Calc 5.8b

Differentiation, integration

Ex 4 p. 392 Integrating a Hyperbolic Function

2

sinh

1 sinh

xdxx Let’s try some things for u

u=1+sinh2x results in du = 2sinch(x)cosh(x)dx. No help.Identities? cosh2x-sinh2x = 1

2

sinh

cosh

xdxx

Can you see power rule?

2cosh sinhx x dx

How about now?

1cosh

1

xC

1

coshC

x

sech x +C

Unlike trig functions, hyperbolic functions are not periodic. Four of the six hyperbolic functions pass the horizontal line test, so their inverses will be functions. With two (hyperbolic cosine and secant) they are one-to-one if we restrict domains to positive real numbers. Since the hyperbolic functions can be written in terms of exponential functions, their inverses can be written in terms of natural log functions.

The hyperbolic secant can be used to define a curve called a tractrix or a pursuit curve

Ex 5 p. 394 A tractrix

1 2 2sechx

y a a xa

Person

A person is holding a rope that is tied to a boat. As the person walks, the boat travels along a curve called a tractrix, given by equation

where a is the length of the boat. If a = 20 feet, find the distance the person must walk to bring the boat 5 feet from the dock.Solution: Notice that the length of y1 is how far the person has walked. That is composed of

2 21 20y y x Inserting y from our equation,

1 2 2 2 21 20sech 20 20

20

xy x x

11 20sech

20

xy When x = 5, this becomes

11

520sech

20y

2-1 1 1

sech x lnx

recallx

2

-1

11 1 41=20sech 20ln4 14

20ln(4 15)

41.27 ft

Ex 6 p. 395 More about a tractrixShow that the boat is always pointing toward the person in the setup from Ex. 5

For a point (x,y) on a tractrix, the slope of the graph gives the direction of the boat. (Refer to picture from Ex 5)

Person

1 2 220sech 2020

xy x

1

2 2 2

2

1 11'( ) 20 20 220 2120 20

y x x xx x

2

2 2 2 2

20'( )

20 20

xy x

x x x

2 2

2 2

(20 )

20

x

x x

2 220 xm

x

The slope of line segment connect (0, y1) with (x, y) is also this quantity (look at pink triangle) so it always points towards the person. It is because of this that it is also called the pursuit curve.

Ex 7 p. 395 Integration Using Inverse Hyperbolic Functions

29 4

dx

x x Let a = 3, u=2x, so du=2dx. This

almost fits one of our new integration patterns.

2

2

2 9 4

dx

x x 21 3 9 4

ln3 2

xC

x

5.8b p. 396/ 39-75 every other odd, 83-91 odd