INVERSE FUNCTIONS

1. 7 INVERSE FUNCTIONS

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

3. 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.

4. INVERSE FUNCTIONS 7.7 Hyperbolic Functions • In this section, we will learn about: • Hyperbolic functions and their derivatives.

5. DEFINITION

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

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

8. HYPERBOLIC FUNCTIONS • The graph of tanh is shown. • It has the horizontal asymptotes y = ±1.

9. 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.

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

11. 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.’

12. 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.

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

14. HYPERBOLIC IDENTITIES • We list some identities here.

15. HYPERBOLIC FUNCTIONS Example 1 • Prove: • cosh2x – sinh2x = 1 • 1 – tanh2x = sech2x

16. HYPERBOLIC FUNCTIONS Example 1 a

17. 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:

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

19. 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 cos2t + sin2 t = 1. • In fact, t can be interpreted as the radian measure of in the figure.

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

21. 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 • cosh2t -sin2t =1 and cosh t≥ 1. • This time, t does not represent the measure of an angle.

22. 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.

23. DERIVATIVES OF HYPERBOLIC FUNCTIONS • The derivatives of the hyperbolic • functions are easily computed. • For example,

24. DERIVATIVES Table 1 • We list the differentiation formulas for • the hyperbolic functions here.

25. DERIVATIVES Equation 1 • Note the analogy with the differentiation • formulas for trigonometric functions. • However, beware that the signs are different in some cases.

26. DERIVATIVES Example 2 • Any of these differentiation rules can • be combined with the Chain Rule. • For instance,

27. 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.

28. INVERSE FUNCTIONS • This figure shows that cosh is not • one-to-one. • However, when restricted to the domain • [0, ∞], it becomes one-to-one.

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

30. INVERSE FUNCTIONS Definition 2 • The remaining inverse hyperbolic functions are defined similarly.

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

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

33. 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.

34. INVERSE FUNCTIONS Eqns. 3, 4, and 5 • In particular, we have:

35. 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

36. INVERSE FUNCTIONS Example 3 • Solving by the quadratic formula, • we get: • Note that ey> 0, but (because ). • So, the minus sign is inadmissible and we have: • Thus,

37. DERIVATIVES Table 6

38. 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.

39. 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.

40. DERIVATIVES E. g. 4—Solution 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 cosh2y -sin2y =1 and cosh y≥ 0, we have: • So,

41. DERIVATIVES E. g. 4—Solution 2 • From Equation 3, we have:

42. DERIVATIVES Example 5 • Find . • Using Table 6 and the Chain Rule, we have:

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