1 / 18

Aim: How do we differentiate and integrate the exponential function?

Aim: How do we differentiate and integrate the exponential function?. Do Now:. Do Now. The Natural Exponential Function. Natural Exponential Function. f -1 (x) = e x. Characteristics of Natural Log Function. Monotonic - increasing. Domain – (0, ). Range – all reals.

Download Presentation

Aim: How do we differentiate and integrate the exponential function?

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Aim: How do we differentiate and integrate the exponential function? Do Now:

  2. Do Now

  3. The Natural Exponential Function Natural Exponential Function f -1(x) = ex Characteristics of Natural Log Function Monotonic - increasing Domain – (0, ) Range – all reals Has an inverse f -1

  4. Definition of Natural Exponential Function Natural Exponential Function if x is rational f -1(x) = ex ln(ex) = x ln e = x(1) = x Natural Log Function The inverse of the natural logarithmic functionf(x) = ln x is called the natural exponential function and is denoted by f -1(x) = ex. That is, y = ex x = ln y ln ex = x

  5. Properties of Natural Exponential Function Natural Exponential Function f -1(x) = ex Natural Log Function • domain – (-, ); range – (0, ) • continuous, increasing, and 1-to-1 • concave up on its entire domain

  6. e2x= 5/4 Divide both sides by 4 ln e2x= ln 5/4 Property of Equality for Ln functions 2x = ln 5/4 Inverse Property of Logs & Expos Problems Solve 4e2x = 5 to 3 decimal places Check: 4e2(0.112) = 5

  7. apply inverse property ln e = 1 Solving Exponential Equations take ln of both sides solve for x

  8. apply inverse property Solving Log Equations expo both sides solve for x

  9. (ex)2 – 3ex + 2 = 0 Quadratic Form (ex – 2)(ex – 1) = 0 Factor (ex – 2) = 0 (ex – 1) = 0 Set factors equal to zero ex = 2 ex = 1 x = ln 2 x = 0 x = 0.693 x = 0 Complicated Problem Solve e2x – 3ex + 2 = 0 Graph to verify

  10. u = 2x - 1 u = -3/x u’ = 2 u’ = 3/x2 Derivatives of Exponential Functions

  11. exis never 0 Model Problem Find the relative extrema of f(x) = xex x + 1 = 0 x = -1

  12. Model Problem When 2nd derivative equals zero. u = -x2/2; u’ = -x x = ±1

  13. Model Problem For 1980 through 1993, the number y of medical doctors in the U.S. can be modeled by y = 476,260e0.026663t where t = 0 represents 1980. At what rate was the number of M.D.’s changing in 1988? When t in 1st derivativeequals 8.

  14. u = 3x + 1 du/dx = 3;du = 3dx Integrals of Exponential Functions multiple and divide by 3

  15. Model Problem u = -x2 du/dx = -2x  du = -2xdx  xdx = du/2 regroup integrand substitute factor out -5/2

  16. Model Problem u = 1/x u = cosx

  17. Model Problem Find the areas

  18. Aim: How do we differentiate and integrate the exponential function? Do Now:

More Related