1. Chapter 3 – Differentiation Rules
3.11 Hyperbolic Functions
3.11 Hyperbolic Functions
Dr. Erickson

2. Definition of the Hyperbolic Functions
Certain even and odd combinations of the exponential functions ex and e-x arise so frequently in mathematics and its applications that they deserve to be given special names.
3.11 Hyperbolic Functions
Dr. Erickson

3. Hyperbolic Identities
The hyperbolic functions satisfy a number of identities that are similar to well-known trig identities. Here are some of those hyperbolic identities.
3.11 Hyperbolic Functions
Dr. Erickson

4. Derivatives of Hyperbolic Functions
Note the analogy with the differentiation formulas for trig functions, but be aware that the signs are different in some cases.
3.11 Hyperbolic Functions
Dr. Erickson

5. Inverse Hyperbolic Functions
The sinh and tanh are one-to-one functions and so they have inverse functions denoted by sinh–1 and tanh–1. The cosh is not one-to-one, but when restricted to the domain [0, ) it becomes one-to-one.
The inverse hyperbolic cosine function is defined as the inverse of this restricted function.
3.11 Hyperbolic Functions
Dr. Erickson

6. Inverse Hyperbolic Functions
We can sketch the graphs of sinh–1, cosh–1, and tanh–1
domain = [1,  ) range = [0, )
domain = range =
domain = (–1, 1) range =
3.11 Hyperbolic Functions
Dr. Erickson

7. Inverse Hyperbolic Functions
Since the hyperbolic functions are defined in terms of exponential functions, it’s not surprising to learn that the inverse hyperbolic functions can be expressed in terms of logarithms.
3.11 Hyperbolic Functions
Dr. Erickson

8. Derivatives of Inverse Hyperbolic Functions
Notice that the derivatives of tanh-1x and coth-1x appear to be identical. But, the domains of these functions have no numbers in common. tanh-1x is defined for |x|<1 and coth-1x is defined for |x|>1
3.11 Hyperbolic Functions
Dr. Erickson

9. Book Example 1- pg # 3
Find the numerical value of each expression.
3.11 Hyperbolic Functions
Dr. Erickson

10. Book Example 2 – pg. 262 # 20
If , find the values of the other hyperbolic functions at x. Note: Must use hyperbolic identities not trig identities.
3.11 Hyperbolic Functions
Dr. Erickson

11. Example 3
The Gateway Arch in St. Louis was designed by Eero Saarinen and constructed using the equation for the central curve of the arch, where x and y are measured in meters and |x|91.20.
Graph the central curve.
What is the height of the arch at its center?
At what points is the height of the arch 100 m?
What is the slope of the arch at the points in part c?
3.11 Hyperbolic Functions
Dr. Erickson

12. Example 4
Find the derivative of the following and simplify where possible.
3.11 Hyperbolic Functions
Dr. Erickson