PP 7.3 - e dx du e e dx d x u u u u u u and then ), ( If...

Info iconThis preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
The Natural Exponential Function
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
y = ln(x) y = exp(x) y = x
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 8
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.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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,
Background image of page 10
With the Chain Rule + = = = C e du
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

Page1 / 14

PP 7.3 - e dx du e e dx d x u u u u u u and then ), ( If...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online