Transcript Hyperbolic Function Power Point

Hyperbolic Functions
Background
• Suspension cables like those of the
Golden Gate Bridge, which support a
constant load per horizontal foot, hang in
parabolas.
Background
• Cables like power line cables, which hang
freely, hang in curves called hyperbolic
cosine curves.
Background
• Besides describing the shapes of hanging
cables, hyperbolic functions describe the
motions of waves in elastic solids, the
temperature distributions in metal cooling
fins, and the motions of falling bodies that
encounter air resistance proportional to
the square of the velocity.
Definitions of Hyperbolic Functions
e e
sinh x 
2
x
x
sinh x
tanh x 
cosh x
cosh x
coth x 
sinh x
e e
cosh x 
2
x
x
1
csc hx 
sinh x
1
sec hx 
cosh x
Graph of Hyperbolic Cosine
Graphs of Hyperbolic Sine
Graphs of Hyperbolic Tangent and Cotangent
Graphs of Hyperbolic Sine and Cosecant
Graphs of Hyperbolic Cosine and Secant
Hyperbolic Identities
sinh( x)   sinh x
cosh h( x)  cosh x
cosh x  sinh x  1
2
2
Hyperbolic Identities
1  tanh x  sec h x
2
2
sinh(x  y)  sinh x cosh y  cosh x sinh y
cosh(x  y)  cosh x cosh y  sinh x sinh y
Derivatives of Hyperbolic Functions
d
(sinh x)  cosh x
dx
d
(cosh x)  sinh x
dx
d
2
(tanh x)  sec h x
dx
Derivatives of Hyperbolic Functions
d
(sec hx )   sec hx tanh x
dx
d
(csc hx )   csc hx coth x
dx
d
2
(coth x)   csc h x
dx
Integrals of Hyperbolic Functions
sinh
u

cosh
u

C

cosh
u

sinh
u

C

sec
hu

tanh
u

C

2
Integrals of Hyperbolic Functions
csc
hu
cot
udu


csc
hu

C

sec
hu
tanh
u


sec
hu

C

csc
hu


coth
u

C

2
Graph of Inverse Hyperbolic Sine
Graph of Inverse Hyperbolic Cosine
Graph of Inverse Hyperbolic Secant
Graph of Inverse Hyperbolic Cosecant
Graph of Inverse Hyperbolic Cotangent
Graph of Inverse Hyperbolic Tangent
Derivatives of Inverse Hyperbolic
Functions
d
1
1
(sinh x) 
2
dx
1 x
d
1
1
(csc h x)  
2
dx
x x 1
d
1
(cosh x) 
dx
1
x 1
2
Derivatives of Inverse Hyperbolic
Functions
d
1
1
(sec h x)  
2
dx
x 1 x
d
1
1
(tanh x) 
2
dx
1 x
d
1
1
(coth x) 
2
dx
1 x
Integrals of Inverse Hyperbolic
Functions



du
u
 sinh
C
2
2
a
a u
du
1 u
 cosh
C
2
2
a
u a
1
1
1 u
2
2
tan

C
if
u

a
 a

du
a
 

2
2
a u
 1 cot1 u  C if u 2  a 2 
 a

a
Integrals of Inverse Hyperbolic
Functions
1
1 u
 u a 2  u 2  a sec h a  C
du
1
1 u
 u u 2  a 2  a csc h a  C
du
0ua
u0