Aim: Differentiating & Integrating Expo Functions Course: Calculus Do Now: Aim: How do we differentiate and integrate the exponential function?

Aim: Differentiating & Integrating Expo Functions Course: Calculus Do Now

Aim: Differentiating & Integrating Expo Functions Course: Calculus The Natural Exponential Function Characteristics of Natural Log Function Monotonic - increasing Domain – (0,  ) Range – all reals Has an inverse f -1 f -1 (x) = e x Natural Exponential Function

Aim: Differentiating & Integrating Expo Functions Course: Calculus Definition of Natural Exponential Function Natural Log Function f -1 (x) = e x Natural Exponential Function The inverse of the natural logarithmic function f(x) = ln x is called the natural exponential function and is denoted by f -1 (x) = e x. That is, y = e x  x = ln y ln(e x ) = x ln e = x(1) = x if x is rational ln e x = x

Aim: Differentiating & Integrating Expo Functions Course: Calculus Properties of Natural Exponential Function Natural Log Function f -1 (x) = e x Natural Exponential Function 1.domain – (- ,  ); range – (0,  ) 2.continuous, increasing, and 1-to-1 3.concave up on its entire domain 4.

Aim: Differentiating & Integrating Expo Functions Course: Calculus Problems Solve 4e 2x = 5 to 3 decimal places ln e 2x = ln 5/4 Property of Equality for Ln functions 2x = ln 5/4 Inverse Property of Logs & Expos e 2x = 5/4 Divide both sides by 4 Check: 4e 2(0.112) = 5

Aim: Differentiating & Integrating Expo Functions Course: Calculus Solving Exponential Equations take ln of both sides solve for x apply inverse property ln e = 1

Aim: Differentiating & Integrating Expo Functions Course: Calculus Solving Log Equations expo both sides solve for x apply inverse property

Aim: Differentiating & Integrating Expo Functions Course: Calculus Complicated Problem Solve e 2x – 3e x + 2 = 0 Quadratic Form (e x ) 2 – 3e x + 2 = 0 Factor (e x – 2)(e x – 1) = 0 Set factors equal to zero (e x – 2) = 0 (e x – 1) = 0 e x = 2 e x = 1 x = ln 2 x = 0 x = x = 0 Graph to verify

Aim: Differentiating & Integrating Expo Functions Course: Calculus Derivatives of Exponential Functions u = 2x - 1 u’ = 2 u = -3/x u’ = 3/x 2

Aim: Differentiating & Integrating Expo Functions Course: Calculus Model Problem Find the relative extrema of f(x) = xe x e x is never 0 x + 1 = 0 x = -1

Aim: Differentiating & Integrating Expo Functions Course: Calculus Model Problem When 2 nd derivative equals zero. u = -x 2 /2; u’ = -x x = ±1

Aim: Differentiating & Integrating Expo Functions Course: Calculus Model Problem For 1980 through 1993, the number y of medical doctors in the U.S. can be modeled by y = 476,260e t where t = 0 represents At what rate was the number of M.D.’s changing in 1988? When t in 1st derivative equals 8.

Aim: Differentiating & Integrating Expo Functions Course: Calculus Integrals of Exponential Functions u = 3x + 1 du/dx = 3;du = 3dx multiple and divide by 3

Aim: Differentiating & Integrating Expo Functions Course: Calculus Model Problem u = -x 2 du/dx = -2x  du = -2xdx  xdx = du/2 regroup integrand substitute factor out -5/2

Aim: Differentiating & Integrating Expo Functions Course: Calculus Model Problem u = 1/x u = cosx

Aim: Differentiating & Integrating Expo Functions Course: Calculus Model Problem Find the areas

Aim: Differentiating & Integrating Expo Functions Course: Calculus Do Now: Aim: How do we differentiate and integrate the exponential function?