 # Thus e x e x 1 x e x 1 x 1 x e x therefore de x d x e

• 59

Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 16 - 20 out of 59 pages.

Thusex(ex1)x=ex(1 + ∆x1)x= ex.Thereforedexdx= ex.(1.11)The function ex(or written as exp(x)) is generally called the natural exponen-tial function, or simply the exponential function. Not only is the exponentialfunction equal to its own derivative, it is the only function (apart from a mul-tiplication constant) that has this property. Because of this, the exponentialfunction plays a central role in mathematics and sciences.
1.3 A Peculiar Number Called e191.3.2 The Natural LogarithmIf ey=x,then by definitiony= logex.The logarithm to the base e is known as the natural logarithm. It appearswith amazing frequency in mathematics and its applications. So we give it aspecial symbol. It is written as lnx. That isy= logex= lnx.Thuselnx=x.Furthermore,ln ex=xln e =x.In this sense, the exponential function and the natural logarithm are inversesof each other.Example 1.3.1.Find the value of ln 10.
201 Complex Numbersfor a very smallt. Since ∆xapproaches zero as a limit, for any fixedx,xxcan certainly be made as small as we wish. Therefore, we can setxx=t, andconcludexx= ln1 +xx.Thus,d(lnx)dx=limx01xln1 +xx=limx01xxx=1x.This in turn meansd (lnx) =dxxordxx= lnx+c,(1.12)wherecis the constant of integration. It is well known that because ofdxn+1dx= (n+ 1)xn,we havexndx=xn+1(n+ 1)+c.This formula holds for all values ofnexcept forn=1, since then thedenominatorn+ 1 is zero. This had been a diﬃcult problem, but now we seethat (1.12) provides the “missing case.”In numerous phenomena, ranging from population growth to the decay ofradioactive material, in which the rate of change of some quantity is propor-tional to the quantity itself. Such phenomenon is governed by the differentialequationdydt=ky,wherekis a constant that is positive ifyis increasing and negative ifyisdecreasing. To solve this equation, we write it asdyy=kdtand then integrate both sides to getlny=kt+c,ory= ekt+c= ektec.Ify0denotes the value ofywhent= 0, theny0= ecandy=y0ekt.This equation is called the law of exponential change.
1.4 The Exponential Function as an Infinite Series211.3.3 Approximate Value of eThe number e is found of such great importance, but what is the numericalvalue of e, which we have, so far, defined as 10(1/2.3025)? We can use our tableof successive square root of 10 to calculate this number. The powers of 10are given in the first column of Table 1.1. If we can find a series of numbersn1, n2, n3, . . .in this column, such that12.3026=n1+n2+n3+· · ·,then1012.3026= 10n1+n2+n3+···= 10n110n210n3· · ·.We can read from the second column of the table 10n1,and 10n2,and 10n3and so on, and multiply them together. Let us do just that.

Course Hero member to access this document

Course Hero member to access this document

End of preview. Want to read all 59 pages?

Course Hero member to access this document

Term
Winter
Professor
N/A
Tags
Leonhard Euler, ei
• • • 