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lecture25

# lecture25 - 7 Asymptotes Lecture 25 8 Translations ex...

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Lecture 25 1 Lecture 25: (Sec. 5.1) Exponential Functions Def. An Exponential function with base b is the function defined by where x is any real number, b > 0 and b 6 = 1. How do we evaluate b x for all x ?

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Lecture 25 2 ex. For f ( x ) = 4 x , find: 1) f ( - 2) 2) f ( 3 2 ) 3) f ( 3) Be sure to notice the difference between these two functions: ex. f ( x ) = 4 x ex. g ( x ) = x 4
Lecture 25 3 Laws of Exponents Let x and y be any real numbers and let a and b be positive. Then, 1) a 0 = 2) a x a y = 3) a x a y = 4) ( a x ) y = 5) ( ab ) x = 6) ( a b ) x = 7) a - x =

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Lecture 25 4 ex. Simplify: 1) ( 4 3 ) - 2 = 2) 25 - 2 3 5 25 1 6 3) 8 2 n +1 ( 1 4 ) n
Lecture 25 5 NOTE: We have another important property of exponents. 8) Let x and y be real numbers and let b > 0, b 6 = 1. If b x = b y , then NOTE: What happens if b = 1? ex. Solve for x : ( 1 27 ) 2 x - 5 = 81 x

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Lecture 25 6 The Graph of an Exponential Function ex. f ( x ) = 2 x ex. f ( x ) = ( 1 2 ) x ex. f ( x ) = 2 - x
Lecture 25 7 Characteristics of Exponential Functions 1) Domain: 1) Domain: 2) Range: 2) Range: 3) x -intercept(s): 3) x -intercept(s): 4) 4) 5) lim x + b x 5) lim x + b - x 6) lim x →-∞ b x 6) lim x →-∞ b -

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Unformatted text preview: 7) Asymptotes: Lecture 25 8 Translations ex. Sketch each function: 1) f ( x ) = 3 x-1 2) f ( x ) = 3-( x +3)-2 Lecture 25 9 The Exponential Function with base e Consider the following table of values (p. 336): m (1 + 1 m ) m 10 2.59374 100 2.70481 1000 2.71692 10,000 2.71815 100,000 2.71827 1,000,000 2.71828 Lecture 25 10 The Base e Def. 1) 2) NOTE: e ≈ Def. the exponential function: Lecture 25 11 The graph of the exponential function : ex. Graph f ( x ) = e x . x y ex. Graph f ( x ) = e-x . Lecture 25 12 NOTE: 1) lim x →∞ e x 2) lim x →-∞ e x 3) lim x →∞ e-x 4) lim x →-∞ e-x Lecture 25 13 ex. Find each horizontal asymptote of the graph of a) f ( x ) = 1 e x-2 b) f ( x ) =-3 4 + e-x...
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