Hyperbolic functions

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Hyperbolic functions
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Hyperbolic functions Hungarian and English
notation
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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

• What are the asymptotes of cosh(x)
• in the infinity (2)
• negative infiniy (2)
• PROVE YOUR STATEMENT!

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Summary cosh

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

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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

120'
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Summary sinh

• What are the asymptotes of cosh(x)
• in the infinity (2)
• negative infiniy (2)
• PROVE YOUR STATEMENT!

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Analogy between trigonometric and hyperbolic
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
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
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HYPERBOLIC FUNCTIONS

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

In the trigonometric case t represents twice the
area of the shaded circular sector
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Identities
Except for the one above. if we have trig-like
functions, it follows that we will have
trig-like identities. For example
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Proof of
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Other identities HW Prove all remainder ones in
your cheatsheet!
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Derivatives
Surprise, this is positive!
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Summary Tanh(x)

• 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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Application of tanh description of ocean waves

• 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.

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Hyperbolic cotangent

• 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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INVERSE HYPERBOLIC FUNCTIONS

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

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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, 8,
• 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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INVERSE FUNCTIONS
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INVERSE FUNCTIONS
(ey)2 2x(ey) 1 0
ey 2x e-y 0
multiplying by ez . e2y 2xey 1 0
(ey)2 2x(ey) 1 0

• ey gt0

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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 lt
1. coth-1x is defined for x gt1.
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Sources

• http//www.mathcentre.ac.uk/resources/workbooks/ma
  thcentre/hyperbolicfunctions.pdf