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INVERSE FUNCTIONS

7. INVERSE FUNCTIONS. INVERSE FUNCTIONS. 7.6 Inverse Trigonometric Functions. In this section, we will learn about: Inverse trigonometric functions and their derivatives. INVERSE TRIGONOMETRIC FUNCTIONS.

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INVERSE FUNCTIONS

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  1. 7 INVERSE FUNCTIONS

  2. INVERSE FUNCTIONS 7.6 Inverse Trigonometric Functions • In this section, we will learn about: • Inverse trigonometric functions • and their derivatives.

  3. INVERSE TRIGONOMETRIC FUNCTIONS • Here, we apply the ideas of Section 7.1 to find the derivatives of the so-called inverse trigonometric functions.

  4. INVERSE TRIGONOMETRIC FUNCTIONS • However, we have a slight difficulty in this task. • As the trigonometric functions are notone-to-one, they don’t have inverse functions. • The difficulty is overcome by restricting the domains of these functions so that they become one-to-one.

  5. INVERSE TRIGONOMETRIC FUNCTIONS • Here, you can see that the sine function y = sin x is not one-to-one. • Use the Horizontal Line Test.

  6. INVERSE TRIGONOMETRIC FUNCTIONS • However, here, you can see that the function f(x) = sin x, , is one-to-one.

  7. INVERSE SINE FUNCTION / ARCSINE FUNCTION • The inverse function of this restricted sine function f exists and is denoted by sin-1 or arcsin. • It is called the inverse sine function or the arcsine function.

  8. INVERSE SINE FUNCTIONS Equation 1 • As the definition of an inverse function states • that • we have: • Thus, if -1 ≤x≤ 1, sin-1x is the number between and whose sine is x.

  9. INVERSE SINE FUNCTIONS Example 1 • Evaluate: • a. • b.

  10. INVERSE SINE FUNCTIONS Example 1 a • We have: • This is because , and lies between and .

  11. INVERSE SINE FUNCTIONS Example 1 b • Let , so . • Then, we can draw a right triangle with angle θ. • So, we deduce from the Pythagorean Theorem that the third side has length .

  12. INVERSE SINE FUNCTIONS Example 1b • This enables us to read from the triangle that:

  13. INVERSE SINE FUNCTIONS Equations 2 • In this case, the cancellation equations • for inverse functions become:

  14. INVERSE SINE FUNCTIONS • The inverse sine function, sin-1, has domain [-1, 1] and range .

  15. INVERSE SINE FUNCTIONS • The graph is obtained from that of • the restricted sine function by reflection • about the line y = x.

  16. INVERSE SINE FUNCTIONS • We know that: • The sine function f is continuous, so the inverse sine function is also continuous. • The sine function is differentiable, so the inverse sine function is also differentiable (from Section 3.4).

  17. INVERSE SINE FUNCTIONS • We could calculate the derivative of sin-1by the formula in Theorem 7 in Section 7.1. • However, since we know that is sin-1 differentiable, we can just as easily calculate it by implicit differentiation as follows.

  18. INVERSE SINE FUNCTIONS • Let y = sin-1x. • Then, sin y = x and –π/2 ≤y≤π/2. • Differentiating sin y = x implicitly with respect to x, we obtain:

  19. INVERSE SINE FUNCTIONS Formula 3 • Now, cos y≥ 0 since –π/2 ≤y≤π/2, so

  20. INVERSE SINE FUNCTIONS Example 2 • If f(x) = sin-1(x2 – 1), find: • the domain of f. • f’(x). • the domain of f’.

  21. INVERSE SINE FUNCTIONS Example 2 a • Since the domain of the inverse sine function is [-1, 1], the domain of f is:

  22. INVERSE SINE FUNCTIONS Example 2 b • Combining Formula 3 with the Chain Rule, we have:

  23. INVERSE SINE FUNCTIONS Example 2 c • The domain of f’ is:

  24. INVERSE COSINE FUNCTIONS Equation 4 • The inverse cosine function is handled similarly. • The restricted cosine function f(x) = cos x, 0 ≤ x ≤ π,is one-to-one. • So, it has an inverse function denoted by cos-1 or arccos.

  25. INVERSE COSINE FUNCTIONS Equation 5 • The cancellation equations are:

  26. INVERSE COSINE FUNCTIONS • The inverse cosine function,cos-1, has domain [-1, 1] and range , and is a continuous function.

  27. INVERSE COSINE FUNCTIONS Formula 6 • Its derivative is given by: • The formula can be proved by the same method as for Formula 3. • It is left as Exercise 17.

  28. INVERSE TANGENT FUNCTIONS • The tangent function can be made one-to-one by restricting it to the interval .

  29. INVERSE TANGENT FUNCTIONS Equation 7 • Thus, the inverse tangent • function is defined as • the inverse of the function • f(x) = tan x, • . • It is denoted by tan-1 or arctan.

  30. INVERSE TANGENT FUNCTIONS E. g. 3—Solution 1 • Simplify the expression cos(tan-1x) • Let y = tan-1x. • Then, tan y = x and .

  31. INVERSE TANGENT FUNCTIONS E. g. 3—Solution 1 • We want to find cos y. • However,since tan y is known, it is easier to find sec y first. • Therefore,

  32. INVERSE TANGENT FUNCTIONS E. g. 3—Solution 1 • Thus,

  33. INVERSE TANGENT FUNCTIONS E. g. 3—Solution 2 • Instead of using trigonometric identities, it is perhaps easier to use a diagram. • If y = tan-1x, then tan y = x. • We can read from the figure (which illustrates the case y > 0) that:

  34. INVERSE TANGENT FUNCTIONS • The inverse tangent function, tan-1 = arctan, has domain and range .

  35. INVERSE TANGENT FUNCTIONS • We know that: • So, the lines are vertical asymptotes of the graph of tan.

  36. INVERSE TANGENT FUNCTIONS • The graph of tan-1 is obtained by reflecting the graph of the restricted tangent function about the line y = x. • It follows that the lines y = π/2and y = -π/2are horizontal asymptotes of the graph of tan-1.

  37. INVERSE TANGENT FUNCTIONS Equations 8 • This fact is expressed by these limits:

  38. INVERSE TANGENT FUNCTIONS Example 4 • Evaluate: • Sincethe first equation in Equations 8 gives:

  39. INVERSE TANGENT FUNCTIONS • Since tan is differentiable, tan-1 is also differentiable. • To find its derivative, let y = tan-1x. • Then, tan y = x.

  40. INVERSE TANGENT FUNCTIONS Equation 9 • Differentiating that latter equation implicitly with respect to x, we have: • Thus,

  41. INVERSE TRIG. FUNCTIONS Equations 10 • The remaining inverse trigonometric functions are not used as frequently and are summarized as follows.

  42. INVERSE TRIG. FUNCTIONS Equations 10

  43. INVERSE TRIG. FUNCTIONS • The choice of intervals for y in the definitions of csc-1 and sec-1 is not universally agreed upon.

  44. INVERSE TRIG. FUNCTIONS • For instance, some authors use in the definition of sec-1. • You can see from the graph of the secant function that both this choice and the one in Equations 10 will work.

  45. DERIVATIVES OF INVERSE TRIG. FUNCTIONS • In the following table, we collect the differentiation formulas for all the inverse trigonometric functions. • The proofs of the formulas for the derivatives of csc-1, sec-1, and cot-1 are left as Exercises 19–21.

  46. DERIVATIVES Table 11

  47. DERIVATIVES • Each of these formulas can be combined with the Chain Rule. • For instance, if u is a differentiable function of x, then

  48. DERIVATIVES Example 5 • Differentiate:

  49. DERIVATIVES Example 5 a

  50. DERIVATIVES Example 5 b

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