INVERSE FUNCTIONS

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INVERSE FUNCTIONS
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INVERSE FUNCTIONS

• Certain even and odd combinations of
• the exponential functions ex and e-x arise so
• frequently in mathematics and its applications
• that they deserve to be given special names.

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INVERSE FUNCTIONS

• In many ways, they are analogous to
• the trigonometric functions, and they have
• the same relationship to the hyperbola that
• the trigonometric functions have to the circle.
• For this reason, they are collectively called
  hyperbolic functions and individually called
  hyperbolic sine, hyperbolic cosine, and so on.

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INVERSE FUNCTIONS
7.7 Hyperbolic Functions
In this section, we will learn
about Hyperbolic functions and their derivatives.
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DEFINITION
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HYPERBOLIC FUNCTIONS

• The graphs of hyperbolic sine and cosine
• can be sketched using graphical addition,
• as in these figures.

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HYPERBOLIC FUNCTIONS

• Note that sinh has domain and
• range , whereas cosh has domain
• and range .

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HYPERBOLIC FUNCTIONS

• The graph of tanh is shown.
• It has the horizontal asymptotes y 1.

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APPLICATIONS

• Some mathematical uses of hyperbolic
• functions will be seen in Chapter 8.
• Applications to science and engineering
• occur whenever an entity such as light,
• Velocity, electricity, or radioactivity is
• gradually absorbed or extinguished.
• The decay can be represented by hyperbolic
  functions.

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APPLICATIONS

• The most famous application is
• the use of hyperbolic cosine to describe
• the shape of a hanging wire.

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APPLICATIONS

• It can be proved that, if a heavy flexible cable
• is suspended between two points at the same
• height, it takes the shape of a curve with
• equation y c a cosh(x/a) called a catenary.
• The Latin word catena means chain.

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APPLICATIONS

• Another application occurs in the
• 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 IDENTITIES

• The hyperbolic functions satisfy
• a number of identities that are similar to
• well-known trigonometric identities.

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HYPERBOLIC IDENTITIES

• We list some identities here.

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HYPERBOLIC FUNCTIONS
Example 1

• Prove
• cosh2x sinh2x 1
• 1 tanh2 x sech2x

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HYPERBOLIC FUNCTIONS
Example 1 a
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HYPERBOLIC FUNCTIONS
Example 1 b

• We start with the identity proved in (a)
• cosh2x sinh2x 1
• If we divide both sides by cosh2x, we get

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HYPERBOLIC FUNCTIONS

• The identity proved in Example 1a
• gives a clue to the reason for the name
• hyperbolic functions, as follows.

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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.
• In fact, t can be interpreted as the radian
  measure of in the figure.

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HYPERBOLIC FUNCTIONS

• For this reason, the trigonometric
• functions are sometimes called
• circular functions.

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HYPERBOLIC FUNCTIONS

• Likewise, 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.
• This time, t does not represent the measure of
  an angle.

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HYPERBOLIC FUNCTIONS

• However, it turns out that t represents twice
• the area of the shaded hyperbolic sector in
• the first figure.
• This is just as in the trigonometric case t
  represents twice the area of the shaded circular
  sector in the second figure.

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DERIVATIVES OF HYPERBOLIC FUNCTIONS

• The derivatives of the hyperbolic
• functions are easily computed.
• For example,

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DERIVATIVES
Table 1

• We list the differentiation formulas for
• the hyperbolic functions here.

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DERIVATIVES
Equation 1

• Note the analogy with the differentiation
• formulas for trigonometric functions.
• However, beware that the signs are different in
  some cases.

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DERIVATIVES
Example 2

• Any of these differentiation rules can
• be combined with the Chain Rule.
• For instance,

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INVERSE HYPERBOLIC FUNCTIONS

• You can see from the figures that sinh
• and tanh are one-to-one functions.
• So, they have inverse functions denoted by
  sinh-1 and 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.

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INVERSE FUNCTIONS

• The inverse hyperbolic cosine
• function is defined as the inverse
• of this restricted function.

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INVERSE FUNCTIONS
Definition 2

• The remaining inverse hyperbolic functions are
  defined similarly.

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INVERSE FUNCTIONS

• By using these figures,
• we can sketch the graphs
• of sinh-1, cosh-1, and
• tanh-1.

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INVERSE FUNCTIONS

• The graphs of sinh-1,
• cosh-1, and tanh-1 are
• displayed.

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INVERSE FUNCTIONS

• Since the hyperbolic functions are defined
• in terms of exponential functions, its not
• surprising to learn that the inverse hyperbolic
• functions can be expressed in terms of
• logarithms.

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INVERSE FUNCTIONS
Eqns. 3, 4, and 5

• In particular, we have

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INVERSE FUNCTIONS
Example 3

• Show that .
• Let y sinh-1 x. Then,
• So, ey 2x e-y 0
• Or, multiplying by ey, e2y 2xey 1 0
• This is really a quadratic equation in ey (ey)2
  2x(ey) 1 0

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INVERSE FUNCTIONS
Example 3

• Solving by the quadratic formula,
• we get
• Note that ey gt 0, but
  (because ).
• So, the minus sign is inadmissible and we have
• Thus,

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DERIVATIVES
Table 6
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DERIVATIVES
Note

• 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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DERIVATIVES

• The inverse hyperbolic functions are
• all differentiable because the hyperbolic
• functions are differentiable.
• The formulas in Table 6 can be proved either by
  the method for inverse functions or by
  differentiating Formulas 3, 4, and 5.

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DERIVATIVES
E. g. 4Solution 1

• Prove that .
• Let y sinh-1 x. Then, sinh y x.
• If we differentiate this equation implicitly
  with respect to x, we get
• As cosh2 y - sin2 y 1 and cosh y 0, we have
• So,

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DERIVATIVES
E. g. 4Solution 2

• From Equation 3, we have

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DERIVATIVES
Example 5

• Find .
• Using Table 6 and the Chain Rule, we have

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DERIVATIVES
Example 6

• Evaluate
• Using Table 6 (or Example 4), we know that an
  antiderivative of is sinh-1x.
• Therefore,