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## Ms. Battaglia AP Calculus

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**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