Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19/latex
\setcounter{section}{19}
\subtitle {Mean value theorem for integrals}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {MittelwertsatzDerIntegralrechnung-f_grad5.png} }
\end{center}
\imagetext {} }
\imagelicense { MittelwertsatzDerIntegralrechnung-f grad5.png } {} {Der Mathekernel} {Commons} {CC-by-sa 3.0} {}
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 \keyword {Mean value theorem for definite integrals} {} claims that, for a continuous function, this \keyword {mean value} {} is in fact obtained by the function somewhere.
\inputfactproof
{Mean value theorem for definite integrals/Riemann/Fact}
{Theorem}
{}
{
\factsituation {Suppose that $[a,b]$ is a
compact interval,
and let
\mathdisp {f \colon [a,b] \longrightarrow \R} { }
be a
continuous function.}
\factconclusion {Then there exists some
\mathrelationchain
{\relationchain
{c
}
{ \in }{ [a,b]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that
\mathrelationchaindisplay
{\relationchain
{ \int_{ a }^{ b } f ( t) \, d t
}
{ =} { f(c)(b-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
}
{
On the
compact interval,
the function $f$ is bounded from above and from below, let
\mathcor {} {m} {and} {M} {}
denote the
minimum
and the
maximum
of the function. Due to
Theorem 11.13
,
they are both obtained. Then, in particular,
\mathrelationchain
{\relationchain
{ m
}
{ \leq }{ f(x)
}
{ \leq }{ M
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mathrelationchain
{\relationchain
{ x
}
{ \in }{ [a,b]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and so
\mathrelationchaindisplay
{\relationchain
{ m(b-a)
}
{ \leq} { \int_{ a }^{ b } f ( t) \, d t
}
{ \leq} { M(b-a)
}
{ } {
}
{ } {
}
}
{}{}{.}
Therefore,
\mathrelationchain
{\relationchain
{ \int_{ a }^{ b } f ( t) \, d t
}
{ = }{ d (b-a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with some
\mathrelationchain
{\relationchain
{d
}
{ \in }{ [m,M]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Due to
the Intermediate value theorem,
there exists a
\mathrelationchain
{\relationchain
{ c
}
{ \in }{ [a,b]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that
\mathrelationchain
{\relationchain
{ f(c)
}
{ = }{ d
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
\subtitle {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
\mathrelationchain
{\relationchain
{a
}
{ < }{b
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and an integrable function
$f \colon [a,b] \rightarrow \R$,
we define
\mathrelationchaindisplay
{\relationchain
{ \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
\mathrelationchain
{\relationchain
{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 \keyword {indefinite integral} {.}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {HauptsatzDerInfinitesitimesrechnung-f_grad5.gif} }
\end{center}
\imagetext {The $x$ from the theorem is $x_0$ in the animation, and $x+h$ in the theorem is the moving $x$ in the animation. The moving point $z$ in the animation is a point which exists by the mean value theorem of definite integrals, applied to $x_0$ and $x$.} }
\imagelicense { HauptsatzDerInfinitesimalrechnung-f grad5.gif } {} {DerMathekernel} {Commons} {CC-by-sa 3.0} {}
The following statement is called \keyword {Fundamental theorem of calculus} {.}
\inputfactproof
{Fundamental theorem of calculus/Riemann/Fact}
{Theorem}
{}
{
\factsituation {Let $I$ denote a real interval, and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
continuous function.
Let
\mathrelationchain
{\relationchain
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and let
\mathrelationchaindisplay
{\relationchain
{ F(x)
}
{ \defeq} { \int_{ a }^{ x } f ( t) \, d t
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
denote the corresponding
integral function.}
\factconclusion {Then $F$ is
differentiable,
and the identity
\mathrelationchaindisplay
{\relationchain
{ F'(x)
}
{ =} { f(x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\factextra {}
}
{
Let $x$ be fixed. The
difference quotient
is
\mathrelationchaindisplay
{\relationchain
{\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
\mathrelationchain
{\relationchain
{c_h
}
{ \in }{ [x,x+h]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with
\mathrelationchaindisplay
{\relationchain
{ f(c_h) \cdot h
}
{ =} { \int_x^{x+h} f(t)dt
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and therefore
\mathrelationchaindisplay
{\relationchain
{ 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)}{.}
\subtitle {Primitive functions}
\inputdefinition
{ }
{
Let
\mathrelationchain
{\relationchain
{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 \definitionword {primitive function}{} for $f$, if $F$ is
differentiable
on $I$ and if
\mathrelationchain
{\relationchain
{ F'(x)
}
{ = }{ f(x)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
}
A primitive function is also called an \keyword {antiderivative} {.} The fundamental theorem of calculus might be rephrased, in connection with Theorem 18.17 , as an existence theorem for primitive functions.
\inputfactproof
{Continuous Function/Primitive function exists/Fact}
{Corollary}
{}
{
\factsituation {Let $I$ denote a real interval, and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
continuous function.}
\factconclusion {Then $f$ has a
primitive function.}
\factextra {}
}
{
Let
\mathrelationchain
{\relationchain
{a
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an arbitrary point. Due to
Theorem 18.17
,
there exists the function
\mathrelationchaindisplay
{\relationchain
{ F(x)
}
{ =} { \int_{ a }^{ x } f ( t) \, d t
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and because of
the Fundamental theorem,
the identity
\mathrelationchain
{\relationchain
{ F'(x)
}
{ = }{ f(x)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds. This means that $F$ is a primitive function for $f$.
\inputfactproof
{Interval/Primitive function/Constant difference/Fact}
{Lemma}
{}
{
\factsituation {Let $I$ denote a real interval, and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
function.}
\factcondition {Suppose that
\mathcor {} {F} {and} {G} {}
are
primitive functions
of $f$.}
\factconclusion {Then \mathl{F-G}{} is a
constant function.}
\factextra {}
}
{
We have
\mathrelationchaindisplay
{\relationchain
{ (F-G)'
}
{ =} { F'-G'
}
{ =} { f-f
}
{ =} { 0
}
{ } {
}
}
{}{}{.}
Therefore, due to
Corollary 15.6
,
the difference \mathl{F-G}{} is constant.
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {GodfreyKneller-IsaacNewton-1689.jpg } }
\end{center}
\imagetext {Isaac Newton (1643-1727)} }
\imagelicense { GodfreyKneller-IsaacNewton-1689.jpg } {Godfrey Kneller} {} {Commons} {PD} {}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Gottfried_Wilhelm_Leibniz_c1700.jpg} }
\end{center}
\imagetext {Gottfried Wilhelm Leibniz (1646-1716)} }
\imagelicense { Gottfried Wilhelm Leibniz c1700.jpg } {Johann Friedrich Wentzel d. Ä.} {AndreasPraefcke} {Commons} {PD} {http://archiv.bbaw.de/archiv/archivbestaende/abteilung-sammlungen/gesamtbestand-des-kunstbesitzes/gelehrtengemaelde/gelehrtengetimesde-seiten/VZLOBO-0031.html}
The following statement is also a version of the fundamental theorem, it is called the \keyword {Newton-Leibniz-formula} {.}
\inputfactproof
{Main theorem of calculus/Riemann/Newton-Leibniz-formula/Fact}
{Corollary}
{}
{
\factsituation {Let $I$ denote a
real interval,
and let
\mathdisp {f \colon I \longrightarrow \R} { }
denote a
continuous function.}
\factcondition {Suppose that $F$ is a
primitive function
for $f$.}
\factconclusion {Then for
\mathrelationchain
{\relationchain
{a,b
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the identity
\mathrelationchaindisplay
{\relationchain
{ \int_{ a }^{ b } f ( t) \, d t
}
{ =} { F(b)- F(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds.}
\factextra {}
}
{
Due to
Theorem 18.17
,
the integral exists. With the
integral function
\mathrelationchaindisplay
{\relationchain
{ G(x)
}
{ \defeq} { \int_{ a }^{ x } f ( t) \, d t
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
we have the relation
\mathrelationchaindisplay
{\relationchain
{ \int_{ a }^{ b } f ( t) \, d t
}
{ =} { G(b)
}
{ =} { G(b) - G(a)
}
{ } {
}
{ } {
}
}
{}{}{.}
Because of
Theorem 19.3
,
the function $G$ is
differentiable
and
\mathrelationchaindisplay
{\relationchain
{ G'(x)
}
{ =} { f(x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds. Hence $G$ is a primitive function for $f$. Due to
Lemma 19.6
,
we have
\mathrelationchain
{\relationchain
{F(x)
}
{ = }{ G(x)+c
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Therefore,
\mathrelationchaindisplay
{\relationchain
{ \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
\mathrelationchaindisplay
{\relationchain
{ \int_{ }^{ } f ( t) \, d t
}
{ =} { F+c
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Here $c$ is called a \keyword {constant of integration} {.} In certain situations, in particular in relation with \keyword {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
\mathrelationchain
{\relationchain
{ a,b
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then one sets
\mathrelationchaindisplay
{\relationchain
{ 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
\mathrelationchain
{\relationchain
{x
}
{ \in }{\R_+
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathcond {a \in \R} {}
{a \neq -1} {}
{} {} {} {,}
is \mathl{{ \frac{ 1 }{ a+1 } }x^{a+1}}{.}
\inputexample{}
{
Suppose that the distance between two masses
\extrabracket {thought of as mass points} {} {}
\mathcor {} {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
\mathrelationchain
{\relationchain
{ 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
\mathrelationchaindisplay
{\relationchain
{ 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
\mathrelationchaindisplay
{\relationchain
{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
\mathcor {} {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 (3).
The primitive function of \mathl{{ \frac{ 1 }{ 1-x^2 } }}{}
\extrabracket {for
\mathrelationchain
{\relationchain
{ x
}
{ \in }{ {]{-1},1[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {}
is \mathl{{ \frac{ 1 }{ 2 } } \ln { \frac{ 1+x }{ 1-x } }}{,} because we have
\mathrelationchainalign
{\relationchainalign
{ { \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
\mathrelationchaindisplay
{\relationchain
{ \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.
\inputexample{}
{
We consider the function
\mathdisp {f \colon \R \longrightarrow \R
, t \longmapsto f(t)} { , }
given by
\mathrelationchaindisplay
{\relationchain
{ 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
\mathrelationchaindisplay
{\relationchain
{ 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
\mathrelationchain
{\relationchain
{t
}
{ \neq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the derivative is
\mathrelationchaindisplay
{\relationchain
{ H'(t)
}
{ =} {t \cos \frac{1}{t^2} + \frac{1}{t} \sin \frac{1}{t^2}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For
\mathrelationchain
{\relationchain
{t
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the
difference quotient
is
\mathrelationchaindisplay
{\relationchain
{ \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
\extrabracket {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
\mathrelationchain
{\relationchain
{t
}
{ \neq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
from the explicit derivative and for
\mathrelationchain
{\relationchain
{t
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
from
\mathrelationchaindisplay
{\relationchain
{ H'(0)-G'(0)
}
{ =} { 0-0
}
{ =} { 0
}
{ } {
}
{ } {
}
}
{}{}{.}
}
\subtitle {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}
{}
{
\factsituation {Let
\mathrelationchain
{\relationchain
{f
}
{ = }{ \sum_{n = 0}^\infty a_n x^n
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote a
power series
which
converges
on \mathl{]-r,r[}{.}}
\factconclusion {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 $f$.}
\factextra {}
}
{Convergent power series/R/Primitive function/Fact/Proof
With the help of this statement, one can sometimes find the Taylor polynomial
\extrabracket {or Taylor series} {} {}
of a function by using the Taylor polynomial of the derivative. We give a typical example.
\inputexample{}
{
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
\mathrelationchaindisplay
{\relationchain
{ { \frac{ 1 }{ x } }
}
{ =} { \sum_{k = 0}^\infty (-1)^k (x-1)^k
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
due to
Theorem 9.13
\extrabracket {which converges for
\mathrelationchain
{\relationchain
{ \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
\mathrelationchain
{\relationchain
{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} { . }
}