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Derivative of an Inverse. 1980 AB Free Response 3.

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Presentation on theme: "Derivative of an Inverse. 1980 AB Free Response 3."— Presentation transcript:

1 Derivative of an Inverse

2 1980 AB Free Response 3

3 Continuity and Differentiability of Inverses 1.If f is continuous in its domain, then its inverse is continuous on its domain. 2.If f is increasing on its domain, then its inverse is increasing on its domain 3.If f is decreasing on its domain, then its inverse is decreasing on its domain 4.If f is differentiable on an interval containing c and f '(c) does NOT equal 0, then the inverse is differentiable at f (c). Let’s investigate this…

4 Differentiability of an Inverse f is differentiable at x = 2. Since f (2) = 6, g(x) is differentiable at x = 6. If f is differentiable at c, the inverse is differentiable at f(c). Example: If f '(c) = 0, the inverse is not differentiable at f(c). Example: f '(0) = 0 Since f (0) = 2, g(x) is not differentiable at x = 2. Reciprocals.

5 The Derivative of an Inverse Assume that f(x) is differentiable and one- to-one on an interval I with inverse g(x). g(x) is differentiable at any x for which f '(g(x)) ≠ 0. In particular: Other Forms:

6 Example 1 A function f and its derivative take on the values shown in the table. xf (x)f '(x) 261/3 683/2 If g is the inverse of f, find g'(6).

7 Example 2 Let f (x) = x 3 + x – 2 and let g be the inverse function. Evaluate g'(0). Note: It is difficult to find an equation for the inverse function g. We NEED the formula to evaluate g'(0). (Solve x 3 + x – 2 = 0 with a calculator or guess and check)

8 2007 AB Free Response 3


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