then the power series
\mathdisp {\sum_{n=1}^\infty \frac{a_{n-1} }{n} x^n} { }
converges also on \mathl{]-r,r[}{,} and represents a
primitive function
for $f$.}
\faktzusatz {}
With the help of this statement, one can sometimes find the Taylor polynomial
\zusatzklammer {or Taylor series} {} {}
of a function by using the Taylor polynomial of the derivative. We give a typical example.
\inputbeispiel{}
{
We would like to determine the Taylor series of the
natural logarithm
in the point $1$. The
derivative
of the natural logarithm equals \mathl{1/x}{,} due to
Corollary 16.6
.
This function has the power series expansion
\mavergleichskettedisp
{\vergleichskette
{ { \frac{ 1 }{ x } }
}
{ =} { \sum_{k = 0}^\infty (-1)^k (x-1)^k
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
due to
Theorem 9.13
\zusatzklammer {which converges for
\mavergleichskette
{\vergleichskette
{ \betrag { x-1 }
}
{ < }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {.}
Therefore, because of
Lemma 19.11
,
the power series expansion of the natural logarithm is
\mathdisp {\sum_{k=1}^\infty { \frac{ (-1)^{k-1} }{ k } } (x-1)^k} { . }
Setting
\mavergleichskette
{\vergleichskette
{z
}
{ = }{ x-1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we may write this series as
\mathdisp {z- { \frac{ z^2 }{ 2 } } + { \frac{ z^3 }{ 3 } } - { \frac{ z^4 }{ 4 } } + { \frac{ z^5 }{ 5 } } - \ldots} { . }
}