Calculus Made Easy/Chapter 14

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Let there be a quantity growing in such a way that the increment of its growth, during a given time, shall always be proportional to its own magnitude. This resembles the process of reckoning interest on money at some fixed rate; for the bigger the capital, the bigger the amount of interest on it in a given time.

Now we must distinguish clearly between two cases, in our calculation, according as the calculation is made by what the arithmetic books call “simple interest,” or by what they call “compound interest.” For in the former case the capital remains fixed, while in the latter the interest is added to the capital, which therefore increases by successive additions.

(1) At simple interest. Consider a concrete case. Let the capital at start be £, and let the rate of interest be per cent. per annum. Then the increment to the owner of the capital will be £ every year. Let him go on drawing his interest every year, and hoard it by putting it by in a stocking, or locking it up in his safe. Then, if he goes on for years, by the end of that time he will have received increments of £ each, or £, making, with the original £, a total of £ in all. His property will have doubled itself in years. If the rate of interest had been per cent., he would have had to hoard for years to double his property. If it had been only per cent., he would have had to hoard for years. It is easy to see that if the value of the yearly interest is of the capital, he must go on hoarding for years in order to double his property.

Or, if be the original capital, and the yearly interest is , then, at the end of n years, his property will be


(2) At compound interest. As before, let the owner begin with a capital of £, earning interest at the rate of per cent. per annum; but, instead of hoarding the interest, let it be added to the capital each year, so that the capital grows year by year. Then, at the end of one year, the capital will have grown to £; and in the second year (still at ) this will earn £ interest. He will start the third year with £, and the interest on that will be £. s.; so that he starts the fourth year with £. s., and so on. It is easy to work it out, and find that at the end of the ten years the total capital will have grown to £. s. d. In fact, we see that at the end of each year, each pound will have earned 110 of a pound, and therefore, if this is always added on, each year multiplies the capital by ; and if continued for ten years (which will multiply by this factor ten times over) will multiply the original capital by . Let us put this into symbols. Put for the original capital; for the fraction added on at each of the operations; and for the value of the capital at the end of the operation. Then


But this mode of reckoning compound interest once a year, is really not quite fair; for even during the first year the £ ought to have been growing. At the end of half a year it ought to have been at least £, and it certainly would have been fairer had the interest for the second half of the year been calculated on £. This would be equivalent to calling it per half-year; with operations, therefore, at each of which the capital is multiplied by . If reckoned this way, by the end of ten years the capital would have grown to £. s.; for


But, even so, the process is still not quite fair; for, by the end of the first month, there will be some interest earned; and a half-yearly reckoning assumes that the capital remains stationary for six months at a time. Suppose we divided the year into parts, and reckon a one-per-cent. interest for each tenth of the year. We now have operations lasting over the ten years; or


which works out to £. s.

Even this is not final. Let the ten years be divided into periods, each of of a year; the interest being per cent. for each such period; then


which works out to £. s. d.

Go even more minutely, and divide the ten years into parts, each of a year, with interest at of per cent. Then


which amounts to £. s. d.

Finally, it will be seen that what we are trying to find is in reality the ultimate value of the expression , which, as we see, is greater than ; and which, as we take larger and larger, grows closer and closer to a particular limiting value. However big you make , the value of this expression grows nearer and nearer to the figure

a number never to be forgotten.

Let us take geometrical illustrations of these things. In Fig. 36, stands for the original value. is the whole time during which the value is growing. It is divided into periods, in each of which there is an equal step up. Here is a constant; and if each step up is of the original , then, by such steps, the height is doubled. If we had taken steps,

Fig. 36.

each of half the height shown, at the end the height would still be just doubled. Or such steps, each of of the original height , would suffice to double the height. This is the case of simple interest. Here is growing till it becomes .

In Fig. 37, we have the corresponding illustration of the geometrical progression. Each of the successive ordinates is to be , that is, times as high as its predecessor. The steps up are not equal, because each step up is now of the ordinate at that part of the curve. If we had literally steps, with for the multiplying factor, the final total would be or times the original . But if only we take n sufficiently large (and the corresponding sufficiently small), then the final value to to which unity will grow will be .

Fig. 37.

Epsilon. To this mysterious number etc., the mathematicians have assigned as a symbol the Greek letter (pronounced epsilon). All schoolboys know that the Greek letter (called pi) stands for etc.; but how many of them know that epsilon means ? Yet it is an even more important number than !

What, then, is epsilon?

Suppose we were to let grow at simple interest till it became ; then, if at the same nominal rate of interest, and for the same time, we were to let grow at true compound interest, instead of simple, it would grow to the value epsilon.

This process of growing proportionately, at every instant, to the magnitude at that instant, some people call a logarithmic rate of growing. Unit logarithmic rate of growth is that rate which in unit time will cause to grow to . It might also be called the organic rate of growing: because it is characteristic of organic growth (in certain circumstances) that the increment of the organism in a given time is proportional to the magnitude of the organism itself.

If we take per cent. as the unit of rate, and any fixed period as the unit of time, then the result of letting grow arithmetically at unit rate, for unit time, will be , while the result of letting grow logarithmically at unit rate, for the same time, will be

A little more about Epsilon. We have seen that we require to know what value is reached by the expression , when becomes indefinitely great. Arithmetically, here are tabulated a lot of values (which anybody can calculate out by the help of an ordinary table of logarithms) got by assuming ; ; ; and so on, up to .

It is, however, worth while to find another way of calculating this immensely important figure.

Accordingly, we will avail ourselves of the binomial theorem, and expand the expression in that well-known way.

The binomial theorem gives the rule that

Putting and , we get

Now, if we suppose n to become indefinitely great, say a billion, or a billion billions, then , , and , etc., will all be sensibly equal to ; and then the series becomes

By taking this rapidly convergent series to as many terms as we please, we can work out the sum to any desired point of accuracy. Here is the working for ten terms:

dividing by 1
dividing by 2
dividing by 3
dividing by 4
dividing by 5
dividing by 6
dividing by 7
dividing by 8
dividing by 9

is incommensurable with , and resembles in being an interminable non-recurrent decimal.

The Exponential Series. We shall have need of yet another series.

Let us, again making use of the binomial theorem, expand the expression , which is the same as when we make indefinitely great.

But, when is made indefinitely great, this simplifies down to the following:

This series is called the exponential series.

The great reason why is regarded of importance is that possesses a property, not possessed by any other function of , that when you differentiate it its value remains unchanged; or, in other words, its differential coefficient is the same as itself. This can be instantly seen by differentiating it with respect to , thus:

which is exactly the same as the original series.

Now we might have gone to work the other way, and said: Go to; let us find a function of , such that its differential coefficient is the same as itself. Or, is there any expression, involving only powers of , which is unchanged by differentiation? Accordingly; let us assume as a general expression that


(in which the coefficients , , , etc. will have to be determined), and differentiate it.


Now, if this new expression is really to be the same as that from which it was derived, it is clear that must ; that ; that ; that , etc.

The law of change is therefore that


If, now, we take for the sake of further simplicity, we have


Differentiating it any number of times will give always the same series over again.

If, now, we take the particular case of , and evaluate the series, we shall get simply

and therefore

when ; that is, thus finally demonstrating that

[Note.–How to read exponentials. For the benefit of those who have no tutor at hand it may be of use to state that is read as “epsilon to the eksth power;” or some people read it “exponential eks.” So is read “epsilon to the pee-teeth-power” or “exponential pee tee.” Take some similar expressions:–Thus, is read “epsilon to the minus two power” or “exponential minus two.” is read “epsilon to the minus ay-eksth” or “exponential minus ay-eks.”]

Of course it follows that remains unchanged if differentiated with respect to . Also , which is equal to , will, when differentiated with respect to , be , because a is constant.

Natural or Naperian Logarithms.

Another reason why is important is because it was made by Napier, the inventor of logarithms, the basis of his system. If is the value of , then is the logarithm, to the base , of . Or, if

The two curves plotted in Fig. 38 and 39 represent these equations.

The points calculated are:

For Fig. 38

For Fig. 39

Fig. 38.

Fig. 39.

It will be seen that, though the calculations yield different points for plotting, yet the result is identical. The two equations really mean the same thing.

As many persons who use ordinary logarithms, which are calculated to base instead of base , are unfamiliar with the “natural” logarithms, it may be worth while to say a word about them. The ordinary rule that adding logarithms gives the logarithm of the product still holds good; or


Also the rule of powers holds good;


But as is no longer the basis, one cannot multiply by or by merely adding or to the index. One can change the natural logarithm to the ordinary logarithm simply by multiplying it by ; or


and conversely,


A Useful Tables of “Napierian Logarithms”
(Also called Natural Logarithms or Hyperbolic Logarithms)

Exponential and Logarithmic Equations.

Now let us try our hands at differentiating certain expressions that contain logarithms or exponentials.

Take the equation:


First transform this into


whence, since the differential of with regard to is the original function unchanged (see p. 143),


and, reverting from the inverse to the original function,


Now this is a very curious result. It may be written


Note that is a result that we could never have got by the rule for differentiating powers. That rule (page 25) is to multiply by the power, and reduce the power by . Thus, differentiating gave us ; and differentiating gave . But differentiating does not give us or , because is itself , and is a constant. We shall have to come back to this curious fact that differentiating gives us when we reach the chapter on integrating.

Now, try to differentiate


that is


we have , since the differential of remains .

This gives


hence, reverting to the original function, we get


Next try


First change to natural logarithms by multiplying by the modulus This gives us




The next thing is not quite so simple. Try this:


Taking the logarithm of both sides, we get

Since is a constant, we get


hence, reverting to the original function.


We see that, since

and .

We shall find that whenever we have an expression such as a function of , we always have the differential coefficient of the function of , so that we could have written at once, from ,

Let us now attempt further examples.


(1) . Let ; then .

; ; hence .

Or thus:


(2) . Let ; then .


Or thus:


(3) .


Check by writing .

(4) . .


(For if and , ,


Check by writing .

(5) . Let ; then .


(6) . Let ; then .


(7) .

(8) .

(For if , let and .

Similarly, if , ) and

(9) .

(10) .

(11) . Let .

(12) .

Try now the following exercises.

Exercises XII. (See page 260 for Answers.)

(1) Differentiate .

(2) Find the differential coefficient with respect to of the expression .

(3) if , find .

(4) Show that if  ; .

(5) If , find .


(6) .

(7) .

(8) .

(9) .

(10) .

(11) .

(12) .

(13) It was shown by Lord Kelvin that the speed of signalling through a submarine cable depends on the value of the ratio of the external diameter of the core to the diameter of the enclosed copper wire. If this ratio is called , then the number of signals that can be sent per minute can be expressed by the formula


where is a constant depending on the length and the quality of the materials. Show that if these are given, will be a maximum if .

(15) Differentiate .

(16) Differentiate .

The Logarithmic Curve.

Let us return to the curve which has its successive ordinates in geometrical progression, such as that represented by the equation .

We can see, by putting , that is the initial height of .

Then when


Also, we see that is the numerical value of the ratio between the height of any ordinate and that of the next preceding it. In Fig. 40, we have taken as ; each ordinate being as high as the preceding one.

Fig. 40.

Fig. 41.

If two successive ordinates are related together thus in a constant ratio, their logarithms will have a constant difference; so that, if we should plot out a new curve, Fig. 41, with values of as ordinates, it would be a straight line sloping up by equal steps. In fact, it follows from the equation, that

Now, since is a mere number, and may be written as , it follows that


and the equation takes the new form


The Die-away Curve.

If we were to take as a proper fraction (less than unity), the curve would obviously tend to sink downwards, as in Fig. 42, where each successive ordinate is of the height of the preceding one.

The equation is still


Fig. 42.

but since is less than one, will be a negative quantity, and may be written ; so that , and now our equation for the curve takes the form


The importance of this expression is that, in the case where the independent variable is time, the equation represents the course of a great many physical processes in which something is gradually dying away. Thus, the cooling of a hot body is represented (in Newton’s celebrated “law of cooling”) by the equation


where is the original excess of temperature of a hot body over that of its surroundings, the excess of temperature at the end of time , and is a constant—namely, the constant of decrement, depending on the amount of surface exposed by the body, and on its coefficients of conductivity and emissivity, etc.

A similar formula,


is used to express the charge of an electrified body, originally having a charge , which is leaking away with a constant of decrement ; which constant depends in this case on the capacity of the body and on the resistance of the leakage-path.

Oscillations given to a flexible spring die out after a time; and the dying-out of the amplitude of the motion may be expressed in a similar way.

In fact serves as a die-away factor for all those phenomena in which the rate of decrease is proportional to the magnitude of that which is decreasing; or where, in our usual symbols, is proportional at every moment to the value that y has at that moment. For we have only to inspect the curve, Fig. 42 above, to see that, at every part of it, the slope is proportional to the height ; the curve becoming flatter as grows smaller. In symbols, thus

or ,

and, differentiating, ;

hence ; or, in words, the slope of the curve is downward, and proportional to and to the constant .

We should have got the same result if we had taken the equation in the form

The Time-constant. In the expression for the “die-away factor” , the quantity is the reciprocal of another quantity known as “the time-constant,” which we may denote by the symbol . Then the die-away factor will be written ; and it will be seen, by making that the meaning of (or of ) is that this is the length of time which it takes for the original quantity (called or in the preceding instances) to die away th part—that is to —of its original value.

The values of and are continually required in different branches of physics, and as they are given in very few sets of mathematical tables, some of the values are tabulated here for convenience.

0 1.0000 1.0000 0.0000
0.10 1.1052 0.9048 0.0952
0.20 1.2214 0.8187 0.1813
0.50 1.6487 0.6065 0.3935
0.75 2.1170 0.4724 0.5276
0.90 2.4596 0.4066 0.5934
1.00 2.7183 0.3679 0.6321
1.10 3.0042 0.3329 0.6671
1.20 3.3201 0.3012 0.6988
1.25 3.4903 0.2865 0.7135
1.50 4.4817 0.2231 0.7769
1.75 5.755 0.1738 0.8262
2.00 7.389 0.1353 0.8647
2.50 12.182 0.0821 0.9179
3.00 20.086 0.0498 0.9502
3.50 33.115 0.0302 0.9698
4.00 54.598 0.0183 0.9817
4.50 90.017 0.0111 0.9889
5.00 148.41 0.0067 0.9933
5.50 244.69 0.0041 0.9959
6.00 403.43 0.00248 0.99752
7.50 1808.04 0.00055 0.99947
10.00 22026.5 0.000045 0.999955

As an example of the use of this table, suppose there is a hot body cooling, and that at the beginning of the experiment (i.e. when ) it is hotter than the surrounding objects, and if the time-constant of its cooling is minutes (that is, if it takes minutes for its excess of temperature to fall to part of ), then we can calculate to what it will have fallen in any given time . For instance, let be minutes. Then , and we shall have to find the value of , and then multiply the original by this. The table shows that is . So that at the end of minutes the excess of temperature will have fallen to .

Further Examples.

(1) The strength of an electric current in a conductor at a time secs. after the application of the electromotive force producing it is given by the expression .

The time constant is .

If , , ; then when is very large the term becomes , and ; also


Its value at any time may be written:


the time-constant being . This means that it takes sec. for the variable term to fall by of its initial value .

To find the value of the current when sec., say, , (from table).

It follows that, after sec., the variable term is , and the actual current is .

Similarly, at the end of sec.,


the variable term is , the current being .

(2) The intensity of a beam of light which has passed through a thickness cm. of some transparent medium is , where is the initial intensity of the beam and is a “constant of absorption.”

This constant is usually found by experiments. If it be found, for instance, that a beam of light has its intensity diminished by in passing through 10 cms. of a certain transparent medium, this means that or , and from the table one sees that very nearly; hence .

To find the thickness that will reduce the intensity to half its value, one must find the value of which satisfies the equality , or . It is found by putting this equation in its logarithmic form, namely,


which gives

centimetres nearly.

(3) The quantity of a radio-active substance which has not yet undergone transformation is known to be related to the initial quantity of the substance by the relation , where is a constant and the time in seconds elapsed since the transformation began.

For “Radium ,” if time is expressed in seconds, experiment shows that . Find the time required for transforming half the substance. (This time is called the “mean life” of the substance.)

We have .

and minutes very nearly.

Exercises XIII. (See page 260 for Answers.)

(1) Draw the curve ; where , , and is given various values from to .

(2) If a hot body cools so that in minutes its excess of temperature has fallen to half the initial amount, deduce the time-constant, and find how long it will be in cooling down to per cent. of the original excess.

(3) Plot the curve .

(4) The following equations give very similar curves:

Draw all three curves, taking millimetres; millimetres.

(5) Find the differential coefficient of with respect to , if

(a) ; (b) ; (c) .

(6) For “Thorium ,” the value of is ; find the “mean life,” that is, the time taken by the transformation of a quantity of “Thorium ” equal to half the initial quantity in the expression


being in seconds.

(7) A condenser of capacity , charged to a potential , is discharging through a resistance of ohms. Find the potential after (a) second; (b) second; assuming that the fall of potential follows the rule .

(8) The charge of an electrified insulated metal sphere is reduced from to units in minutes. Find the coefficient of leakage, if ; being the initial charge and being in seconds. Hence find the time taken by half the charge to leak away.

(9) The damping on a telephone line can be ascertained from the relation , where is the strength, after seconds, of a telephonic current of initial strength ; is the length of the line in kilometres, and is a constant. For the Franco-English submarine cable laid in , . Find the damping at the end of the cable ( kilometres), and the length along which is still of the original current (limiting value of very good audition).

(10) The pressure of the atmosphere at an altitude kilometres is given by ; being the pressure at sea-level ( millimetres).

The pressures at , and kilometres being , , respectively, find in each case. Using the mean value of , find the percentage error in each case.

(11) Find the minimum or maximum of .

(12) Find the minimum or maximum of .

(13) Find the minimum or maximum of .