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