# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19/latex

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\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.
\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} {.}
\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.
\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 $f$.
\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} {.}
\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.
\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 } {Convergent power series/R/Primitive function/Fact/Proof
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} { . }