5.10 hyperbolic functions
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/1.jpg)
5.10 Hyperbolic Functions
Greg Kelly, Hanford High School, Richland, Washington
![Page 2: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/2.jpg)
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.
![Page 3: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/3.jpg)
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.
![Page 4: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/4.jpg)
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.
![Page 5: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/5.jpg)
Hyperbolic Sine: sinh2
x xe ex
(pronounced “cinch x”)
Hyperbolic Cosine:
(pronounced “kosh x”)
cosh2
x xe ex
![Page 6: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/6.jpg)
First, an easy one:
Now, if we have “trig-like” functions, it follows that we will have “trig-like” identities.
![Page 7: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/7.jpg)
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
![Page 8: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/8.jpg)
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.
![Page 9: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/9.jpg)
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!
![Page 10: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/10.jpg)
So,
sinh cosh d
u u udx
cosh sinh d
u u udx
![Page 11: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/11.jpg)
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
![Page 12: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/12.jpg)
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
![Page 13: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/13.jpg)
Another example of a catenary is the Gateway Arch in St. Louis, Missouri.
![Page 14: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/14.jpg)
Another example of a catenary is the Gateway Arch in St. Louis, Missouri.
![Page 15: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/15.jpg)
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.
![Page 16: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/16.jpg)
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.
![Page 17: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/17.jpg)
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
![Page 18: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/18.jpg)
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.
![Page 19: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/19.jpg)
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
![Page 20: 5.10 Hyperbolic Functions](https://reader033.vdocuments.us/reader033/viewer/2022061405/568140d1550346895dac9dab/html5/thumbnails/20.jpg)
Homework
5.10 (page 403)
#1,3
15-27 odd
39-47 odd