Hyperbolic%20function%20arose%20from%20comparison%20of%20the%20area%20of%20a%20semicircular%20regions%20with%20the%20area%20of%20a%20region%20under%20a%20hyperbola. - PowerPoint PPT Presentation

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Hyperbolic%20function%20arose%20from%20comparison%20of%20the%20area%20of%20a%20semicircular%20regions%20with%20the%20area%20of%20a%20region%20under%20a%20hyperbola.

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Hyperbolic function arose from comparison of the area of a semicircular regions with the area of a region under ... Derivatives and integrals of Hyperbolic functions. – PowerPoint PPT presentation

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Title: Hyperbolic%20function%20arose%20from%20comparison%20of%20the%20area%20of%20a%20semicircular%20regions%20with%20the%20area%20of%20a%20region%20under%20a%20hyperbola.


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) v(1x2)
and the circle g(x) v(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
  • 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.
  •  
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