Section6.3 - Math 2B Dr. C. Famiglietti * Section 6.3: The...

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Math 2 B Dr. C. Famiglietti ************************************************************************ Section 6.3: The Exponential Function The inverse of the natural logarithmic function (presented in Section 6.1) is the natural exponential function y = e x , where e 2.71828 Since the two are inverse functions, the graph of y = e x can be obtained from the graph of y = ln x by reflecting the graph of y = ln x about the line y = x . The graph of y = e x is shown here and has the following characteristics: Domain: −∞ , ( ) Range: 0, ( ) Increasing throughout its domain Concave upward Passes through 0,1 ( ) since e 0 = 1 lim x →∞ e x = lim x →−∞ e x = 0 . The graph of y = e x is a reflection of the graph of y = e x about the y-axis and is therefore, decreasing throughout its domain with lim x →∞ e x = 0 and lim x →−∞ e x = .
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Properties of the natural exponential function, y = e x 1. e ln x = x , for x > 0 . 2. ln e x ( ) = x , for −∞ ≤ x ≤ ∞ . Derivative of the natural exponential function d dx e x ( ) = e x To derive the differentiation formula (without using the definition of the derivative) for
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Section6.3 - Math 2B Dr. C. Famiglietti * Section 6.3: The...

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