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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.

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

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  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:

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