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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 Chapter 4 The Exponential and Natural Logarithm Functions

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 55 Exponential Functions The Exponential Function e x Differentiation of Exponential Functions The Natural Logarithm Function The Derivative ln x Properties of the Natural Logarithm Function Chapter Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 55 § 4.1 Exponential Functions

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 55 Exponential Functions Properties of Exponential Functions Simplifying Exponential Expressions Graphs of Exponential Functions Solving Exponential Equations Section Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 55 Exponential Function DefinitionExample Exponential Function: A function whose exponent is the independent variable

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 55 Properties of Exponential Functions

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 55 Simplifying Exponential ExpressionsEXAMPLE SOLUTION Write each function in the form 2 kx or 3 kx, for a suitable constant k. (a) We notice that 81 is divisible by 3. And through investigation we recognize that 81 = 3 4. Therefore, we get (b) We first simplify the denominator and then combine the numerator via the base of the exponents, 2. Therefore, we get

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 55 Graphs of Exponential Functions Notice that, no matter what b is (except 1), the graph of y = b x has a y-intercept of 1. Also, if 0 1, then the function is increasing.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 55 Solving Exponential EquationsEXAMPLE SOLUTION Solve the following equation for x. This is the given equation. Factor. Simplify. Since 5 x and 6 – 3x are being multiplied, set each factor equal to zero. 5 x ≠ 0.

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