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147. Higher Mean Value Theorems. In the preceding chapter (§ 125) we proved that if has a derivative throughout the interval then
where ; or that, if has a derivative throughout , then(1) |
where . This we proved by considering the function
which vanishes when and when .Let us now suppose that has also a second derivative throughout , an assumption which of course involves the continuity of the first derivative , and consider the function
This function also vanishes when and when ; and its derivative is and this must vanish (§ 121) for some value of between and (exclusive of and ). Hence there is a value of , between and , and therefore capable of representation in the form , where , for whichIf we put we obtain the equation
(2) |
which is the standard form of what may be called the Mean Value Theorem of the second order.
The analogy suggested by (1) and (2) at once leads us to formulate the following theorem:
Taylor’s or the General Mean Value Theorem. If is a function of which has derivatives of the first orders throughout the interval , then
where ; and if then
where .
The proof proceeds on precisely the same lines as were adopted before in the special cases in which and . We consider the function
whereThis function vanishes for and ; its derivative is
and there must be some value of between and for which the derivative vanishes. This leads at once to the desired result.In view of the great importance of this theorem we shall give at the end of this chapter another proof, not essentially distinct from that given above, but different in form and depending on the method of integration by parts.
Examples LV. 1. Suppose that is a polynomial of degree . Then is identically zero when , and the theorem leads to the algebraical identity
2. By applying the theorem to , and supposing and positive, obtain the result
[Since
we can verify the result by showing that can be put in the form , or that , as is evidently the case.]3. Obtain the formula
the corresponding formula for , and similar formulae involving powers of extending up to .
4. Show that if is a positive integer, and a positive integer not greater than , then
Show also that, if the interval does not include , the formula holds for all real values of and all positive integral values of ; and that, even if or , the formula still holds if is positive.5. The formula is not true if and . [For and ; it is evident that the conditions for the truth of the Mean Value Theorem are not satisfied.]
6. If , , , then the equation
is satisfied by . [This example shows that the result of the theorem may hold even if the conditions under which it was proved are not satisfied.]7. Newton’s method of approximation to the roots of equations. Let be an approximation to a root of an algebraical equation , the actual root being . Then
so thatIt follows that in general a better approximation than is
If the root is a simple root, so that , we can, when is small enough, find a positive constant such that for all the values of which we are considering, and then, if is regarded as of the first order of smallness, is of the first order of smallness, and the error in taking as the root is of the second order.8. Apply this process to the equation , taking as the first approximation. [We find , , which is quite a good approximation, in spite of the roughness of the first. If now we repeat the process, taking , we obtain , which is correct to places of decimals.]
9. By considering in this way the equation , where is small, show that approximately, the error being of the fourth order.
10. Show that the error in taking the root to be , where is the argument of every function, is in general of the third order.
11. The equation , where is small, has a root nearly equal to . Show that is a better approximation, and a better still. [The method of Exs. 7–10 does not depend on being an algebraical equation, so long as and are continuous.]
12. Show that the limit when of the number which occurs in the general Mean Value Theorem is , provided that is continuous.
[For is equal to each of
where as well as lies between and . Hence But if we apply the original Mean Value Theorem to the function , taking in place of , we find where also lies between and . Hence from which the result follows, since and tend to the same limit as .]13. Prove that as , provided that is continuous. [Use equation (2) of §147.]
14. Show that, if the is continuous for , then
where and as .7315. Show that if
where and tend to zero as , then , , …, . [Making we see that . Now divide by and afterwards make . We thus obtain ; and this process may be repeated as often as is necessary. It follows that if , and the first derivatives of are continuous, then .]148. Taylor’s Series. Suppose that is a function all of whose differential coefficients are continuous in an interval surrounding the point . Then, if is numerically less than , we have
where , for all values of . Or, if we haveNow let us suppose, in addition, that we can prove that as . Then
This expansion of is known as Taylor’s Series. When the formula reduces to
which is known as Maclaurin’s Series. The function is known as Lagrange’s form of the remainder.The reader should be careful to guard himself against supposing that the continuity of all the derivatives of is a sufficient condition for the validity of Taylor’s series. A direct discussion of the behaviour of is always essential.
Examples LVI. 1. Let . Then all the derivatives of are continuous for all values of . Also for all values of and . Hence in this case , which tends to zero as (Ex. XXVII. 12) whatever value may have. It follows that
for all values of and . In particular for all values of . Similarly we can prove that2. The Binomial Series. Let , where is any rational number, positive or negative. Then and Maclaurin’s Series takes the form
When is a positive integer the series terminates, and we obtain the ordinary formula for the Binomial Theorem with a positive integral exponent. In the general case
and in order to show that Maclaurin’s Series really represents for any range of values of when is not a positive integer, we must show that for every value of in that range. This is so in fact if , and may be proved, when , by means of the expression given above for , since if , and as (Ex. XXVII. 13). But a difficulty arises if , since and if ; knowing only that , we cannot be assured that is not quite small and quite large.In fact, in order to prove the Binomial Theorem by means of Taylor’s Theorem, we need some different form for , such as will be given later (§162).
149. Applications of Taylor’s Theorem. A. Maxima and minima. Taylor’s Theorem may be applied to give greater theoretical completeness to the tests of Ch. VI, §§ 122–123, though the results are not of much practical importance. It will be remembered that, assuming that has derivatives of the first two orders, we stated the following as being sufficient conditions for a maximum or minimum of at : for a maximum, , ; for a minimum, , . It is evident that these tests fail if as well as is zero.
Let us suppose that the first derivatives
are continuous, and that all save the last vanish when . Then, for sufficiently small values of , In order that there should be a maximum or a minimum this expression must be of constant sign for all sufficiently small values of , positive or negative. This evidently requires that should be even. And if is even there will be a maximum or a minimum according as is negative or positive.Thus we obtain the test: if there is to be a maximum or minimum the first derivative which does not vanish must be an even derivative, and there will be a maximum if it is negative, a minimum if it is positive.
Examples LVII. 1. Verify the result when , being a positive integer, and .
2. Test the function , where and are positive integers, for maxima and minima at the points , . Draw graphs of the different possible forms of the curve .
3. Test the functions , , , …, , , , … for maxima or minima at .
150. B. The calculation of certain limits. Suppose that and are two functions of whose derivatives and are continuous for and that and are both equal to zero. Then the function
is not defined when . But of course it may well tend to a limit as .Now
where lies between and ; and similarly , where also lies between and . Thus We must now distinguish four cases.(1) If neither nor is zero, then
(2) If , , then
(3) If , , then becomes numerically very large as : but whether tends to or , or is sometimes large and positive and sometimes large and negative, we cannot say, without further information as to the way in which as .
(4) If , , then we can as yet say nothing about the behaviour of as .
But in either of the last two cases it may happen that and have continuous second derivatives. And then
where again and lie between and ; so that
We can now distinguish a variety of cases similar to those considered above. In particular, if neither second derivative vanishes for , we haveIt is obvious that this argument can be repeated indefinitely, and we obtain the following theorem: suppose that and and their derivatives, so far as may be wanted, are continuous for . Suppose further that and are the first derivatives of and which do not vanish when . Then
(1) if , ;
(3) if , and is even, either or , the sign being the same as that of ;
(4) if and is odd, either or , as , the sign being the same as that of , while if the sign must be reversed.
This theorem is in fact an immediate corollary from the equations
Examples LVIII. 1. Find the limit of
as . [Here the functions and their first derivatives vanish for , and , .]2. Find the limits as of
3. Find the limit of as . [Put .]
4. Prove that
being any integer; and evaluate the corresponding limits involving .5. Find the limits as of
6. , , as .
151. C. The contact of plane curves. Two curves are said to intersect (or cut) at a point if the point lies on each of them. They are said to touch at the point if they have the same tangent at the point.
Let us suppose now that , are two functions which possess derivatives of all orders continuous for , and let us consider the curves , . In general and will not be equal. In this case the abscissa does not correspond to a point of intersection of the curves. If however , the curves intersect in the point , . Let us suppose this to be the case. Then in order that the curves should not only cut but touch at this point it is obviously necessary and sufficient that the first derivatives , should also have the same value when .
The contact of the curves in this case may be regarded from a different point of view. In the figure the two
curves are drawn touching at , and is equal to , or, since , , to
where lies between and . Hence when . In other words, when the curves touch at the point whose abscissa is , the difference of their ordinates at the point whose abscissa is is at least of the second order of smallness when is small.The reader will easily verify that when the curves cut and do not touch, so that is then of the first order of smallness only.
It is evident that the degree of smallness of may be taken as a kind of measure of the closeness of the contact of the curves. It is at once suggested that if the first derivatives of and have equal values when , then will be of th order of smallness; and the reader will have no difficulty in proving that this is so and that
We are therefore led to frame the following definition:Contact of the th order. If , , …, , but , then the curves , will be said to have contact of the th order at the point whose abscissa is .
The preceding discussion makes the notion of contact of the th order dependent on the choice of axes, and fails entirely when the tangent to the curves is parallel to the axis of . We can deal with this case by taking as the independent and as the dependent variable. It is better, however, to consider and as functions of a parameter . An excellent account of the theory will be found in Mr Fowler’s tract referred to on p. 1537, or in de la Vallée Poussin’s Cours d’Analyse, vol. ii, pp. 396 et seq.
Examples LIX. 1. Let , so that is a straight line. The conditions for contact at the point for which are , . If we determine and so as to satisfy these equations we find , , and the equation of the tangent to at the point is
or . Cf. Ex. XXXIX. 5.2. The fact that the line is to have simple contact with the curve completely determines the line. In order that the tangent should have contact of the second order with the curve we must have , i.e. . A point at which the tangent to a curve has contact of the second order is called a point of inflexion.
3. Find the points of inflexion on the graphs of the functions , , , , , .
4. Show that the conic cannot have a point of inflexion. [Here and
suffixes denoting differentiations. Thus at a point of inflexion or or But this is inconsistent with the equation of the conic unless or ; and this is the condition that the conic should degenerate into two straight lines.]5. The curve has one or three points of inflexion according as the roots of are real or complex.
[The equation of the curve can, by a change of origin (cf. Ex. XLVI. 15), be reduced to the form
where , are real or conjugate. The condition for a point of inflexion will be found to be , which has one or three real roots according as is positive or negative, i.e. according as and are real or conjugate.]6. Discuss in particular the curves , , .
7. Show that when the curve of Ex. 5 has three points of inflexion, they lie on a straight line. [The equation can be put in the form , so that the points of inflexion lie on the line or .]
8. Show that the curves , have each infinitely many points of inflexion.
9. Contact of a circle with a curve. Curvature.74 The general equation of a circle, viz.
(1) |
contains three arbitrary constants. Let us attempt to determine them so that the circle has contact of as high an order as possible with the curve at the point , where . We write , for , . Differentiating the equation of the circle twice we obtain
If the circle touches the curve then the equations (1) and (2) are satisfied when , , . This gives . If the contact is of the second order then the equation (3) must also be satisfied when . Thus ; and hence we find
The circle which has contact of the second order with the curve at the point is called the circle of curvature, and its radius the radius of curvature. The measure of curvature (or simply the curvature) is the reciprocal of the radius: thus the measure of curvature is , or
10. Verify that the curvature of a circle is constant and equal to the reciprocal of the radius; and show that the circle is the only curve whose curvature is constant.
11. Find the centre and radius of curvature at any point of the conics , .
12. In an ellipse the radius of curvature at is , where is the semi-diameter conjugate to .
13. Show that in general a conic can be drawn to have contact of the fourth order with the curve at a given point .
[Take the general equation of a conic, viz.
and differentiate four times with respect to . Using suffixes to denote differentiation we obtainIf the conic has contact of the fourth order, then these five equations must be satisfied by writing , , , , , , for , , , , , . We have thus just enough equations to determine the ratios .]
14. An infinity of conics can be drawn having contact of the third order with the curve at . Show that their centres all lie on a straight line.
[Take the tangent and normal as axes. Then the equation of the conic is of the form , and when is small one value of may be expressed (Ch. V, Misc. Ex. 22) in the form
where with . But this expression must be the same as where with , and so , , in virtue of the result of Ex. LV. 15. But the centre lies on the line .]15. Determine a parabola which has contact of the third order with the ellipse at the extremity of the major axis.
16. The locus of the centres of conics which have contact of the third order with the ellipse at the point is the diameter . [For the ellipse itself is one such conic.]
152. Differentiation of functions of several variables. So far we have been concerned exclusively with functions of a single variable , but there is nothing to prevent us applying the notion of differentiation to functions of several variables , , ….
Suppose then that is a function of two75 real variables and , and that the limits
exist for all values of and in question, that is to say that possesses a derivative or with respect to and a derivative or with respect to . It is usual to call these derivatives the partial differential coefficients of , and to denote them by or or simply , or , . The reader must not suppose, however, that these new notations imply any essential novelty of idea: ‘partial differentiation’ with respect to is exactly the same process as ordinary differentiation, the only novelty lying in the presence in of a second variable independent of .In what precedes we have supposed and to be two real variables entirely independent of one another. If and were connected by a relation the state of affairs would be very different. In this case our definition of would fail entirely, as we could not change into without at the same time changing . But then would not really be a function of two variables at all. A function of two variables, as we defined it in Ch. II, is essentially a function of two independent variables. If depends on , is a function of , say ; and then
is really a function of the single variable . Of course we may also represent it as a function of the single variable . Or, as is often most convenient, we may regard and as functions of a third variable , and then , which is of the form , is a function of the single variable .Examples LX. 1. Prove that if , , so that , , then
2. Account for the fact that and . [When we were considering a function of one variable it followed from the definitions that and were reciprocals. This is no longer the case when we are dealing with functions of two variables. Let (Fig. 46) be the point or . To find we must increase , say by an increment , while keeping constant. This brings to . If along we take , the increment of is , say; and . If on the other hand we want to calculate , and
being now regarded as functions of and , we must increase by , say, keeping constant. This brings to , where : the corresponding increment of is , say; and
Now :76 but . Indeed it is easy to see from the figure that but so thatThe fact is of course that and are not formed upon the same hypothesis as to the variation of .]
3. Prove that if then .
4. Find , , … when , . Express , as functions of , and find , , ….
5. Find , … when , , ; express , , in terms of , , and find , ….
[There is of course no difficulty in extending the ideas of the last section to functions of any number of variables. But the reader must be careful to impress on his mind that the notion of the partial derivative of a function of several variables is only determinate when all the independent variables are specified. Thus if , , , and being the independent variables, then . But if we regard as a function of the variables , , and , so that , then .]
153. Differentiation of a function of two functions. There is a theorem concerning the differentiation of a function of one variable, known generally as the Theorem of the Total Differential Coefficient, which is of very great importance and depends on the notions explained in the preceding section regarding functions of two variables. This theorem gives us a rule for differentiating
with respect to .Let us suppose, in the first instance, that is a function of the two variables and , and that , are continuous functions of both variables (§ 107) for all of their values which come in question. And now let us suppose that the variation of and is restricted in that lies on a curve
where and are functions of with continuous differential coefficients , . Then reduces to a function of the single variable , say . The problem is to determine .Suppose that, when changes to , and change to and . Then by definition
But, by the Mean Value Theorem,
where and each lie between and . As , and , and , : also
Hence where we are to put , after carrying out the differentiations with respect to and . This result may also be expressed in the formExamples LXI. 1. Suppose , , so that the locus of is the circle . Then
where and are to be put equal to and after carrying out the differentiations.
We can easily verify this formula in particular cases. Suppose, e.g., that . Then , , and it is easily verified that , which is obviously correct, since .
2. Verify the theorem in the same way when (a) , , ; (b) , , .
3. One of the most important cases is that in which is itself. We then obtain
where is to be replaced by after differentiation.It was this case which led to the introduction of the notation , . For it would seem natural to use the notation for either of the functions and , in one of which is put equal to before and in the other after differentiation. Suppose for example that and . Then , but .
The distinction between the two functions is adequately shown by denoting the first by and the second by , in which case the theorem takes the form
though this notation is also open to objection, in that it is a little misleading to denote the functions and , whose forms as functions of are quite different from one another, by the same letter in and .4. If the result of eliminating between , is , then
5. If and are functions of , and and are the polar coordinates of , then , , dashes denoting differentiations with respect to .
154. The Mean Value Theorem for functions of two variables. Many of the results of the last chapter depended upon the Mean Value Theorem, expressed by the equation
or as it may be written, if ,Now suppose that is a function of the two independent variables and , and that and receive increments , or , respectively: and let us attempt to express the corresponding increment of , viz.
in terms of , and the derivatives of with respect to and .Let . Then
where . But, by § 153,Hence finally
which is the formula desired. Since , are supposed to be continuous functions of and , we havewhere and tend to zero as and tend to zero. Hence the theorem may be written in the form
(1) |
where and are small when and are small.
The result embodied in (1) may be expressed by saying that the equation
is approximately true; i.e. that the difference between the two sides of the equation is small in comparison with the larger of and .77 We must say ‘the larger of and ’ because one of them might be small in comparison with the other; we might indeed have or .It should be observed that if any equation of the form is ‘approximately true’ in this sense, we must have , . For we have
where , , , all tend to zero as and tend to zero; and so where and tend to zero. Hence, if is any assigned positive number, we can choose so that for all values of and numerically less than . Taking we obtain , or , and, as may be as small as we please, this can only be the case if . Similarly .155. Differentials. In the applications of the Calculus, especially in geometry, it is usually most convenient to work with equations expressed not, like equation (1) of § 154, in terms of the increments , , of the functions , , , but in terms of what are called their differentials , , .
Let us return for a moment to a function of a single variable . If is continuous then
(1) |
where as : in other words the equation
(2) |
is ‘approximately’ true. We have up to the present attributed no meaning of any kind to the symbol standing by itself. We now agree to define by the equation
(3) |
If we choose for the particular function , we obtain
(4) |
so that
(5) |
If we divide both sides of (5) by we obtain
(6) |
where denotes not, as heretofore, the differential coefficient of , but the quotient of the differentials , . The symbol thus acquires a double meaning; but there is no inconvenience in this, since (6) is true whichever meaning we choose.
The equation (5) has two apparent advantages over (2). It is exact and not merely approximate, and its truth does not depend on any assumption as to the continuity of . On the other hand it is precisely the fact that we can, under certain conditions, pass from the exact equation (5) to the approximate equation (2), which gives the former its importance. The advantages of the ‘differential’ notation are in reality of a purely technical character. These technical advantages are however so great, especially when we come to deal with functions of several variables, that the use of the notation is almost inevitable.
When is continuous, we have
when . This is sometimes expressed by saying that is the principal part of when is small, just as we might say that is the ‘principal part’ of when is small.We pass now to the corresponding definitions connected with a function of two independent variables and . We define the differential by the equation
(7) |
Putting and in turn, we obtain
which is the exact equation corresponding to the approximate equation (1) of § 154. Here again it is to be observed that the former is of importance only for reasons of practical convenience in working and because the latter can in certain circumstances be deduced from it.
One property of the equation (9) deserves special remark. We saw in §153 that if , and being not independent but functions of a single variable , so that is also a function of alone, then
Multiplying this equation by and observing that we obtain which is the same in form as (9). Thus the formula which expresses in terms of and is the same whether the variables and are independent or not. This remark is of great importance in applications.It should also be observed that if is a function of the two independent variables and , and
then , . This follows at once from the last paragraph of §154.It is obvious that the theorems and definitions of the last three sections are capable of immediate extension to functions of any number of variables.
Examples LXII. 1. The area of an ellipse is given by , where , are the semiaxes. Prove that
and state the corresponding approximate equation connecting the increments of the axes and the area.2. Express , the area of a triangle , as a function of (i) , , , (ii) , , , and (iii) , , , and establish the formulae
where is the radius of the circumcircle.
3. The sides of a triangle vary in such a way that the area remains constant, so that may be regarded as a function of and . Prove that
[This follows from the equations
4. If , , vary so that remains constant, then
and so[Use the formulae , …, and the facts that and are constant.]
5. If is a function of and , which are functions of and , then
[We have
Substitute for and in the first equation and compare the result with the equation6. Let be a function of and , and let , , be defined by the equations
Then may be expressed as a function of and . Express , in terms of , . [Let these differential coefficients be denoted by , and , . Then , or Comparing this equation with we see that7. If
then8. Differentiation of implicit functions. Suppose that and its derivative are continuous in the neighbourhood of the point , and that
Then we can find a neighbourhood of throughout which has always the same sign. Let us suppose, for example, that is positive near . Then is, for any value of sufficiently near to , and for values of sufficiently near to , an increasing function of in the stricter sense of §95. It follows, by the theorem of §108, that there is a unique continuous function which is equal to when and which satisfies the equation for all values of sufficiently near to .Let us now suppose that possesses a derivative which is also continuous near . If , , , we have
where and tend to zero with and . Thus or9. The equation of the tangent to the curve , at the point , , is
156. Definite Integrals and Areas. It will be remembered that, in Ch. VI, § 145, we assumed that, if is a continuous function of , and is the
graph of , then the region shown in Fig. 47 has associated with it a definite number which we call its area. It is clear that, if we denote and by and , and allow to vary, this area is a function of , which we denote by .
Making this assumption, we proved in § 145 that , and we showed how this result might be used in the calculation of the areas of particular curves. But we have still to justify the fundamental assumption that there is such a number as the area .
We know indeed what is meant by the area of a rectangle, and that it is measured by the product of its sides. Also the properties of triangles, parallelograms, and polygons proved by Euclid enable us to attach a definite meaning to the areas of such figures. But nothing which we know so far provides us with a direct definition of the area of a figure bounded by curved lines. We shall now show how to give a definition of which will enable us to prove its existence.78
Let us suppose continuous throughout the interval , and let us divide up the interval into a number of sub-intervals by means of the points of division , , , …, , where
Further, let us denote by the interval , and by the lower bound (§ 102) of in , and let us write say.It is evident that, if is the upper bound of in , then . The aggregate of values of is therefore, in the language of § 80, bounded above, and possesses an upper bound which we will denote by . No value of exceeds , but there are values of which exceed any number less than .
In the same way, if is the upper bound of in , we can define the sum
It is evident that, if is the lower bound of in , then . The aggregate of values of is therefore bounded below, and possesses a lower bound which we will denote by . No value of is less than , but there are values of less than any number greater than .
It will help to make clear the significance of the sums and if we observe that, in the simple case in which increases steadily from to , is and is . In this case is the total area of the rectangles shaded in Fig. 48, and is the
area bounded by a thick line. In general and will still be areas, composed of rectangles, respectively included in and including the curvilinear region whose area we are trying to define.
We shall now show that no sum such as can exceed any sum such as . Let , be the sums corresponding to one mode of subdivision, and , those corresponding to another. We have to show that and .
We can form a third mode of subdivision by taking as dividing points all points which are such for either , or , . Let , be the sums corresponding to this third mode of subdivision. Then it is easy to see that
(1) |
For example, differs from in that at least one interval which occurs in is divided into a number of smaller intervals
so that a term of is replaced in by a sum where , , … are the lower bounds of in , , …. But evidently , , …, so that the sum just written is not less than . Hence and the other inequalities (1) can be established in the same way. But, since , it follows that which is what we wanted to prove.It also follows that . For we can find an as near to as we please and an as near to as we please,79 and so would involve the existence of an and an for which .
So far we have made no use of the fact that is continuous. We shall now show that , and that the sums , tend to the limit when the points of division are multiplied indefinitely in such a way that all the intervals tend to zero. More precisely, we shall show that, given any positive number , it is possible to find so that whenever for all values of .
There is, by Theorem II of § 106, a number such that
whenever every is less than . Hence But and all the three terms on the right-hand side are positive, and therefore all less than . As is a constant, it must be zero. Hence and , , as was to be proved.We define the area of as being the common limit of and , that is to say . It is easy to give a more general form to this definition. Consider the sum
where denotes the value of at any point in . Then plainly lies between and , and so tends to the limit when the intervals tend to zero. We may therefore define the area as the limit of .157. The definite integral. Let us now suppose that is a continuous function, so that the region bounded by the curve , the ordinates and , and the axis of , has a definite area. We proved in Ch. VI, § 145, that if is an ‘integral function’ of , i.e. if
then the area in question is .As it is not always practicable actually to determine the form of , it is convenient to have a formula which represents the area and contains no explicit reference to . We shall write
The expression on the right-hand side of this equation may then be regarded as being defined in either of two ways. We may regard it as simply an abbreviation for , where is some integral function of , whether an actual formula expressing it is known or not; or we may regard it as the value of the area , as directly defined in § 156.
The number
is called a definite integral; and are called its lower and upper limits; is called the subject of integration or integrand; and the interval the range of integration. The definite integral depends on and and the form of the function only, and is not a function of . On the other hand the integral function is sometimes called the indefinite integral of .The distinction between the definite and the indefinite integral is merely one of point of view. The definite integral is a function of , and may be regarded as a particular integral function of . On the other hand the indefinite integral can always be expressed by means of a definite integral, since
But when we are considering ‘indefinite integrals’ or ‘integral functions’ we are usually thinking of a relation between two functions, in virtue of which one is the derivative of the other. And when we are considering a ‘definite integral’ we are not as a rule concerned with any possible variation of the limits. Usually the limits are constants such as and ; and
is not a function at all, but a mere number.It should be observed that the integral , having a differential coefficient , is a fortiori a continuous function of .
Since is continuous for all positive values of , the investigations of the preceding paragraphs supply us with a proof of the actual existence of the function , which we agreed to assume provisionally in §128.
158. Area of a sector of a circle. The circular functions. The theory of the trigonometrical functions , , etc., as usually presented in text-books of elementary trigonometry, rests on an unproved assumption. An angle is the configuration formed by two straight lines , ; there is no particular difficulty in translating this ‘geometrical’ definition into purely analytical terms. The assumption comes at the next stage, when it is assumed that angles are capable of numerical measurement, that is to say
that there is a real number associated with the configuration, just as there is a real number associated with the region of Fig. 47. This point once admitted, and may be defined in the ordinary way, and there is no further difficulty of principle in the elaboration of the theory. The whole difficulty lies in the question, what is the which occurs in and ? To answer this question, we must define the measure of an angle, and we are now in a position to do so. The most natural definition would be this: suppose that is an arc of a circle whose centre is and whose radius is unity, so that . Then , the measure of the angle, is the length of the arc . This is, in substance, the definition adopted in the text-books, in the accounts which they give of the theory of ‘circular measure’. It has however, for our present purpose, a fatal defect; for we have not proved that the arc of a curve, even of a circle, possesses a length. The notion of the length of a curve is capable of precise mathematical analysis just as much as that of an area; but the analysis, although of the same general character as that of the preceding sections, is decidedly more difficult, and it is impossible that we should give any general treatment of the subject here.
We must therefore found our definition on the notion not of length but of area. We define the measure of the angle as twice the area of the sector of the unit circle.
Suppose, in particular, that is and that is , where . The area is a function of , which we may denote by . If we write for , is the point , and we have
Differentiating with respect to , we find Thus the analytical equivalent of our definition would be to define by the equation and the whole theory of the circular functions could be worked out from this starting point, just as the theory of the logarithm is worked out from a similar definition in Ch. IX. See Appendix III.Examples LXIII. Calculation of the definite from the indefinite integral. 1. Show that
and in particular that2. , .
3. , .
[There is an apparent difficulty here owing to the fact that is a many valued function. The difficulty may be avoided by observing that, in the equation
must denote an angle lying between and . For the integral vanishes when and increases steadily and continuously as increases. Thus the same is true of , which therefore tends to as . In the same way we can show that as . Similarly, in the equation where , denotes an angle lying between and . Thus, if and are both numerically less than unity, we have5. if , except when , when the value of the integral is , which is the limit of as .
6. , .
7. , if . [For the form of the indefinite integral see Exs. LIII. 3, 4. If then the subject of integration has an infinity between and . What is the value of the integral when is negative and ?]
8. , if and are positive. What is the value of the integral when and have opposite signs, or when both are negative?
9. Fourier’s integrals. Prove that if and are positive integers then
is always equal to zero, and are equal to zero unless , when each is equal to .10. Prove that and are each equal to zero except when , when each is equal to ; and that
according as is odd or even.159. Calculation of the definite integral from its definition as the limit of a sum. In a few cases we can evaluate a definite integral by direct calculation, starting from the definitions of §§ 156 and 157. As a rule it is much simpler to use the indefinite integral, but the reader will find it instructive to work through a few examples.
Examples LXIV. 1. Evaluate by dividing into equal parts by the points of division , , , …, , and calculating the limit as of
[This sum is
which tends to the limit as . Verify the result by graphical reasoning.]
2. Calculate in the same way.
3. Calculate , where , by dividing into parts by the points of division , , , …, , , where . Apply the same method to the more general integral .
4. Calculate and by the method of Ex. 1.
5. Prove that as .
[This follows from the fact that
which tends to the limit as , in virtue of the direct definition of the integral.]6. Prove that . [The limit is .]
160. General properties of the definite integral. The definite integral possesses the important properties expressed by the following equations.80
This follows at once from the definition of the integral by means of the integral function , since . It should be observed that in the direct definition it was presupposed that the upper limit is greater than the lower; thus this method of definition does not apply to the integral when . If we adopt this definition as fundamental we must extend it to such cases by regarding the equation (1) as a definition of its right-hand side.
The reader will find it an instructive exercise to write out formal proofs of these properties, in each case giving a proof starting from () the definition by means of the integral function and () the direct definition.
The following theorems are also important.
(6) If when , then .
We have only to observe that the sum of §156 cannot be negative. It will be shown later (Misc. Ex. 41) that the value of the integral cannot be zero unless is always equal to zero: this may also be deduced from the second corollary of §121.
(7) If when , then
This follows at once if we apply (6) to
and
.
This follows from (7). For we can take to be the least and the greatest value of in . Then the integral is equal to , where lies between and . But, since is continuous, there must be a value of for which (§100).
If is the integral function, we can write the result of (8) in the form
so that (8) appears now to be only another way of stating the Mean Value Theorem of §125. We may call (8) the First Mean Value Theorem for Integrals.(9) The Generalised Mean Value Theorem for integrals. If is positive, and and are defined as in (7), then
and where is defined as in (8).This follows at once by applying Theorem (6) to the integrals
The reader should formulate for himself the corresponding result which holds when is always negative.(10) The Fundamental Theorem of the Integral Calculus. The function
has a derivative equal to .This has been proved already in § 145, but it is convenient to restate the result here as a formal theorem. It follows as a corollary, as was pointed out in § 157, that is a continuous function of .
Examples LXV. 1. Show, by means of the direct definition of the definite integral, and equations (1)–(5) above, that
being an integer. [The truth of these equations will appear geometrically intuitive, if the graphs of the functions under the sign of integration are sketched.]2. Prove that is equal to or to according as is odd or or even. [Use the formula , the last term being or .]
3. Prove that is equal to or to according as is odd or even.
4. If , and is a positive integer not greater than , then
If then the value of each of the last two integrals is zero. [Use Ex. LXIII. 9.]5. If , and is a positive integer not greater than , then
If then the value of the last integral is zero. [Use Ex. LXIII. 10.]6. Prove that if and are positive then
[Use Ex. LXIII. 8 and Ex. 1 above.]
7. If when , then .
8. Prove that
9.81If then
[The first inequality follows from the fact that , the second from the fact that .]
10. Prove that
11. Prove that if , and hence that
12. Prove that
[Put : then replace by and by .]
13. If and are positive acute angles then
If , then the integral lies between and .14. Prove that
[If is the sum considered at the end of §156, and the corresponding sum formed from the function , then .]
15. If , then
161. Integration by parts and by substitution. It follows from § 138 that
This formula is known as the formula for integration of a definite integral by parts.Again, we know (§ 133) that if is the integral function of , then
Hence, if , , we have which is the formula for the transformation of a definite integral by substitution.The formulae for integration by parts and for transformation often enable us to evaluate a definite integral without the labour of actually finding the integral function of the subject of integration, and sometimes even when the integral function cannot be found. Some instances of this will be found in the following examples. That the value of a definite integral may sometimes be found without a knowledge of the integral function is only to be expected, for the fact that we cannot determine the general form of a function in no way precludes the possibility that we may be able to determine the difference between two of its particular values. But as a rule this can only be effected by the use of more advanced methods than are at present at our disposal.
2. More generally,
where
3. Prove that
4. Prove that if and are positive then
[Integrate by parts and use Ex. LXIII. 8.]
5. If
then[Integrate repeatedly by parts.]
6. Prove by integration by parts that if
where and are positive integers, then , and deduce that7. Prove that if
then . Hence evaluate the integral for all positive integral values of .[Put and integrate by parts.]
8. Deduce from the last example that lies between and .
9. Prove that if
then . [Write for and integrate by parts.]10. Deduce that is equal to
according as is odd or even.11. The Second Mean Value Theorem. If is a function of which has a differential coefficient of constant sign for all values of from to , then there is a number between and such that
[Let . Then
by the generalised Mean Value Theorem of §160: i.e.
which is equivalent to the result given.]12. Bonnet’s form of the Second Mean Value Theorem. If is of constant sign, and and have the same sign, then
where lies between and . [For , where lies between and , and so is the value of for a value of such as . The important case is that in which .]Prove similarly that if and have the same sign, then
where lies between and . [Use the function . It will be found that the integral can be expressed in the form The important case is that in which .]13. Prove that
if . [Apply the first formula of Ex. 12, and note that the integral of over any interval whatever is numerically less than .]14. Establish the results of Ex. LXV. 1 by means of the rule for substitution. [In (i) divide the range of integration into the two parts , , and put in the first. In (ii) use the substitution to obtain the first equation: to obtain the second divide the range into two equal parts and use the substitution . In (iii) divide the range into equal parts and use the substitutions , , ….]
15. Prove that
16. Prove that
17. Prove that
[Put .]
18. Prove that
19. Show by means of the transformation that
20. Show by means of the substitution that
when is a positive integer and , and evaluate the integral when , , .21. If and are positive integers then
[Put , and use Ex. 6.]
162. Proof of Taylor’s Theorem by Integration by Parts. We shall now give the alternative form of the proof of Taylor’s Theorem to which we alluded in § 147.
Let be a function whose first derivatives are continuous, and let
Then
and so If now we write for , and transform the integral by putting , we obtain whereNow, if is any positive integer not greater than , we have, by Theorem (9) of § 160,
where . Hence
If we take we obtain Lagrange’s form of (§ 148). If on the other hand we take we obtain Cauchy’s form, viz.
163. Application of Cauchy’s form to the Binomial Series. If , where is not a positive integer, then Cauchy’s form of the remainder is
Now is less than unity, so long as , whether is positive or negative; and is less than a constant for all values of , being in fact less than if and than if . Hence
say. But as , by Ex. XXVII. 13, and so . The truth of the Binomial Theorem is thus established for all rational values of and all values of between and . It will be remembered that the difficulty in using Lagrange’s form, in Ex. LVI. 2, arose in connection with negative values of .164. Integrals of complex functions of a real variable. So far we have always supposed that the subject of integration in a definite integral is real. We define the integral of a complex function of the real variable , between the limits and , by the equations
and it is evident that the properties of such integrals may be deduced from those of the real integrals already considered.There is one of these properties that we shall make use of later on. It is expressed by the inequality
This inequality may be deduced without difficulty from the definitions of §§ 156 and 157. If has the same meaning as in § 156, and are the values of and at a point of , and , then we haveand so
while The result now follows at once from the inequalityIt is evident that the formulae (1) and (2) of § 162 remain true when is a complex function .
1. Verify the terms given of the following Taylor’s Series:
2. Show that if and its first derivatives are continuous, and , and is the value of which occurs in Lagrange’s form of the remainder after terms of Taylor’s Series, then
where as . [Follow the method of Ex. LV. 12.]3. Verify the last result when . [Here .]
4. Show that if has derivatives of the first three orders then
where . [Apply to the functionarguments similar to those of §147.]
5. Show that under the same conditions
6. Show that if has derivatives of the first five orders then
7. Show that under the same conditions
8. Establish the formulae
where lies between and , and where and lie between the least and greatest of , , . [To prove (ii) consider the function which vanishes when , , and . Its first derivative, by Theorem B of §121, must vanish for two distinct values of lying between the least and greatest of , , ; and its second derivative must therefore vanish for a value of satisfying the same condition. We thus obtain the formula The reader will now complete the proof without difficulty.]9. If is a function which has continuous derivatives of the first orders, of which the first vanish when , and when , then when .
Apply this result to
and deduce Taylor’s Theorem.10. If , , and so on, and has derivatives of the first orders, then
where lies between and . Deduce that if is continuous then as . [This result has been stated already when , in Ex. LV. 13.]11. Deduce from Ex. 10 that as , being any rational number and any positive integer. In particular prove that
12. Suppose that is a function of with continuous derivatives of at least the first four orders, and that , , so that
where as . Establish the formula where as , for that value of which vanishes with ; and prove that as .13. The coordinates of the centre of curvature of the curve , , at the point , are given by
and the radius of curvature of the curve is dashes denoting differentiations with respect to .14. The coordinates of the centre of curvature of the curve , at the point , are given by
15. Prove that the circle of curvature at a point will have contact of the third order with the curve if at that point. Prove also that the circle is the only curve which possesses this property at every point; and that the only points on a conic which possess the property are the extremities of the axes. [Cf. Ch. VI, Misc. Ex. 10 (iv).]
16. The conic of closest contact with the curve , at the origin, is . Deduce that the conic of closest contact at the point of the curve is
where .17. Homogeneous functions.84 If then is unaltered, save for a factor , when , , , … are all increased in the ratio . In these circumstances is called a homogeneous function of degree in the variables , , , …. Prove that if is homogeneous and of degree then
This result is known as Euler’s Theorem on homogeneous functions.18. If is homogeneous and of degree then , , … are homogeneous and of degree .
19. Let be an equation in and (e.g. ), and let be the form it assumes when made homogeneous by the introduction of a third variable in place of unity (e.g. ). Show that the equation of the tangent at the point of the curve is
where , , denote the values of , , when , , .20. Dependent and independent functions. Jacobians or functional determinants. Suppose that and are functions of and connected by an identical relation
(1) |
Differentiating (1) with respect to and , we obtain
(2) |
and, eliminating the derivatives of ,
(3) |
where , , , are the derivatives of and with respect to and . This condition is therefore necessary for the existence of a relation such as (1). It can be proved that the condition is also sufficient; for this we must refer to Goursat’s Cours d’ Analyse, vol. i, pp. 125 et seq.
Two functions and are said to be dependent or independent according as they are or are not connected by such a relation as (1). It is usual to call the Jacobian or functional determinant of and with respect to and , and to write
Similar results hold for functions of any number of variables. Thus three functions , , of three variables , , are or are not connected by a relation according as
does or does not vanish for all values of , , .21. Show that and are independent unless .
22. Show that can be expressed as a product of two linear functions of , , and if and only if
[Write down the condition that and should be connected with the given function by a functional relation.]
23. If and are functions of and , which are themselves functions of and , then
Extend the result to any number of variables.24. Let be a function of whose derivative is and which vanishes when . Show that if , , then , and hence that and are connected by a functional relation. By putting , show that this relation must be . Prove in a similar manner that if the derivative of is , and , then must satisfy the equation
25. Prove that if then
26. Show that if a functional relation exists between
then must be a constant. [The condition for a functional relation will be found to be27. If , , and are connected by a functional relation then is independent of .
28. If , , are the equations of three circles, rendered homogeneous as in Ex. 19, then the equation
represents the circle which cuts them all orthogonally.29. If , , are three functions of such that
vanishes identically, then we can find constants , , such that vanishes identically; and conversely. [The converse is almost obvious. To prove the direct theorem let , …. Then , …, and it follows from the vanishing of the determinant that , …; and so that the ratios are constant. But .]30. Suppose that three variables , , are connected by a relation in virtue of which (i) is a function of and , with derivatives , , and (ii) is a function of and , with derivatives , . Prove that
[We have
The result of substituting for in the first equation is which can be true only if , .]31. Four variables , , , are connected by two relations in virtue of which any two can be expressed as functions of the others. Show that
where denotes the derivative of , when expressed as a function of and , with respect to .32. Find , , , so that the first four derivatives of
vanish when ; and , , , , , so that the first six derivatives of vanish when .33. If , , and , then
the inverse tangent lying between and .8534. Evaluate the integral . For what values of is the integral a discontinuous function of ?
[The value of the integral is if , and if , being any integer; and if is a multiple of .]
35. If when , , and
then according as is positive or negative. In the latter case the inverse tangent lies between and . [It will be found that the substitution reduces the integral to the form .]36. Prove that
37. If then
38. If , , then
where is the positive acute angle whose cosine is .39. If , then
40. Prove that if then
the inverse tangent lying between and .41. If is continuous and never negative, and , then for all values of between and . [If were equal to a positive number when , say, then we could, in virtue of the continuity of , find an interval throughout which ; and then the value of the integral would be greater than .]
42. Schwarz’s inequality for integrals. Prove that
[Use the definitions of §§156 and 157, and the inequality
(Ch. I, Misc. Ex. 10).]43. If
then is a polynomial of degree , which possesses the property that if is any polynomial of degree less than . [Integrate by parts times, where is the degree of , and observe that .]44. Prove that
if , but that if then the value of the integral is .45. If is a polynomial of degree , which possesses the property that
if is any polynomial of degree less than , then is a constant multiple of .[We can choose so that is of degree : then
and so Now apply Ex. 41.]46. Approximate Values of definite integrals. Show that the error in taking as the value of the integral is less than , where is the maximum of in the interval ; and that the error in taking is less than . [Write in Exs. 4 and 5.] Show that the error in taking
as the value is less than , where is the maximum of . [Use Ex. 6. This rule, which gives a very good approximation, is known as Simpson’s Rule. It amounts to taking one-third of the first approximation given above and two-thirds of the second.]Show that the approximation assigned by Simpson’s Rule is the area bounded by the lines , , , and a parabola with its axis parallel to and passing through the three points on the curve whose abscissae are , , .
It should be observed that if is any cubic polynomial then , and Simpson’s Rule is exact. That is to say, given three points whose abscissae are , , , we can draw through them an infinity of curves of the type ; and all such curves give the same area. For one curve , and this curve is a parabola.
47. If is a polynomial of the fifth degree, then
and being the roots of the equation .48. Apply Simpson’s Rule to the calculation of from the formula . [The result is . If we divide the integral into two, from to and to , and apply Simpson’s Rule to the two integrals separately, we obtain . The correct value is .]
49. Show that
50. Calculate the integrals
to two places of decimals. [In the last integral the subject of integration is not defined when : but if we assign to it, when , the value , it becomes continuous throughout the range of integration.]up | next | prev | ptail | top |