5.10 Hyperbolic Functions
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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.

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5.10 Hyperbolic Functions

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5 10 hyperbolic functions

5.10 Hyperbolic Functions

Greg Kelly, Hanford High School, Richland, Washington


Objectives

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.


5 10 hyperbolic functions

Consider the following two functions:

These functions show up frequently enough that they

have been given names.


5 10 hyperbolic functions

The behavior of these functions shows such remarkable

parallels to trig functions, that they have been given

similar names.


5 10 hyperbolic functions

Hyperbolic Sine:

(pronounced “cinch x”)

Hyperbolic Cosine:

(pronounced “kosh x”)


5 10 hyperbolic functions

Now, if we have “trig-like” functions, it follows that we will have “trig-like” identities.

First, an easy one:


5 10 hyperbolic functions

Note that this is similar to but not the same as:

I will give you a sheet with the formulas on it to use on the test.

Don’t memorize these formulas.


5 10 hyperbolic functions

Derivatives can be found relatively easily using the definitions.

Surprise, this is positive!


5 10 hyperbolic functions

So,


5 10 hyperbolic functions

Find the derivative.


5 10 hyperbolic functions

A hanging cable makes a shape called a catenary.

Even though it looks like a parabola, it is not a parabola!

(for some constant a)

Length of curve calculation:


5 10 hyperbolic functions

Another example of a catenary is the Gateway Arch in St. Louis, Missouri.


5 10 hyperbolic functions

Another example of a catenary is the Gateway Arch in St. Louis, Missouri.


5 10 hyperbolic functions

If air resistance is proportional to the square of velocity:

y is the distance the object falls in t seconds.

A and B are constants.


5 10 hyperbolic functions

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.

semi-truck

Another example is a boat attached to a rope being pulled by a person walking along the shore.

boat


5 10 hyperbolic functions

A third application is the tractrix.

(pursuit curve)

a

Both of these situations (and others) can be modeled by:

semi-truck

a

boat


5 10 hyperbolic functions

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.


Homework

Homework

5.10 (page 403)

#1,3

15-27 odd

39-47 odd

p


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