Post on 10-Jan-2016
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5.10 Hyperbolic Functions
Greg Kelly, Hanford High School, Richland, Washington
Objectives
• Develop properties of hyperbolic functions.
• Differentiate and integrate hyperbolic functions.
• Develop properties of inverse hyperbolic functions.
• Differentiate and integrate functions involving inverse hyperbolic functions.
Consider the following two functions:
2 2
x x x xe e e ey y
These functions show up frequently enough that theyhave been given names.
2 2
x x x xe e e ey y
The behavior of these functions shows such remarkableparallels to trig functions, that they have been given similar names.
Hyperbolic Sine: sinh2
x xe ex
(pronounced “cinch x”)
Hyperbolic Cosine:
(pronounced “kosh x”)
cosh2
x xe ex
First, an easy one:
Now, if we have “trig-like” functions, it follows that we will have “trig-like” identities.
2 2cosh sinh 1x x 2 2
12 2
x x x xe e e e
2 2 2 22 2
14 4
x x x xe e e e
41
4
1 1
2 2cosh sinh 1x x
Note that this is similar to but not the same as:
2 2sin cos 1x x
I will give you a sheet with the formulas on it to use on the test.
Don’t memorize these formulas.
Derivatives can be found relatively easily using the definitions.
sinh cosh2 2
x x x xd d e e e ex x
dx dx
cosh sinh2 2
x x x xd d e e e ex x
dx dx
Surprise, this is positive!
So,
sinh cosh d
u u udx
cosh sinh d
u u udx
Find the derivative.
sinh cosh d
u u udx
cosh sinh d
u u udx
2( ) sinh( 3)f x x
2( ) cosh( 3) 2f x x x
( ) ln coshf x x
1( ) sinh
coshf x x
x tanh x
Even though it looks like a parabola, it is not a parabola!
A hanging cable makes a shape called a catenary.
coshx
y b aa
(for some constant a)
sinhdy x
dx a
Length of curve calculation:2
1d
c
dydx
dx
21 sinhd
c
xdx
a
2coshd
c
xdx
a
coshd
c
xdx
a
sinhd
c
xa
a
Another example of a catenary is the Gateway Arch in St. Louis, Missouri.
Another example of a catenary is the Gateway Arch in St. Louis, Missouri.
If air resistance is proportional to the square of velocity:
ln cosh y A Bty is the distance the
object falls in t seconds.A and B are constants.
boat
semi-truck
A third application is the tractrix.(pursuit curve)
An example of a real-life situation that can be modeled by a tractrix equation is a semi-truck turning a corner.
Another example is a boat attached to a rope being pulled by a person walking along the shore.
boat
semi-truck
A third application is the tractrix.(pursuit curve)
Both of these situations (and others) can be modeled by:
1 2 2 sechx
y a a xa
a
a
The word tractrix comes from the Latin tractus, which means “to draw, pull or tow”. (Our familiar word “tractor” comes from the same root.)
Other examples of a tractrix curve include a heat-seeking missile homing in on a moving airplane, and a dog leaving the front porch and chasing person running on the sidewalk.
sinh(2 )u x
2cosh(2 )sinh (2 )x x dx
2cosh(2 )2cosh(2 )
dux u
x
21
2u du
31
6u C
3sinh (2 )
6
xC
2cosh(2 )
2cosh(2 )
du x dx
dudx
x
Homework
5.10 (page 403)
#1,3
15-27 odd
39-47 odd