5.4 Differentiation and Integration of “E” 2012 The Natural Exponential Function The function f(x) = ln x is increasing on its entire domain, and therefore.

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Presentation transcript:

5.4 Differentiation and Integration of “E” 2012

The Natural Exponential Function The function f(x) = ln x is increasing on its entire domain, and therefore it has an inverse function f –1. The domain of f –1 is the set of all reals, and the range is the set of positive reals, as shown in Figure Figure 5.19

So, for any real number x, If x happens to be rational, then Because the natural logarithmic function is one-to-one, you can conclude that f –1 (x) and e x agree for rational values of x. The Natural Exponential Function

The following definition extends the meaning of e x to include all real values of x. The inverse relationship between the natural logarithmic function and the natural exponential function can be summarized as follows. The Natural Exponential Function

5.4 Exponential Functions: Differentiation and Integration Solve for x

5.4 Exponential Functions: Differentiation and Integration

Chain Rule

5.4 Exponential Functions: Differentiation and Integration

Always Positive

5.4 Exponential Functions: Differentiation and Integration

Page ,34, 35 – 51 odd, 57, 58, 65, 69, 85 – 91odd, odd