nyperbolic trigðnometric functI8ns" are specinc 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 x = 2 the "hyperbolic sine function" is given by et sinh x = - e and the "hyperbolic tangent function" is given by sinh x tanh x cosh x (a) Use the definitions above to show that d tanh x dx 1 cosh? x (b) water with depth d is given by The velocity of a water wave with length L moving across a body of gL V 2 (*) V = tanh 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 L→∞ Hint: Find lim v² first, and then use that to answer the question above.
nyperbolic trigðnometric functI8ns" are specinc 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 x = 2 the "hyperbolic sine function" is given by et sinh x = - e and the "hyperbolic tangent function" is given by sinh x tanh x cosh x (a) Use the definitions above to show that d tanh x dx 1 cosh? x (b) water with depth d is given by The velocity of a water wave with length L moving across a body of gL V 2 (*) V = tanh 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 L→∞ Hint: Find lim v² first, and then use that to answer the question above.
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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Transcribed Image Text: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
-x
et
cosh x
the "hyperbolic sine function" is given by
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 x
dx
1
%3D
cosh? x
(b)
water with depth d is given by
The velocity of a water wave with length L moving across a body of
gL
tanh
2nd
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.
L→00
6.
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