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Hyperbolic functions

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Hyperbolic functions * Hyperbolic functions Hungarian and English notation Groupwork in 4 groups For each function : - find domain - discuss parity - find limits at ... – PowerPoint PPT presentation

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Title: Hyperbolic functions


1
Hyperbolic functions
2
Hyperbolic functions Hungarian and English
notation
3
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

4
Summary cosh
  • What are the asymptotes of cosh(x)
  • in the infinity (2)
  • negative infiniy (2)
  • PROVE YOUR STATEMENT!

5
Summary cosh
  • Application of the use of hyperbolic cosine to
    describe the shape of a hanging wire/chain.

6
Background
  • So, cables like power line cables, which hang
    freely, hang in curves called hyperbolic cosine
    curves.

7
Chaincurve-catentity
8
Background
  • Suspension cables like
  • those of the Golden Gate
  • Bridge, which support a
  • constant load per horizontal
  • foot, hang in parabolas.

9
Which shape do you suppose in this case?
10
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'
11
Summary sinh
  • What are the asymptotes of cosh(x)
  • in the infinity (2)
  • negative infiniy (2)
  • PROVE YOUR STATEMENT!

12
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
13
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
14
Identities
Except for the one above. if we have trig-like
functions, it follows that we will have
trig-like identities. For example
15
Proof of
16
Other identities HW Prove all remainder ones in
your cheatsheet!
17
Derivatives
Surprise, this is positive!
18
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!

19
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.

20
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!

21
Summary Hyperbolic Functions
22
INVERSE HYPERBOLIC FUNCTIONS
  • The sinh is one-to-one function. So, it has
    inverse function denoted by sinh-1

23
INVERSE HYPERBOLIC FUNCTIONS
The tanh is one-to-one function. So, it has
inverse function denoted by tanh-1
24
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

25
Inverse hyperbolic functions
HW. Define the inverse of the coth(x) function
26
INVERSE FUNCTIONS
27
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

28
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.
29
Sources
  • http//www.mathcentre.ac.uk/resources/workbooks/ma
    thcentre/hyperbolicfunctions.pdf
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