rogawskilt4e_lectureslides_ch07.pptx - Chapter 7...

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Chapter 7: EXPONENTIAL AND LOGARITHMIC FUNCTIONS Contents: 7.1 The Derivative of f ( x ) = b x and the Number e 7.2 Inverse Functions 7.3 Logarithmic Functions and Their Derivatives 7.4 Applications of Exponential and Logarithmic Functions 7.5 L’Hôpital’s Rule 7.6 Inverse Trigonometric Functions 7.7 Hyperbolic Functions
7.1 The Derivative of f ( x ) = b x and the Number e (1 of 17) An exponential function is a function of the form f ( x ) = b x , where b > 0 and b ≠ 1. The number b is called the base . Some examples are f ( x ) = 2 x , g ( x ) = (1.4) x , and h ( x ) = 10 x . The case b = 1 is excluded because f ( x ) = 1 x is a constant function. Calculators give good decimal approximations to values of exponential functions: 2 4 = 16 2 −3 = 0.125 (1.4) 0.8 ≈ 1.309 10 4.6 ≈ 39,810.717
Three properties of exponential functions should be singled out from the start (see Figure 1 for the case b = 2 ): Exponential functions are positive: b x > 0 for all x . The range of f ( x ) = b x is the set of all positive real numbers. f ( x ) = b x is increasing if b > 1 and decreasing if 0 < b < 1. 7.1 The Derivative of f ( x ) = b x and the Number e (2 of 17)
7.1 The Derivative of f ( x ) = b x and the Number e (3 of 17) If b > 1, the exponential function f ( x ) = b x is not merely increasing but is rapidly increasing. Although the term “rapid increase” is perhaps subjective, the following precise statement is true: For all n , if x is positive and large enough, then f ( x ) = b x increases more rapidly than the power function g ( x ) = x n . For example, Figure 2 shows that f ( x ) = 3 x eventually overtakes and increases faster than the power functions g ( x ) = x 3 , g ( x ) = x 4 , and g ( x ) = x 5 .

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