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Differentiating the Inverse

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Objectives Students will be able to Calculate the inverse of a function. Determine if a function has an inverse. Differentiate the inverse of a given function. Find the equation of the tangent line to the inverse of a given function at a given point.

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Example 1 Find the inverse of the function Algebraically verify that the function f(x) and the function f -1 (x) are inverses of each other.

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Example 2-1 Determine if the function f(x) is one-to-one, continuous, and strictly increase or strictly decreasing.

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Example 2-2 Find the inverse g(x) for the function f(x) below

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Example 2-3 Find the derivative of f(x) (below) and g(x) (the inverse of f (x)).

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Inverse Function Theorem If f is continuous and strictly increasing (or strictly decreasing) in an interval I, then f has an inverse function g, which is continuous and strictly increasing (strictly decreasing) in the interval f(I). If x 0 is an interior point of I and f‘ (x 0 )≠0, then g is differentiable at y 0 = f(x 0 ) and

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Example 2-4 Verify that the Inverse Function Theorem applies to the functions f(x) and g(x) at x 0 = 2

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Example 2-5 Find the equation of the tangent line to the curve g(x) at point (7, 2)

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