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7 INVERSE FUNCTIONS 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. 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. INVERSE FUNCTIONS 7.7 Hyperbolic Functions In this section, we will learn about: Hyperbolic functions and their derivatives. DEFINITION e e sinh x 2 x x e e cosh x 2 x x sinh x tanh x cosh x 1 csc h x sinh x 1 sec h x cosh x cosh x coth x sinh x HYPERBOLIC FUNCTIONS The graphs of hyperbolic sine and cosine can be sketched using graphical addition, as in these figures. HYPERBOLIC FUNCTIONS Note that sinh has domain and range , whereas cosh has domain and range [1, ) . HYPERBOLIC FUNCTIONS The graph of tanh is shown. It has the horizontal asymptotes y = ±1. 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. APPLICATIONS The most famous application is the use of hyperbolic cosine to describe the shape of a hanging wire. 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.’ 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 gL 2 d v tanh 2 L where g is the acceleration due to gravity. HYPERBOLIC IDENTITIES The hyperbolic functions satisfy a number of identities that are similar to well-known trigonometric identities. HYPERBOLIC IDENTITIES We list some identities here. sinh( x) sinh x cosh( x) cosh x cosh x sinh x 1 2 2 1 tanh x sech x 2 2 sinh( x y ) sinh x cosh y cosh x sinh y cosh( x y ) cosh x cosh y sinh x sinh y HYPERBOLIC FUNCTIONS Prove: a. cosh2x – sinh2x = 1 b. 1 – tanh2 x = sech2x Example 1 HYPERBOLIC FUNCTIONS Example 1 a 2 2 e e e e cosh x sinh x 2 2 2x 2 x 2x 2 x e 2e e 2e 4 4 4 1 4 x 2 2 x x x 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: sinh 2 x 1 1 2 cosh x cosh 2 x or 1 tanh 2 x sec h 2 x HYPERBOLIC FUNCTIONS The identity proved in Example 1a gives a clue to the reason for the name ‘hyperbolic’ functions, as follows. 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 POQ in the figure. HYPERBOLIC FUNCTIONS For this reason, the trigonometric functions are sometimes called circular functions. 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. 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. DERIVATIVES OF HYPERBOLIC FUNCTIONS The derivatives of the hyperbolic functions are easily computed. For example, x x x x d d e e e e (sinh x) cosh x dx dx 2 2 DERIVATIVES Table 1 We list the differentiation formulas for the hyperbolic functions here. d (sinh x) cosh x dx d (cosh x) sinh x dx d (tanh x) sec h 2 x dx d (csc h x) csc h x coth x dx d (sec h x) sec h x tanh x dx d (coth x) csc h 2 x dx DERIVATIVES Equation 1 Note the analogy with the differentiation formulas for trigonometric functions. However, beware that the signs are different in some cases. d (sinh x) cosh x dx d (cosh x) sinh x dx d (tanh x) sec h 2 x dx d (csc h x) csc h x coth x dx d (sec h x) sec h x tanh x dx d (coth x) csc h 2 x dx DERIVATIVES Example 2 Any of these differentiation rules can be combined with the Chain Rule. For instance, d d sinh x (cosh x ) sinh x x dx dx 2 x 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. INVERSE FUNCTIONS This figure shows that cosh is not one-to-one. However, when restricted to the domain [0, ∞], it becomes one-to-one. INVERSE FUNCTIONS The inverse hyperbolic cosine function is defined as the inverse of this restricted function. INVERSE FUNCTIONS Definition 2 1 sinh y x 1 cosh y x and y 0 1 tanh y x y sinh x y cosh x y tanh x The remaining inverse hyperbolic functions are defined similarly. INVERSE FUNCTIONS By using these figures, we can sketch the graphs of sinh-1, cosh-1, and tanh-1. INVERSE FUNCTIONS The graphs of sinh-1, cosh-1, and tanh-1 are displayed. INVERSE FUNCTIONS Since the hyperbolic functions are defined in terms of exponential functions, it’s not surprising to learn that the inverse hyperbolic functions can be expressed in terms of logarithms. INVERSE FUNCTIONS Eqns. 3, 4, and 5 In particular, we have: 1 x ln x 1 sinh x ln x x 1 cosh 1 2 x 2 1 x tanh x ln 1 x 1 1 2 x x 1 1 x 1 INVERSE FUNCTIONS 1 Example 3 Show that sinh x ln x x 1 . 2 y y e e Let y = sinh-1 x. Then, x sinh y 2 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 INVERSE FUNCTIONS Example 3 Solving by the quadratic formula, we get: y 2 x 4 x 2 4 e x x 1 2 Note that ey > 0, but x (because x x2 1 ). 2 x2 1 0 So, the minus sign is inadmissible and we have: y 2 y ln( e ) ln x x 1 Thus, e y x x2 1 DERIVATIVES Table 6 d 1 1 (sinh x) dx 1 x2 d (csc h 1 x) dx x d 1 cosh x dx d 1 1 (sec h x) dx x 1 x2 d 1 1 (coth x) dx 1 x2 1 x2 1 d 1 1 (tanh x) dx 1 x2 1 x2 1 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 | < 1. coth-1x is defined for | x | >1. 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. DERIVATIVES E. g. 4—Solution 1 d 1 1 Prove that . (sinh x) 2 dx 1 x Let y = sinh-1 x. Then, sinh y = x. If we differentiate this equation implicitly with respect to x, we get: cosh y dy 1 dx As cosh2 y - sin2 y = 1 and cosh y ≥ 0, we have: cosh y 1 sinh 2 y 1 1 1 So, dy dx cosh y 1 sinh y 2 1 x2 DERIVATIVES E. g. 4—Solution 2 From Equation 3, we have: d d 1 2 sinh x ln x x 1 dx dx 1 d x x2 1 x x 2 1 dx x 1 x x2 1 x2 1 1 x x2 1 x x 1 2 x 1 2 1 x 1 2 DERIVATIVES Example 5 d 1 tanh (sin x) . Find dx Using Table 6 and the Chain Rule, we have: d 1 1 d tanh (sin x) (sin x) 2 dx 1 (sin x) dx 1 cos x 2 1 sin x cos x sec x 2 cos x DERIVATIVES Evaluate Example 6 1 dx 0 1 x 2 . Using Table 6 (or Example 4), we know that an antiderivative of 1 1 x 2 is sinh-1x. Therefore, 1 1 dx 1 0 1 x 2 sinh x 0 sinh1 1 ln 1 2 [from Equation 3]