Chapter4 - Chapter 4 The Exponential and Natural L ogarithm...

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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 Definition Example Exponential Function : A function whose exponent is the independent variable ( 29 x y 3 - =
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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 Expressions EXAMPLE EXAMPLE SOLUTION 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 ( 29 ( 29 x x x b a - + 2 2 2 81 1 1 5 2 ( 29 ( 29 . 3 3 3 3 1 81 1 2 2 4 2 4 2 4 2 x x x x x - - - = = = = (b) We first simplify the denominator and then combine the numerator via the base of the exponents, 2. Therefore, we get ( 29 . 2 2 2 2 2 2 2 6 1 1 5 1 1 5 1 5 x x x x x x x = = = - - + - + - +
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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 < b < 1, the function is decreasing. If b > 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 Equations EXAMPLE EXAMPLE SOLUTION SOLUTION Solve the following equation for x . ( 29 0 5 4 5 3 2 = + - x x x ( 29 0 5 4 5 3 2 = + - x x x This is the given equation. ( 29 [ ] 0 4 3 2 5 = + - x x Factor. [ ] 0 3 6 5 = - x x Simplify. 0 3 6 0 5 = - = x x Since 5 x and 6 – 3 x are being multiplied , set each factor equal to zero. x x = 2 0 5 5 x ≠ 0.
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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS , 11e – Slide 10 of 55 § 4.2 The Exponential Function e x
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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS , 11e – Slide 11 of 55 e The Derivatives of 2 x , b x , and e x Section Outline
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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS , 11e – Slide 12 of 55
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Chapter4 - Chapter 4 The Exponential and Natural L ogarithm...

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