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Hyperbolic functions

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Hyperbolic functionsHungarian and English notation

2)cosh(

xx eex

2)sinh(

xx eex

xx

xx

ee

ee

xcosh

xsinh)xtanh(

xx

xx

ee

ee)xcoth(

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Groupwork in 4 groups

For each function :

- find domain

- discuss parity

- find limits at the endpoints of the domain

-find zeros if any

-find intervals such that the function is cont.

-find local and global extremas if any

-find range

-find asymptotes

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Summary: cosh

2)cosh(

xx eex

What are the asymptotes of cosh(x)

-in the infinity (2)

-negative infiniy (2)

PROVE YOUR STATEMENT!

xcosh)x(hcosh:EVEN

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Application of the use of hyperbolic cosine to describe the shape of a hanging wire/chain.

Summary: cosh

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Background

So, cables like power line cables, which hang freely, hang in curves called hyperbolic cosine curves.

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Chaincurve-catentity

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Background

Suspension cables like

those of the Golden Gate

Bridge, which support a

constant load per horizontal

foot, hang in parabolas.

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Which shape do you suppose in this case?

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Application: we will solve it SOON!

Electric wires suspended between two towers form a catenary with the equation

If the towers are 120 ft apart, what is the length of the suspended wire?• Use the arc length formula

60cosh60

xy

120'

21 '( )b

i

a

L f x dx

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What are the asymptotes of cosh(x)

-in the infinity (2)

-negative infiniy (2)

PROVE YOUR STATEMENT!

2

ee)xsinh(

xx

Summary: sinh

xsinh)xsinh(:ODD

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Analogy between trigonometric and hyperbolic functions

If t is any real number, then the point P(cos t, sin t) lies on the unit circle x2 + y2 = 1 because cos2 t + sin2 t = 1.T is the OPQ angle measured in radian Trigonometric functions are also called CIRCULAR functions

If t is any real number, then the point P(cosh t, sinh t) lies on the right branch of the hyperbola x2 - y2 = 1 because cosh2 t - sin2 t = 1 and cosh t ≥ 1. t does not represent the measure of an angle. HYPERBOLIC functions

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It turns out that t represents twice the area of the shaded hyperbolic sector

HYPERBOLIC FUNCTIONS

In the trigonometric case t represents twice the area of the shaded circular sector

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Identities

sinh cosh xx x e Except for the one above. if we have “trig-like” functions, it follows that we will have “trig-like” identities. For example:

2 2cosh sinh 1x x 2 2sin cos 1x x

xsinhxcoshx2cosh

xcoshxsinh2x2sinh22

xsinxcosx2cos

xcosxsin2x2sin22

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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

Proof of

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Other identitiesHW: Prove all remainder ones in your cheatsheet!

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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!

Derivatives

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Summary: Tanh(x)

xx

xx

ee

ee

)xcosh(

)xsinh()xtanh(

What are the asymptotes of tanh(x)

-in the infinity (2)

-In the negative infiniy (2)

PROVE YOUR STATEMENT!

Find the derivative!

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The velocity of a water wave with length L moving across a body of water with depth d is modeled by the function

where g is the acceleration due to gravity.

2tanh

2

gL dv

L

Application of tanh: description of ocean waves

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Hyperbolic cotangent

coshcoth

sinh

x x

x x

x e ex

x e e

What are the asymptotes of cotanh(x)

-in the infinity (2)

-In the negative infiniy (2)

-At 0?

PROVE YOUR STATEMENT!

Find the derivative!

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Summary: Hyperbolic Functions

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The sinh is one-to-one function. So, it has inverse function denoted by sinh-1

INVERSE HYPERBOLIC FUNCTIONS

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INVERSE HYPERBOLIC FUNCTIONS

The tanh is one-to-one function. So, it has inverse function denoted by tanh-1

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INVERSE FUNCTIONS

This figure shows that cosh is not one-to-one.However, when restricted to the domain [0, ∞],

it becomes one-to-one.

The inverse hyperbolic cosine function is defined as the inverse

of this restricted function

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Inverse hyperbolic functions

HW.: Define the inverse of the coth(x) function

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1 2

1 2

1 12

sinh ln 1

cosh ln 1 1

1tanh ln 1 1

1

x x x x

x x x x

xx x

x

INVERSE FUNCTIONS

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ey >0 1 2sinh ln 1x x x

sinh2

y ye ex y

INVERSE FUNCTIONS

222 4 41

2y x xe x x

2 1 0x x

ey – 2x – e-y = 0

multiplying by ez . e2y – 2xey – 1 = 0

(ey)2 – 2x(ey) – 1 = 0

(ey)2 – 2x(ey) – 1 = 0

1 2sinh ln 1x x x

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1 1

2 2

1 1

2 2

1 12 2

1 1(sinh ) (csch )

1 1

1 1cosh (sech )

1 11 1

(tanh ) (coth )1 1

d dx x

dx dxx x x

d dx x

dx dxx x xd d

x xdx dxx x

DERIVATIVES

The formulas for the derivatives of

tanh-1x and coth-1x appear to be

identical.

However, the domains of these

functions have no numbers in common:

tanh-1x is defined for | x | < 1.coth-1x is defined for | x | >1.

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Sources:

http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf