Derivatives of Inverse Functions

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Presentation transcript:

Derivatives of Inverse Functions AP Calculus AB

Terminology If R = f(T) ... resistance is a function of temperature, Then T = f -1(R) ... temperature is the inverse function of resistance. f -1(R) is read "f-inverse of R“ is not an exponent it does not mean reciprocal

Continuity and Differentiability Given f(x) a function Domain is an interval I If f has an inverse function f -1(x) then … If f(x) is continuous on its domain, then f -1(x) is continuous on its domain

Continuity and Differentiability Furthermore … If f(x) is differentiable at c and f '(c) ≠ 0 then f -1(x) is differentiable at f(c) f(x) f -1(x) Note the counter example f(x) not differentiable here f -1(x) not differentiable here

Derivative of an Inverse Function Given f(x) a function Domain is an interval I If f(x) has an inverse g(x) then g(x) is differentiable for any x where f '(g(x)) ≠ 0 And … f '(g(x)) ≠ 0

We Gotta Try This! Given g(2) = 2.055 and So Note that we did all this without actually taking the derivative of f -1(x)

Consider This Phenomenon For (2.055, 2) belongs to f(x) (2, 2.055) belongs to g(x) What is f '(2.055)? How is it related to g'(2)? By the definition they are reciprocals

Derivatives of Inverse Trig Functions Note further patterns on page 177

Practice Find the derivative of the following functions

More Practice Given Find the equation of the line tangent to this function at

Assignment Lesson 3.6 Page 179 Exercises 1 – 49 EOO, 67, 69