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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