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...
Related questions
Question
3.11-Number 16, please
, [12](#), [13](#), [14](#), [15](#), [16](#), [17](#), [18](#), [19](#), [20](#), [21](#), [22](#), and [23](#) with proofs of the identity.
---
11. **\(\sinh(-x) = - \sinh x\)**
*(This shows that \(\sinh\) is an odd function)*
12. **\(\cosh(-x) = \cosh x\)**
*(This shows that \(\cosh\) is an even function)*
13. **\(\cosh x + \sinh x = e^x\)**
14. **\(\cosh x - \sinh x = e^{-x}\)**
15. **\(\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y\)**
16. **\(\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y\)**
These foundational identities help understand the behavior of hyperbolic functions, analogous to trigonometric identities, but applied to hyperbolic angles. Explore each to enhance your comprehension of how hyperbolic sine (\(\sinh\)) and cosine (\(\cosh\)) function similarly to their trigonometric counterparts while exhibiting unique properties such as even and odd functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c1d0d01-ade3-4f63-bcf8-3821e33828ca%2Fb1be5851-cb96-4ab5-9fa9-2989245be15b%2Ft8vlhud_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hyperbolic Identities and Functions
9. **Write \( \sinh(\ln x) \) as a rational function of \( x \).**
*Answer* (dropdown for additional information)
10. **Write \( \cosh(4 \ln x) \) as a rational function of \( x \).**
Links to deeper explanations: [11](#), [12](#), [13](#), [14](#), [15](#), [16](#), [17](#), [18](#), [19](#), [20](#), [21](#), [22](#), and [23](#) with proofs of the identity.
---
11. **\(\sinh(-x) = - \sinh x\)**
*(This shows that \(\sinh\) is an odd function)*
12. **\(\cosh(-x) = \cosh x\)**
*(This shows that \(\cosh\) is an even function)*
13. **\(\cosh x + \sinh x = e^x\)**
14. **\(\cosh x - \sinh x = e^{-x}\)**
15. **\(\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y\)**
16. **\(\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y\)**
These foundational identities help understand the behavior of hyperbolic functions, analogous to trigonometric identities, but applied to hyperbolic angles. Explore each to enhance your comprehension of how hyperbolic sine (\(\sinh\)) and cosine (\(\cosh\)) function similarly to their trigonometric counterparts while exhibiting unique properties such as even and odd functions.
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