MATH 221

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 30 July 2014

Lecture 6

The exponential function is the function ex x ex x2 such that dexdx=ex ande0=1. Figure out what ex is:

Suppose ex=a0+a1x+ a2x2+a3x3 +a4x4+. Then e0=a0+0+0+0+ =1. So a0=1. dexdx = a1+2a2x+3a3 x2+4a4x3+ ex = a0+a1x+ a2x2+a3x3 +a4x4+ So a1=a0, 2a2=a1, 3a3=a2, 4a4=a3, So a0=1, a1=1, a2=12, a3=13·2, a4=14·3·2, So ex=1+x+x22 +x33·2+ x44·3·2+ x55·4·3·2 +. So ex=1+x+x22! +x33!+x44! +x55!+. Factorials 7! = 7·6·5·4·3·2 ·1=5040, 5! = 5·4·3·2·1=120, 3! = 3·2·1. So e1 = 1+1+12+16+ 124+1120+1720 +=2.7818 e-3 = 1+(-3)+ (-3)22+ (-3)33!+ (-3)44!+ = 1-3+92-276+ 8124-243120+= So we can evaluate ex for any number x.

By the chain rule d(e2+x)dx =e2+xd(2+x)dx =e2+x·1=e2+x and e2+0=e2. What could e2+x be? If e2+x=b0+ b1x+b2x2 +b3x3+ then e2+0=b0+0+0+ =e2. So b0=e2. de2+xdx= b1+2b2x+3b3 x2+4b4x3+c.= e2+x. So b0=b1, 2b2=b1, 3b3=b2, 4b4=b3, So b0=e2, b1=e2, b2=e22, b3=e23·2, b4=e24·3·2, So e2+x = e2+e2x+ e2x22+ e2x33·2+ e2x44·3·2+ = e2 ( 1+x+x22+ x33!+ x44!+ ) = e2ex. Similarly e6342+x=e6342ex and ey+x=eyex Since e-x+x=e-xex and e-x+x=e0=1 we have e-xex=1. So e-x=1ex e10x = ex+x+x+x+x+x+x+x+x+x = ex ex+x+x+x+x+x+x+x+x = ex ex ex+x+x+x+x+x+x+x = ex ex ex ex+x+x+x+x+x+x = =ex ex ex ex ex ex ex ex ex ex = (ex)10. Similarly e6342x = ex+x+x+x+x++x = ex ex ex ex ex ex ex = (ex)6342. In general, (ex)y=exy

Logarithms

If x f f(x) is a function then the inverse function to f is the function that undoes f.

The logarithm lnx is the inverse function to ex. x ex ex ln7 7 ex lnx x 7 ln7 So eln2=2, lne6762=6762, lne7=7, eln7=7, eln(3+2πi)=3+2πi, elnJ=J. Since 1=e0, ln1=0 In general, If y=ex then lny=lnex=x. Since exey=ex+y ln(ab)=ln (elnaelnb)= ln(elna+lnb)= lna+lnb. Since e-x=1ex, ln(1a)=ln (1elna)= ln(e-lna) =-lna. Since enx=(ex)n, ln(an)=ln ((elna)n) =ln(enlna)= nlna.

Find dydx when y=lnx.

Well ey=x. So deydx=dxdx. So eydydx=1. So dydx=1ey =1x. So dlnxdx=1x

Notes and References

These are a typed copy of Lecture 6 from a series of handwritten lecture notes for the class MATH 221 given on September 18, 2000.

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