exponential_slides - Recall that if f is one-to-one… Note...

Info icon This preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
The natural logarithm is a one-to-one function. A) true B) False Hint: think of the graph of ln(x). Question
Image of page 1

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

View Full Document Right Arrow Icon
Horizontal Line Test. If f is one-to-one, then each horizontal line meets the graph of f in at most one point. ln(x) is one-to-one
Image of page 2
ln(x) is one-to-one ln(x) is invertible Define the exponential function e x to be the inverse of the natural logarithm ln(x) =ln(x) =e x
Image of page 3

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

View Full Document Right Arrow Icon
=ln(x) =e x Note: The inverse function f -1 undoes whatever f does… So we get: ln(e x ) = x e ln(x) = x
Image of page 4
The expression e ln(3a)+ln(6b) is equivalent to: A) 3a+6b B) (3a)(6b) Question
Image of page 5

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

View Full Document Right Arrow Icon
Find all the solution of the equation ln(x 2 -3e 2 )=2. A) ± 2e B) I don’t know Question
Image of page 6
e x e y = e x+y Properties of the exponential function e x /e y = e x-y (e x ) r = e rx r rational
Image of page 7

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

View Full Document Right Arrow Icon
A) 1 B) 2 C) 4 D) 5 The exponential function e x is the inverse of the logarithm ln(x). Identify the graph of e x. Question
Image of page 8
…the graph of f -1 is obtained by flipping the graph of f about the line y=x
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
Image of page 11

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

View Full Document Right Arrow Icon
Image of page 12
Image of page 13

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

View Full Document Right Arrow Icon
Image of page 14
Image of page 15

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

View Full Document Right Arrow Icon
Image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Recall that if f is one-to-one… Note: lim x !" e x = + " lim x !#" e x = y=e x B) -Find the value of value of the limit A) + C) 0 Question lim x ! 2 + e 3/(2 " x ) ! ! B) -Find the value of value of the limit A) + C) 0 Question lim x ! 2 e 3/(2 " x ) 2 ! ! A) B) e x C) ln(x) The exponential function e x is the inverse of the logarithm function f(x)=ln(x). The derivative of ln(x) is f’(x)=1/x. Find the derivative of e x =f-1 (x). Hint: d dx f ! 1 ( x ) = 1 f '( f ! 1 ( x )) 1 1/ x = x Question (e x )’=e x • Chain rule: (e f(x) )’=e f(x) f’(x) • Example: (e tan(x) )’=e tan(x) sec 2 (x) The derivative of e x is e x : B) ln(x) A) e x C) 1 Question ! e x dx = ! e x dx = e x 3 ! e x dx = e 3 " e = e 3 " 1. Substitution rule: e f ( x ) ! f '( x ) dx = e f ( x ) + C...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern