5-4 Exponential Functions: Differentiation and IntegrationObjective: Differentiate and integrate natural exponential functions. Ms. Battaglia AP Calculus
Definition of the Natural Exponential Function The inverse function of the natural logarithmic function f(x)=lnx is called the natural exponential function and is denoted by f -1(x) = ex That is, y = ex if and only if x = lny Inverse Relationship ln(ex) = x and elnx = x
Solving Exponential Equations Solve 7 = ex+1
Solving a Logarithmic Equation Solve ln(2x – 3) = 5
Operations with Exponential Functions Let a and b be any real numbers.
Properties of the Natural Exponential Function • The domain of f(x)=ex is (-∞, ∞), and the range is (0, ∞) • The function f(x)=ex is continuous, and one-to-one on its entire domain. • The graph of f(x)=ex is concave upward on its entire domain. • and
Derivatives of the Natural Exponential Function Let u be a differentiable function of x.
Locating Relative Extrema Find the relative extrema of f(x)=xex.
The Standard Normal Probability Density Function Show that the standard normal probability density function has points of inflection when x=1, -1
Shares Traded The numbers y of shares traded (in millions) on the NY Stock Exchange from 1990 through 2005 can be modeled by where t represents the year, with t=0 corresponding to 1990. At what rate was the number of shares traded changing in 2000?
Integration Rules for Exponential Functions Let u be a differentiable function of x. 1. 2.
Classwork/Homework • AB: Read 5.4 Page 358 #3, 4, 11-14, 37, 38, 39-57 odd • BC: HW: Read 5.4 Page 358 #1-11 odd, 39-59 odd,63,69,79,99, 103,105,111,121,125