Explanation Given: The given curve is as below. f ( x ) = y = e 2 x − e x (1) Calculation: (a) Calculate the domain. From the equation (1), the function is defined for all real values of x (there is no restriction on the value of x ). Hence, the domain of f ( x ) is all real values of x . For the polynomial, the domain is ( − ∞ , ∞ ) . (b) Calculate the intercepts. Calculate the y –intercept. Substitute 0 for the value of x in equation (1). f ( x ) = e 2 x − e x f ( 0 ) = e 2 ( 0 ) − e ( 0 ) = 1 − 1 = 0 The y –intercept is y = 0 . The x –intercept is x = 0 . (c) Calculate the symmetry. Substitute − x for x in the equation (1). f ( − x ) = e 2 ( − x ) − e − x = e − 2 ( x ) − e − x f ( x ) ≠ − f ( x ) The function is neither even nor odd. Therefore, the curve has is not symmetric about y - axis or origin. (d) Calculate asymptotes. Apply limit of x tends to ∞ ( x → ∞ ) in the equation (1). lim x → ∞ f ( x ) = e 2 x − e x = e 2 ∞ − e ∞ = ∞ Apply limit of x tends to − ∞ ( x → ∞ ) in the equation (1). lim x → − ∞ f ( x ) = e 2 ( − ∞ ) − e − ∞ = 0 It has horizontal asymptote as y = 0 . (e) Calculate the intervals. f ( x ) = e 2 x − e x Differentiate the equation (1) with respect to x . f ' ( x ) = 2 e 2 x − e x (2) Equate equation (2) to zero. 2 e 2 x − e x = 0 e x ( 2 e x − 1 ) = 0 e x = 0 or ( 2 e x − 1 ) = 0 e x = 1 2 x = ln 1 2 x = − 0.693 Consider the limit ( − ∞ , − 0.693 ) . Substitute 1 2 for x in f ' ( x ) . f ' ( − 2 ) = 2 e 2 ( − 2 ) − e ( − 2 ) = − 0.098 < 0 The function f shows a decreasing behavior for domain ( − ∞ , − 0.693 ) . Consider the limit ( − 0.693 , ∞ ) . Substitute 2 for x in f ' ( x ) . f ' ( 2 ) = 2 e 2 ( 2 ) − e ( 2 ) = 101

Bundle: Single Variable Calculus: Early Transcendentals, Loose-leaf Version, 8th + Webassign Printed Access Card For Calculus, Multi-term Courses, Life Of Edition

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ISBN: 9780357008034

Author: Stewart

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Chapter 4.5, Problem 46E

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Chapter 4.5, Problem 45E

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Bundle: Single Variable Calculus: Early Transcendentals, Loose-leaf Version, 8th + Webassign Printed Access Card For Calculus, Multi-term Courses, Life Of Edition

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