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# PP 7.3 - e dx du e e dx d x u u u u u u and then If...

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The Natural Exponential Function

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What we Know y = ln(x) ln(x) is one-to-one Domain = (0, ∞) Range = (-∞, ∞) ln(1) = 0, ln(e) = 1 x = 0 vertical asymptote
The Inverse Function Hence ln(x) has an inverse function: exp(x) = y ↔ln(y) = x Also, exp(ln(x)) = x and ln(exp(x)) = x ln(1) = 0 → exp(0) = 1 ln(e) = 1 → exp(1) = e

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y = ln(x) y = exp(x) y = x
Notes x x x e e x e x e x e = = = = = )) exp(ln( ) exp( 1 ) ln( since ) ln( ) ln( x e x f x x f = = - ) ( then ), ln( ) ( if Hence, 1 function. l exponentia natural the called is x e

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x e x f = ) ( Domain = (-∞, ∞) Range = (0, ∞) 1 0 = e y = 0 asymptote Increasing 0 , all ) ln( ) ln( = = x x e x x e x x 0 lim , lim = = -∞ x x x x e e
Solving Equations Solve each equation. 5 ) 1 ln( . 1 = + x 1 1 5 ) 1 ln( 5 5 - = + = = + e x x e x 5 . 2 2 = + x e 2 ) 5 ln( ) 5 ln( 2 5 2 - = = + = + x x e x

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Laws of Exponents y x y x e e e + = . 1 y x y x e e e - = . 2 xy y x e e = ) ( . 3
Proof of Law 1 y x y x y x e e ab e ab b a y x b y a x e b e a = = = + = + = = = = + ) ln( ) ln( ) ln( ). ln( ), ln( Then . , Let Students should try the proofs of Laws 2 and 3.

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Derivatives and Integrals x x x x e e dx d e y dx dy dx dy y y x e y = = = = = = 1 1 ) ln( C e dx e x x + = Hence,
With the Chain Rule + = = =

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Unformatted text preview: e dx du e e dx d x u u u u u u and then ), ( If Examples ) 7 ( ) ( . 1 2 2 + = x e x f x ) 2 ( ) 7 )( 2 ( ) ( 2 2 2 x e x e x f x x + + = ′ Find the derivative of each function. x x xe x e 2 2 2 2 ) 7 ( 2 + + = )) ln( cos( ) ( . 2 5 x e x f x = )) ln( ( )) ln( sin( ) ( 5 5 x e dx d x e x f x x-= ′ +-= x e x e x e x x x 1 ) ln( ) 5 ( )) ln( sin( 5 5 5 +-= x x x e e x x 1 ) ln( 5 )) ln( sin( 5 5 Examples Evaluate each integral. ∫ dx xe x 2 5 . 1 ∫ ∫ = = = 2 2 5 2 5 , ) 2 ( 2 x u du e dx x e u x C e C e x u + = + = 2 2 5 2 5 ∫ 4 3 . 2 dx e x ∫ ∫ = = = 12 3 1 4 3 3 1 3 , ) 3 ( x u du e dx e u x ) 1 ( 12 3 1 12 3 1-= = e e u...
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PP 7.3 - e dx du e e dx d x u u u u u u and then If...

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