7 INVERSE FUNCTIONS INVERSE FUNCTIONS 7 6 Inverse
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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 Here, we apply the ideas of Section 7. 1 to find the derivatives of the so-called inverse trigonometric functions.
INVERSE TRIGONOMETRIC FUNCTIONS However, we have a slight difficulty in this task. § As the trigonometric functions are not one-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.
INVERSE TRIGONOMETRIC FUNCTIONS Here, you can see that the sine function y = sin x is not one-to-one. § Use the Horizontal Line Test.
INVERSE TRIGONOMETRIC FUNCTIONS However, here, you can see that the function f(x) = sin x, is one-to-one. ,
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.
INVERSE SINE FUNCTIONS Equation 1 As the definition of an inverse function states that we have: § Thus, if -1 ≤ x ≤ 1, sin-1 x is the number between and whose sine is x.
INVERSE SINE FUNCTIONS Evaluate: a. b. Example 1
INVERSE SINE FUNCTIONS Example 1 a We have: § This is because and lies between , and .
INVERSE SINE FUNCTIONS Let , so Example 1 b . § Then, we can draw a right triangle with angle θ. § So, we deduce from the Pythagorean Theorem that the third side has length.
INVERSE SINE FUNCTIONS Example 1 b § This enables us to read from the triangle that:
INVERSE SINE FUNCTIONS Equations 2 In this case, the cancellation equations for inverse functions become:
INVERSE SINE FUNCTIONS The inverse sine function, sin-1, has domain [-1, 1] and range .
INVERSE SINE FUNCTIONS The graph is obtained from that of the restricted sine function by reflection about the line y = x.
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).
INVERSE SINE FUNCTIONS We could calculate the derivative of sin-1 by 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.
INVERSE SINE FUNCTIONS Let y = sin-1 x. § Then, sin y = x and –π/2 ≤ y ≤ π/2. § Differentiating sin y = x implicitly with respect to x, we obtain:
INVERSE SINE FUNCTIONS Formula 3 Now, cos y ≥ 0 since –π/2 ≤ y ≤ π/2, so
INVERSE SINE FUNCTIONS If f(x) = sin-1(x 2 – 1), find: (a) the domain of f. (b) f ’(x). (c) the domain of f ’. Example 2
INVERSE SINE FUNCTIONS Example 2 a Since the domain of the inverse sine function is [-1, 1], the domain of f is:
INVERSE SINE FUNCTIONS Example 2 b Combining Formula 3 with the Chain Rule, we have:
INVERSE SINE FUNCTIONS The domain of f ’ is: Example 2 c
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.
INVERSE COSINE FUNCTIONS Equation 5 The cancellation equations are:
INVERSE COSINE FUNCTIONS The inverse cosine function, cos-1, has domain [-1, 1] and range and is a continuous function. ,
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.
INVERSE TANGENT FUNCTIONS The tangent function can be made one-to-one by restricting it to the interval.
INVERSE TANGENT FUNCTIONS 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. Equation 7
INVERSE TANGENT FUNCTIONS E. g. 3—Solution 1 Simplify the expression cos(tan-1 x) § Let y = tan-1 x. § Then, tan y = x and .
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,
INVERSE TANGENT FUNCTIONS Thus, E. g. 3—Solution 1
INVERSE TANGENT FUNCTIONS E. g. 3—Solution 2 Instead of using trigonometric identities, it is perhaps easier to use a diagram. § If y = tan-1 x, then tan y = x. § We can read from the figure (which illustrates the case y > 0) that:
INVERSE TANGENT FUNCTIONS The inverse tangent function, tan-1 = arctan, has domain range . and
INVERSE TANGENT FUNCTIONS We know that: § So, the lines are vertical asymptotes of the graph of tan.
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/2 and y = -π/2 are horizontal asymptotes of the graph of tan-1.
INVERSE TANGENT FUNCTIONS Equations 8 This fact is expressed by these limits:
INVERSE TANGENT FUNCTIONS Example 4 Evaluate: § Since the first equation in Equations 8 gives:
INVERSE TANGENT FUNCTIONS Since tan is differentiable, tan-1 is also differentiable. To find its derivative, let y = tan-1 x. § Then, tan y = x.
INVERSE TANGENT FUNCTIONS Equation 9 Differentiating that latter equation implicitly with respect to x, we have: Thus,
INVERSE TRIG. FUNCTIONS Equations 10 The remaining inverse trigonometric functions are not used as frequently and are summarized as follows.
INVERSE TRIG. FUNCTIONS Equations 10
INVERSE TRIG. FUNCTIONS The choice of intervals for y in the definitions of csc-1 and sec-1 is not universally agreed upon.
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.
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.
DERIVATIVES Table 11
DERIVATIVES Each of these formulas can be combined with the Chain Rule. § For instance, if u is a differentiable function of x, then
DERIVATIVES Differentiate: Example 5
DERIVATIVES Example 5 a
DERIVATIVES Example 5 b
INVERSE TRIG. FUNCTIONS Example 6 Prove the identity § Although calculus is not needed to prove this, the proof using calculus is quite simple.
INVERSE TRIG. FUNCTIONS Example 6 If f(x) = tan-1 x + cot-1 x , then for all values of x. § Therefore f(x) = C, a constant.
INVERSE TRIG. FUNCTIONS Example 6 To determine the value of C, we put x = 1. § Then, § Thus, tan-1 x + cot-1 x = π/2.
INTEGRATION FORMULAS Equations 12 & 13 Each of the formulas in Table 11 gives rise to an integration formula. § The two most useful of these are:
INTEGRATION FORMULAS Example 7 Find: § If we write the integral resembles Equation 12 and the substitution u = 2 x is suggested. § This gives du = 2 dx; so dx = du/2.
INTEGRATION FORMULAS When x = 0, u = 0. When x = ¼, u = ½. Thus, Example 7
INTEGRATION FORMULAS Example 8 Evaluate: § To make the given integral more like Equation 13, we write: § This suggests that we substitute u = x/a.
INTEGRATION FORMULAS Example 8 Then, du = dx/a, dx = a du, and
INTEGRATION FORMULAS E. g. 8—Formula 14 Thus, we have the formula
INTEGRATION FORMULAS Example 9 Find: § We substitute u = x 2 because then du = 2 x dx and we can use Equation 14 with a = 3:
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