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(a) Use the definitions above to show that d tanh x dr 1 cosh r

(a) Use the definitions above to show that d tanh x dr 1 cosh r

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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6. The "hyperbolic trigonometric functions" are specific combinations of ex- ponential functions that appear in application frequently enough that they have their own names. For example, the "hyperbolic cosine function" is given by et +e- cosh r the "hyperbolic sine function" is given by - e sinh x = 2 and the "hyperbolic tangent function" is given by sinh x tanh x = cosh x (a) Use the definitions above to show that d tanh a dx 1 cosh? x (b) The velocity of a water wave with length L moving across a body of water with depth d is given by gL ( 2nd tanh 27 V = L where g represents the acceleration due to gravity (a positive constant). What happens to the velocity as the wave length increases without bound? That is, evaluate lim v Hint: Find lim v² first, and then use that to answer the question above. L00
Transcribed Image Text:6. The "hyperbolic trigonometric functions" are specific combinations of ex- ponential functions that appear in application frequently enough that they have their own names. For example, the "hyperbolic cosine function" is given by et +e- cosh r the "hyperbolic sine function" is given by - e sinh x = 2 and the "hyperbolic tangent function" is given by sinh x tanh x = cosh x (a) Use the definitions above to show that d tanh a dx 1 cosh? x (b) The velocity of a water wave with length L moving across a body of water with depth d is given by gL ( 2nd tanh 27 V = L where g represents the acceleration due to gravity (a positive constant). What happens to the velocity as the wave length increases without bound? That is, evaluate lim v Hint: Find lim v² first, and then use that to answer the question above. L00
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