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The Exponential Function

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The Exponential Function The Exp onential Function Supp ose the newspap er headlines read The cost of living went up this year Can we translate this information into an equation Let V denote the value of a dollar in terms of the real go o ds it can buy whatever economists mean by that Let the elapsed time t b e measured in years y Then supp ose that V is a function of t V t which function we would like to know explicitly Call now t and let the initial value of the dollar now b e V which we could take to b e if we disregard ination at earlier times Then our news item can b e written V V whereas V y V V This formula can b e rewritten in terms of the changes in the dep endent and indep endent variables V V y V and t y V V t where it is now to b e understo o d that V is measured in dollars and t is measured in years That is the average time rate of change of V is prop ortional to the value of V at the b eginning of the time interval and the constant of prop ortionality is y By y or inverse years we mean the p er year rate of change This is almost like a derivative If only t were innitesimally small it would b e a derivative Since were just trying to describ e the qualitative b ehaviour lets make an approximation assume that t year is close enough to an innitesimal time interval and that the ab ove formula for the ination rate can b e turned into an instantaneous rate of change dV V dt This means that the dollar in your p o cket right now will b e worth only in one second Well this is interesting but we cannot go any further with it until we ask a crucial question What will happ en if this go es on That is supp ose we assume that equation is not just a temp orary situation but represents a consistent and ubiquitous prop erty of the function V t the real value of your dollar bill as a function of time Applying the ddt op erator to b oth sides of Eq gives d d V dV dV d V or dt dt dt dt dt 1 Since our dollar will b e worth less a year from now we should really call it deation 2 The error intro duced by this approximation is not very serious 3 Banks insurance companies trade unions and governments all pretend that they dont assume this but they would all go bankrupt if they didnt assume it But dV dt is given by If we substitute that formula into the ab ove equation we get d V V V dt That is the rate of change of the rate of change is always p ositive or the negative rate of change is getting less negative all the time In general whenever we have a p ositive second derivative of a function as is the case here the curve is concave upwards Similarly if the second derivative were negative the curve would b e concave downwards So by noting the initial value of V which is formally written V but in this case equals and by applying our understanding of the graphical meaning of the rst derivative slop e and the second derivative curvature we can visualize the function V t pretty well It starts out with a maximum downward slop e and then starts to level o as time increases This general trend con tinues indefinitely Note that while the function always decreases it never reaches zero This is b ecause the closer it gets to zero the slower it decreases see Eq This is a very cute feature that makes this function esp ecially fun to imagine over long times We can also apply our analytical understanding to the formulas and for the derivatives every time we take still another derivative the result is still prop ortional to V the constant of prop ortionality just picks up another factor of This is a really neat feature of this function namely that we can write down all its derivatives with almost no eort dV V dt d V V V dt d V V V dt d V V V dt n V d n V for any n n dt This is a pretty nifty function What is it That is can we write it down in terms of familiar things like t t t and so on First note that Eq can b e written in the form n d V n k V where k n dt A simpler version would b e where k giving n d W W n dt 4 A p olitician trying to obfuscate the issue might say The rate of decrease is decreasing W t b eing the function satisfying this criterion We should p erhaps try guring out this simpler problem rst and then come back to V t Lets try expressing W t then as a linear combination of such terms For starters we will try a third order p olynomial ie we allow terms up to t W t a a t a t a t Then dW a a t a t dt follows by simple dierentiation a single word for taking the derivative Now these two equations have similarlo oking righthand sides provided that we pretend not to notice that term in t in the rst one and provided the constants a ob ey the rule a na ie a a a a n n n and a a But we cant really neglect that t term To b e sure its co ecient a is smaller than any of the rest so if we had to neglect anything it might b e the b est choice but were trying to b e precise right How precise Well precise enough In that case would we b e precise enough if we added a term a t preserving the rule ab out co ecients a a No Then how ab out a t And so on No matter how precise an agreement with Eq we demand we can always take enough terms using this pro cedure to achieve the desired precision Even if you demand innite precision we just just take an innite numb er of terms 1 X a n n a t where a n a or a W t n n n n n n Now supp ose we give W t the initial value If we want a dierent initial value we can just multiply the whole series by that value without aecting Eq Well W tells us that a In that case a also and a and a and a and so on If we dene the factorial notation n n n n n read n factorial and dene we can express our function W t very simply 1 n X t W t n n We could also write a more abstract version of this function in terms of a generalized variable x 1 n X x W x n n Lets do this and then dene x k t and set V t V W x Then by the Chain Rule for derivatives dW dx dV V dt dx dt 5 Linear combination means we multiply each term by a simple constant and add them up 6 The Chain Rule for derivatives says that if z is an explicit function of y z y and y is an explicit function of x y x then z is an implicit function of x and its derivative with resp ect to x is given by dz dy dz dx dy dx d and since k t k we have dt dV k V W k V dt By rep eating this we obtain Eq Thus 1 n X k t V t V W k t V n n where k in the present case This is a nice description we can always calculate the value of this function to any desired degree of accuracy by including as many terms as we need until the change pro duced by adding the next term is to o small to worry us But it is a little clumsy to keep writing down such an unwieldy formula every time you want to refer to this function esp ecially if it is going to b e as p opular as we claim After all mathematics is the art of precise abbreviation So we give W x from Eq a sp ecial name the exp onential function which we write as either x exp x or e In FORTRAN it is represented as EXPX It is equal to the numb er e th raised to the x p ower In our case we have x t so that our answer is t V t V e which is a lot easier to write down than Eq x Now the choice of notation e is not arbitrary There are a lot of rules we know how to use on a numb er raised to a p ower One is that x e x e You can easily determine that this rule also works for the denition in Eq The inverse of this function the p ower to which one must raise e to obtain a sp ecied numb er is called the natural logarithm or ln function We write x if W e then x lnW or x x lne A handy application of this denition is the rule x x lny x y e or y exp x lny 7 This is exactly what a scientic hand calculator do es when you push the function key whose name will b e revealed momentarily 8 Now you know which key it is on a calculator Before we return to our original function is there anything more interesting ab out the natural logarithm than that it is the inverse of the exp onential function And what is so allred sp ecial ab out e the base of the natural log Well it can easily b e shown that the derivative of lnx is a very simple and familiar function dlnx dx x This is p erhaps the most useful feature of lnx b ecause it gives us a direct connection b etween the exp onential function and a function whose derivative is x The handy and versatile rule r dx r r x is valid for any value of r including r but it do esnt help us with this task dx Why It also explains what is so sp ecial ab out the numb er e Summary The Exp onential Functions x x Figure The functions e e lnx and x plotted on the same graph over the range from x to x Note that ln is undened There is no nite p ower to which we can raise e and get zero Similarly x is undened at x while x x Also ln b ecause any numb er raised to the zeroth p ower equals you can easily check this against the denitions and ln where any p ositive numb er less than is negative However there is no such thing as the natural logarithm of any negative numb er Our formula for the real value of your dollar as a function of time is the only function which will satisfy the dierential equation from which we started The exp onential function is one of the most useful of all for solving a wide variety of dierential equations For now just rememb er this 9 Watch for this phrase Whenever someone says It can easily b e shown they mean This is p ossible
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