Title: Hyperbolic%20function%20arose%20from%20comparison%20of%20the%20area%20of%20a%20semicircular%20regions%20with%20the%20area%20of%20a%20region%20under%20a%20hyperbola.
1Section 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.
2Graph of the hyperbola f(x) v(1x2)
and the circle g(x) v(1
x2)
3Definitions of the hyperbolic functions
4Graph of f(x) sinh (x)Domain (-?, ?)Range
(-?, ?)
5Graph of f(x) cosh (x)Domain (-?, ?)Range
1, ?)
6Graph of f(x) tanh (x)Domain (-?, ?)Range
(-1, 1)
7Graph of f(x) csch (x)Domain (-?, 0)?
(0,?)Range (-?, 0)? (0,?)
8Graph of f(x) sech (x)Domain (-?, ?)Range
(0, 1
9Graph of f(x) coth (x)Domain (-?, 0)? (0,?)
Range (-?, -1)? (1,?)
10Hyperbolic Identities
11Theorem 5.18 Derivatives and integrals of
Hyperbolic functions.
12Theorem 5.18 Derivatives and integrals of
Hyperbolic functions.
13Find the derivative of the function.
14Find the derivative of the function.
- 1. f(x) cosh (x - 2)
- f(x) 1 sinh (x -2)
15Find the derivative of the function.
16Find the derivative of the function.
- 2. y tanh(3x2 - 1)
- y 6x sech2 (3x2 -1)
17Find the derivative of the function.
18Find 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
19Find the integral
20Find the integral
21Find the integral
22Find the integral
- 2. ? sech2 (2x - 1) dx
- Let u 2x 1
- du 2 dx
- du/2 dx
- Now Substiture
- ?sech2 u du/2
- ½ ?sech2 u du
- ½ tan u C
- ½ tan (2x 1) C
23Find the integral
24Find the integral
25Inverse Hyperbolic Functions
26Differentiation involving inverse hyperbolic
functions.
27Integration involving inverse hyperbolic
functions.