Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19
- Mean value theorem for integrals
For a Riemann-integrable function , one may consider
as the mean height of the function, since this value, multiplied with the length of the interval, yields the area below the graph of . The Mean value theorem for definite integrals claims that, for a continuous function, this mean value is in fact obtained by the function somewhere.
Theorem
Suppose that is a compact interval, and let
be a continuous function. Then there exists some such that
Proof
On the compact interval, the function is bounded from above and from below, let and denote the minimum and the maximum of the function. Due to Theorem 11.13 , they are both obtained. Then, in particular, for all , and so
Therefore, with some . Due to the Intermediate value theorem there exists a such that .
- 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 and an integrable function , we define
Definition
Let denote a real interval, let
denote a Riemann-integrable function, and let . Then the function
This function is also called the indefinite integral.
The following statement is called Fundamental theorem of calculus.
Theorem
Let denote a real interval, and let
denote a continuous function. Let , and let
denote the corresponding integral function. Then is differentiable, and the identity
holds for all .
Proof
Let be fixed. The difference quotient is
We have to show that for , the limit exists and equals . Because of the Mean value theorem for definite integrals for every , there exists a with
and therefore
For , converges to , and because of the continuity of , also converges to .
- Primitive functions
Definition
Let denote an interval, and let
denote a function. A function
is called a primitive function for , if is differentiable on and if holds for all
.A primitive function is also called an antiderivative. The fundamental theorem of calculus might be rephrased, in connection with Theorem 18.17 , as an existence theorem for primitive functions.
Corollary
Proof
Let be an arbitrary point. Due to Theorem 18.17 , there exists the function
and because of the Fundamental theorem the identity holds. This means that is a primitive function for .
Lemma
Let denote a real interval, and let
denote a function. Suppose that and are primitive functions of . Then is a constant function.
Proof
The following statement is also a version of the fundamental theorem, it is called the Newton-Leibniz-formula.
Corollary
Let denote a real interval, and let
denote a continuous function. Suppose that is a primitive function for . Then for , the identity
holds.
Proof
Due to Theorem 18.17 , the integral exists. With the integral function
we have the relation
Because of Theorem 19.3 , the function is differentiable and
holds. Hence is a primitive function for . Due to Lemma 19.6 , we have . Therefore,
Since a primitive function is only determined up to an additive constant, we sometimes write
Here is called a constant of integration. In certain situations, in particular in relation with differential equations, this constant is determined by further conditions.
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 , where and , , is .
Example
Suppose that the distance between two masses (thought of as mass points) and is . Because of gravitation, this system contains a certain potential energy. How is this potential energy changing, when we move these masses to a distance ?
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 between the masses, equals
where denotes the constant of gravitation. Therefore, the energy needed to increase the distance from to , equals
Hence it is possible to assign a value to the difference between the potential energies for the two distances and , though it is not possible to assign an absolute value to the potential energy for a given distance.
The primitive function of the function is the natural logarithm.
The primitive function of the exponential function is the exponential function itself.
The primitive function of is , the primitive function of is .
The primitive function of is , due to Theorem 16.20 .
The primitive function of (for ) is , because we have
Caution! Integration rules are only applicable for functions, which are defined on the whole interval. In particular, the following is not true
since we integrate over a point where the function is not defined.
Example
We consider the function
given by
- Failed to parse (unknown function "\begin{cases}"): {\displaystyle {{}} f(t) := \begin{cases} 0 \text{ for } t <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19]] __NOINDEX__ 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 . However, still has a primitive function. To see this, we consider the function
- Failed to parse (unknown function "\begin{cases}"): {\displaystyle {{}} H(t) := \begin{cases} 0 \text{ for } t <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19]] __NOINDEX__ 0, \\ \frac{ t^2}{2} \cos \frac{1}{t^2} \text{ for } t \neq 0 \, .\end{cases} \, }
This function is differentiable. For , the derivative is
For , the difference quotient is
For , the limit exists and equals , so that is differentiable everywhere (but not continuously differentiable). The first summand in is continuous, and therefore, due to Theorem 18.17 , it has a primitive function . Hence is a primitive function for . This follows for from the explicit derivative and for from
- Primitive functions for power series
We recall that the derivative of a convergent power series is obtained by derivating the summands.
Lemma
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle {{}} f = \sum_{n <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19]] __NOINDEX__ 0}^\infty a_n x^n} denote a power series which converges on . Then the power series
converges also on , and represents a primitive function for .
Proof
With the help of this statement, one can sometimes find the Taylor polynomial
(or Taylor series)
of a function by using the Taylor polynomial of the derivative. We give a typical example.
Example
We would like to determine the Taylor series of the natural logarithm in the point . The derivative of the natural logarithm equals , due to Corollary 16.6 . This function has the power series expansion
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle {{}} \frac{ 1 }{ x } = \sum_{k <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 19]] __NOINDEX__ 0}^\infty (-1)^k (x-1)^k \, , }
due to Theorem 9.13 (which converges for ). Therefore, because of Lemma 19.11 , the power series expansion of the natural logarithm is
Setting , we may write this series as
<< | Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >> PDF-version of this lecture Exercise sheet for this lecture (PDF) |
---|