54 Exponential Functions: Differentiation and Integration
f x xln is called the natural exponential function and is denoted by f x e 1 x That is, yex if and only if xy ln Properties of the Natural Exponential Function: 1 The domain of f x ex , is f f , and the range is 0,f 2 The function f x ex is continuous, increasing, and one-to-one on its entire domain 3 The graph of f x ex
exp(x) = inverse of ln(x
Last day, we saw that the function f (x) = lnx is one-to-one, with domain (0;1) and range (1 ;1) We can conclude that f (x) has an inverse function which we call the natural exponential function and denote (temorarily) by f 1(x) = exp(x), The de nition of inverse functions gives us the following: y = f 1(x) if and only if x = f (y)
(25 points) f x
1 (25 points) Sketch a graph of the function f(x) = xln(x) x Your work should include: - Domain - Intercepts - Symmetry - Asymptotes (no Slant asymptotes, though) - Intervals of increase/decrease/local max/min - Concavity and inflection points (1) Domain: x>0 (2) No y intercepts, x intercept x= e(f(x) = 0 ,xln(x) x= 0 ,xln(x) = x,ln(x) = 1
Section 109: Applications of Taylor Polynomials f x a
The derivatives of f(x) are f(x) = ln(x) f(3) = ln(3) f0(x) = 1 x f0(3) = 1 3 f00(x) = 1 x2 f00(3) = 1 9 f000(x) = 2 x3 f000(3) = 2 27: So the third degree Taylor polynomial is T 3(x) = ln(3) + 1 3 (x 3) 1 18 (x 3)2 + 1 81 (x 3)3: (b) Use Taylor’s Inequality to estimate the accuracy of the approximation f(x) ˇT 3(x) for 2 x 4 The fourth
AP Calculus BC - College Board
6 The Maclaurin series for ln(l + x) is given by x 2 x 3 x 4 +1X n x--+---+ ·+(-lf -+ · 2 3 4 n On its interval of convergence, this series converges to lo ( 1 + x ) Let f be the function defined, by 'f(x) = x1n(1 + 1} (a) Write the first four nonzero terms and the general term of the Maclaurin series for f +-ti\ · ·r--,/\- ; k
59 Representations of Functions as a Power Series
Given the function f(x) = X1 0 c n(x a)n, whose domain is the interval of convergence Theorem 5 7 If a power series P c n(x a)n has radius R > 0, then f(x) = X1 0 = c 0 + c 1(x a) + c 2(x a)2 + c 3(x a)3 + ::: is di erentiable on (a R;a+ R), and 1 f0(x) = c 1 + 2c 2(x a) + 3c 3(x a)2 + 4c 4(x a)3 + ::: = X nc n(x a)n 1 2 Z f(x) dx = c+ c 0
Approximating functions by Taylor Polynomials
2(x) has the same first and second derivative that f (x) does at the point x = a 4 3 Higher Order Taylor Polynomials We get better and better polynomial approximations by using more derivatives, and getting higher degreed
Absolute Maximum and Minimum - Texas A&M University
2 Find the absolute maximum and absolute minimum values of f on the given interval f(x)=x+ 9 x on [0 2,12] 3 Find the absolute minimum and absolute maximum values of f on the given interval
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exp(x) = inverse of ln(x) Last day, we saw that the functionf(x) = lnxis one-to-one, with domain (0;1) and range (1;1). We can conclude thatf(x) has an inverse function which we call the natural exponential function and denote (temorarily) byf1(x) = exp(x), The denition of inverse functions gives us the following: y=f1(x) if and only ifx=f(y) y= exp(x) if and only ifx= ln(y)The cancellation laws give us: f
1(f(x)) =xandf(f1(x)) =x
exp(lnx) =xand ln(exp(x)) =x: Annette PilkingtonNatural Logarithm and Natural ExponentialGraph of exp(x)
We can draw the graph ofy=exp(x) by re
ecting the graph ofy=ln(x) in the liney=x.He2,2LH2,e2LH1,0LH0,1LHe,1LH1,eLHe-7,-7LH-7,e-7Ly=expHxL=exy=lnHxL-5510-10-55101520Wehave that the graphy= exp(x) is
one-to-one and continuous with domain (1;1) and range (0;1).Note that exp(x)>0 for all values of
x. We see that exp(0) = 1 since ln1 = 0 exp(1) =esince lne= 1; exp(2) =e2since ln(e2) = 2; exp(7) =e7since ln(e7) =7:In fact for any rational numberr, we
have exp(r) =ersince ln(er) =rlne= r; by the laws of Logarithms. Annette PilkingtonNatural Logarithm and Natural ExponentialDenition ofex.
DenitionWhenxis rational or irrational, we deneexto be exp(x). e x= exp(x)Note:This agrees with denitions ofexgiven elsewhere (as limits), since the denition is the same whenxis a rational number and the exponential function is continuous. Restating the above properties given above in light of this new interpretation of the exponential function, we get:Whenf(x) = ln(x),f1(x) =exand
e x=yif and only if lny=xe lnx=xand lnex=xAnnette PilkingtonNatural Logarithm and Natural ExponentialSolving Equations
We can use the formula below to solve equations involving logarithms and exponentials. e lnx=xand lnex=xExampleSolve forxif ln(x+ 1) = 5I Applying the exponential function to both sides of the equation ln(x+ 1) = 5, we get e ln(x+1)=e5IUsing the fact that elnu=u, (with u=x+ 1), we get
x+ 1 =e5;or x=e51:ExampleSolve forxifex4= 10I
Applying the natural logarithm function to both sides of the equation e x4= 10, we get ln(ex4) = ln(10)I Using the fact thatln(eu) =u, (with u=x4) , we get x4 = ln(10);or x= ln(10) + 4:Annette PilkingtonNatural Logarithm and Natural ExponentialLimits
From the graph we see that
lim x!1ex= 0;limx!1ex=1:ExampleFind the limit limx!1ex10ex1.I As it stands, this limit has an indeterminate form since both numerator and denominator approach innity as x! 1I We modify a trick from Calculus 1 and divide (both Numertor and denominator) by the highest power of e xin the denominator. lim x!1e x10ex1= limx!1e x=ex(10ex1)=exI = lim x!1110(1=ex)=110 Annette PilkingtonNatural Logarithm and Natural ExponentialRules of exponentials
The following rules of exponents follow from the rules of logarithms: e x+y=exey;exy=exe y;(ex)y=exy:Proofsee notes for detailsExampleSimplifyex2e2x+1(ex)2.I
e x2e2x+1(ex)2=ex2+2x+1e 2xI =ex2+2x+12x=ex2+1Annette PilkingtonNatural Logarithm and Natural ExponentialDerivatives
ddx ex=exd dx eg(x)=g0(x)eg(x)ProofWe use logarithmic dierentiation. Ify=ex, we have lny=xand dierentiating, we get 1y dydx = 1 ordydx =y=ex. The derivative on the right follows from the chain rule.ExampleFindddx
esin2xIUsing the chain rule, we get
ddx esin2x=esin2xddx sin2xI=esin2x2(sinx)(cosx) = 2(sinx)(cosx)esin2xAnnette PilkingtonNatural Logarithm and Natural Exponential