5.10 Hyperbolic Functions
5.8 Even Answers:
 -  
68) False: The range of y  arcsin x is  2 , 2 
2
70)
 
2
2
False: arcsin 0  arccos 0  0     1
2
5.10 Hyperbolic Functions
Definition of the Hyperbolic Functions
e e
sinh x 
2
x
x
1
csch x 
,x  0
sinh x
x
1
sech x 
cosh x
e e
cosh x 
2
x
sinh x
tanh x 
cosh x
1
coth x 
,x 0
tanh x
Problem
Problem
Problem 1
Problem 2
5.10 Hyperbolic Functions
Many of the trigonometric identities have corresponding
hyperbolic identities.
sin 2x  2sin x cos x
2sinh x cosh x 
Substitute and see
if we get sinh2x
 e x  e x  e x  e x 
2


 2  2 
e e

2
2x
2 x
 sinh 2x
Definitions
5.10 Hyperbolic Functions
Many of the trigonometric identities have corresponding
hyperbolic identities.
cos x  sin x  1
2
2

e

e


e

e
2
2
cosh x  sinh x  
  2
 2  
x
e 2e

4
2x
4

4
1
x
2 x
2
x
e 2e

4
2x
x
2 x



2
5.10 Hyperbolic Functions
5.10 Hyperbolic Functions
Problem 1
Problem 2
5.10 Hyperbolic Functions
Differentiation of Hyperbolic Functions
d
sinh  x 2  3  

dx 
2 x cosh  x  3
2
d
u ' sinh x
 tanh x
ln cosh x   
dx
u cosh x
Derivative of sinhx and coshx
5.10 Hyperbolic Functions
Differentiation of Hyperbolic Functions
d
 x sinh x  cosh x  
dx
sinh x  x cosh x  sinh x  x cosh x
Derivative of sinhx and coshx
5.10 Hyperbolic Functions
Find the relative extrema of f ( x)  ( x  1) cosh x  sinh x.
f '( x)  cosh x  ( x  1)sinh x  cosh x
 ( x  1) sinh x  0
 x  1, 0
2nd Derivative Test:
f ''( x)  sinh x  ( x  1) cosh x
f ''(1)  0  1,  sinh1 is a relative min.
f ''(0)  0  (0,  cosh 0  sinh 0) is a relative max.
  0, 1
Graph of coshx
Graph of sinhx
5.10 Hyperbolic Functions
Power cables are suspended between two towers, forming
the catenary shown in the picture. The equation for this
catenary is
x
y  a cosh .
a
The distance between the two towers is 2b. Find the slope
of the catenary at the point where the cable meets the righthand tower.
5.10 Hyperbolic Functions
x
y  a cosh .
a
5.10 Hyperbolic Functions
x
y  a cosh .
a
 x  1 
 x
y '  a sinh     sinh  
 a  a 
a
b
@ (b, a cosh ), the slope (from the left) is
a
b
m  sinh .
a
Graph
5.10 Hyperbolic Functions
Solve  cosh  2 x  sinh  2 x  dx
2
u ?
u  sinh 2x
du  2cosh 2xdx
1 2
 1  u
  u du    
2
 2  3
3
3

C

3
sinh 2 x
u
C
 C 
6
6
5.10 Hyperbolic Functions
Read pages 396-397
Problem
5.10 Hyperbolic Functions
A person is holding a rope that is tied to a boat. As the
person walks along the dock, the boat travels along a
tractrix, given by the equation
x
2
2
y  a sec h
 a x
a
1
where a is the length of the rope. If a  20 feet, find the
distance the person must walk to bring the boat 5 feet from
the dock.
Problem 1
Problem 2
x
2
2
y  a sec h
 a x
a
1
5.10 Hyperbolic Functions
Picture
The distance the person has walked is
y1  y  20  x
2
2

1 x
2
2 
2
2
  20sec h
 20  x   20  x
20


x
 20sec h
20
1
When x  5, this distance is
2
5
1

1

1/
4
   20ln 4  4 1  1/16
y1  20sec h 1
 20 ln
20
1/ 4

Sech^-1=?
 41.27 feet

5.10 Hyperbolic Functions
Problem
5.10 Hyperbolic Functions
Problem
5.10 Hyperbolic Functions
Show that the boat is always pointing towards the person.
For a point  x, y  on a tractrix, the slope of the graph gives
the direction of the boat.
d 
1 x
2
2
y' 
20sec
h

20

x


dx 
20



1
 1 
 1   2 x 

 20   
  

2
2
2
 20    x / 20  1   x / 20    2   20  x 


Picture
d/dx[sech^-1]=?


1
 1 
 1   2 x 

 20   
  

2
2
2
 20    x / 20  1   x / 20    2   20  x 




20
x 1   x / 20 
2
20
2
400 x
x

400 400


x
20  x
2
2
x
20  x
2

2
20
x
202  x 2
20

x
20  x
2
2


20

x
202  x 2
20
20
x 20  x
20


202  x 2
2
2
2
x

2
x
2
20  x

x
2
2
x 20  x
2
2

x 20  x
2
x
2
2


m   202  x 2 / x
 the boat is always pointing
towards the person.
5.10 Hyperbolic Functions
Evaluate
x
dx
4  9x
Integrals
2
1 a a u


ln

C
 u a2  u 2 a
u
du
a2

u  3x
2
2
du  3dx
u 2 u
2
du
1 2  4  9x
  ln
C
2
3x
2
2
5.10 Hyperbolic Functions
HW 5.10/1-77 EOO
Quiz :
Evaluate  cosh
2
 x  1 sinh  x  1 dx