Transcript Hyperbolic Functions Samantha Chieu Dennis Wong Danny Hang Period 4 Calculus BC Mr. Bourbois Origin of Hyperbolic Functions • Hyperbolic functions originated from the comparison of the area of a.

Hyperbolic Functions
Samantha Chieu
Dennis Wong
Danny Hang
Period 4
Calculus BC
Mr. Bourbois
Origin of Hyperbolic Functions
• Hyperbolic functions
originated from the
comparison of the area
of a semicircular region
and the area of a
hyperbola.
1

1 x 2 dx  12 [x 1 x 2  sin 1 x]11  2
1
1

1

1 x 2 dx  12 [x 1 x 2  sinh 1 x]11  2.296
Formal definition and example for sinh(x)
• Sinh(x)=
• Y1= (ex)/2
• Y2= -(e-x)/2
Formal definition and example for cosh(x)
• Cosh(x)=
• Y1= (ex)/2
• Y2= (e-x)/2
Formal definition and example for tanh(x)
• Tanh(x)=
• Y1= 1
• Y2= -1
Formal definition and example for csch(x)
• Csch(x)=
Formal definition and example for sech(x)
• Sech(x)=
Formal definition and example for coth(x)
• Coth(x)=
• Y1= 1
• Y2= -1
Derivatives of Hyperbolic Equations
d [sinh(x)] cosh(x)
dx
d [cosh(x)] sinh(x)
dx
d [tanh(x)] sec h2(x)
dx
d [csc h(x)] csc h(x)coth(x)
dx
d [sec h(x)] sec h(x)tanh(x)
dx
d [coth(x)] csc h2(x)
dx
Integrals of Hyperbolic Equations
 (cosh x)dx  sinh(x)C
 (sinhx)dx  cosh(x) C
2
sech
(x)dx  tanh(x)C

 csch(x)coth(x)dx  csch(x)C
 sech(x)tan(x)dx  sech(x)C
2
csch
(x) dx  coth(x)C

Special Identities
• cosh2(x) – sinh2(x)= 1
• sinh(2x)= 2sinh(x)cosh(x)
• cosh(2x)= cosh2(x) + sinh2(x)
= 1 + 2sinh2(x)
= 2cosh2(x) – 1
Real Life
Example
Say we have a long piece of
rope. By fixing its ends to a
set place in space and
letting the rest hang, we
would create a natural
hyperbolic cosine (cosh)
curve.
This idea is used a lot in
the maintenance of power
lines when deciding how
much cable to place in
between each pole.
FIN
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