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

is called the integral function for for the starting point .

This function is also called the indefinite integral.

The from the theorem is in the animation, and in the theorem is the moving in the animation. The moving point in the animation is a point which exists by the mean value theorem of definite integrals, applied to and .

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

Let denote a real interval, and let

denote a continuous function. Then has a primitive function.

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  

We have

Therefore, due to Corollary 15.6 , the difference is constant.


Isaac Newton (1643-1727)
Gottfried Wilhelm Leibniz (1646-1716)

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.


Notation  

Let denote a real interval, and

a primitive function for a function . Suppose that . Then one sets

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

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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

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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

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converges also on , and represents a primitive function for .

Proof

This proof was not presented in the lecture.


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

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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


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