1. Section 5.8 Hyperbolic Functions:
Hyperbolic function arose from comparison of the area of a semicircular regions with the area of a region under a hyperbola.

2. Graph of the hyperbola f(x) = √(1+x2) and the circle g(x) = √(1 – x2)

3. Definitions of the hyperbolic functions

4. Graph of f(x) = sinh (x) Domain: (-, ) Range: (-, )

5. Graph of f(x) = cosh (x) Domain: (-, ) Range: [1, )

6. Graph of f(x) = tanh (x) Domain: (-, ) Range: (-1, 1)

7. Graph of f(x) = csch (x) Domain: (-, 0) (0,) Range: (-, 0) (0,)

8. Graph of f(x) = sech (x) Domain: (-, ) Range: (0, 1]

9. Graph of f(x) = coth (x) Domain: (-, 0) (0,) Range: (-, -1) (1,)

10. Hyperbolic Identities:

11. Theorem 5.18: Derivatives and integrals of Hyperbolic functions.

12. Theorem 5.18: Derivatives and integrals of Hyperbolic functions.

13. Find the derivative of the function.
1. f(x) = cosh (x - 2)

14. Find the derivative of the function.
1. f(x) = cosh (x - 2)
f’(x) = 1∙ sinh (x -2)

15. Find the derivative of the function.
2. y = tanh(3x2 - 1)

16. Find the derivative of the function.
2. y = tanh(3x2 - 1)
y’ = 6x∙ sech2 (3x2 -1)

17. Find the derivative of the function.
3. g(x) = x cosh x –sinh x

18. Find the derivative of the function.
3. g(x) = x cosh x –sinh x
g’(x) = x ∙ sinh x + cosh x – cosh x
g’(x) = x ∙ sinh x

19. Find the integral

20. Find the integral

21. Find the integral
2. ∫ sech2 (2x - 1) dx

22. Find the integral 2. ∫ sech2 (2x - 1) dx Let u = 2x – 1 du = 2 dx
Now Substiture
∫sech2 u du/2
= ½ ∫sech2 u du
= ½ tan u + C
= ½ tan (2x – 1) + C

23. Find the integral

24. Find the integral

25. Inverse Hyperbolic Functions

26. Differentiation involving inverse hyperbolic functions.

27. Integration involving inverse hyperbolic functions.