Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 7 - Transcendental Functions문제 7.7.1
7장, 문제 7.7.1

Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
1. sinh x = -3/4

검증된 단계별 안내
1
Recall the fundamental identity for hyperbolic functions: \(\cosh^{2}x - \sinh^{2}x = 1\).
Substitute the given value \(\sinh x = -\frac{3}{4}\) into the identity: \(\cosh^{2}x - \left(-\frac{3}{4}\right)^{2} = 1\).
Simplify the equation to solve for \(\cosh^{2}x\): \(\cosh^{2}x - \frac{9}{16} = 1\) which leads to \(\cosh^{2}x = 1 + \frac{9}{16}\).
Take the positive square root to find \(\cosh x\) because \(\cosh x\) is always positive: \(\cosh x = \sqrt{1 + \frac{9}{16}}\).
Use the definitions of the other hyperbolic functions in terms of \(\sinh x\) and \(\cosh x\) to find the remaining five functions: - \(\tanh x = \frac{\sinh x}{\cosh x}\) - \(\coth x = \frac{\cosh x}{\sinh x}\) - \(\sech x = \frac{1}{\cosh x}\) - \(\csch x = \frac{1}{\sinh x}\)

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definitions of Hyperbolic Functions

Hyperbolic functions sinh x and cosh x are defined as sinh x = (e^x - e^(-x))/2 and cosh x = (e^x + e^(-x))/2. Understanding these definitions helps relate the functions to exponential expressions and is fundamental for calculating other hyperbolic functions.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral

Hyperbolic Identity: cosh²x - sinh²x = 1

This identity is analogous to the Pythagorean identity in trigonometry and allows finding cosh x when sinh x is known, or vice versa. It ensures the relationship between cosh and sinh values is consistent and is essential for solving for unknown hyperbolic functions.
추천 영상:
03:39
Integrals of Natural Exponential Functions (e^x)

Definitions of Other Hyperbolic Functions

The remaining hyperbolic functions—tanh x, coth x, sech x, and csch x—are defined in terms of sinh x and cosh x, such as tanh x = sinh x / cosh x. Knowing these definitions allows computation of all hyperbolic functions once sinh x and cosh x are determined.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
관련 실천
교과서 질문

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

27. y = θ(sin(lnθ) + cos(lnθ))

22
views
교과서 질문

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

23. y = (x²+1)sech(ln x)

(Hint: Before differentiating, express in terms of exponentials and simplify.)

21
views
교과서 질문

77. The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve x=2/√(y+1) is revolved about the y-axis to generate a solid. Find the volume of the solid.

6
views
교과서 질문

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

47. lim (t → ∞) (e^t + t²) / (e^t - t)

34
views
교과서 질문

In Exercises 5–8, show that each function is a solution of the given initial value problem.

7. Differential Equation: xy' + y = -sin(x), x>0

Initial condition: y(π/2) = 0

Solution candidate: y = cos(x)/x

20
views
교과서 질문

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

75. lim (x → ∞) e^(x²) / (x e^x)

32
views