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2014 Derivatives of Inverse Functions

2014 Derivatives of Inverse Functions. AP Calculus. Monotonic – always increasing or always decreasing. Inverses. Existence of an Inverse: If f(x) is one-to-one on its domain D , then f is called invertible. Further, Domain of f = Range of f -1

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2014 Derivatives of Inverse Functions

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  1. 2014 Derivatives of Inverse Functions AP Calculus

  2. Monotonic – always increasing or always decreasing Inverses Existence of an Inverse: If f(x) is one-to-one on its domain D , then f is called invertible. Further, Domain of f = Range of f -1 Range of f = Domain of f -1 One-to One Functions: A function f(x) is one-to one (on its domain D) if for every x there exists only one y and for every y there exists only one x Horizontal line test.

  3. Find the inverse Switch x and y multiply distribute Collect y factor divide

  4. Find the inverse

  5. for x ≥ 2 makes it monotonic

  6. REVIEW: Inverse Functions If f(x) is a function and ( x, y) is a point on f(x) , then the inverse f -1(x) contains the point ( y, x) To find f -1(x) Reverse the x and y and resolve for y. (a,b) (b,a) Theorem: f and g are inverses iff f(g(x)) = g(f(x)) = x

  7. Restricting the Domain: If a function is not one-to-one the domain can be restricted to portions that are one-to-one.

  8. Restricting the Domain: If a function is not one-to-one the domain can be restricted to portions that are one-to-one. Increasing ( Decreasing Increasing (3, Has an inverse on each interval

  9. Find the derivative of the inverse by implicit differentiation ( without solving for f -1 (x) ) Remember : f -1 (x) = f (y) ; therefore, find

  10. f(a,b) =m (a,b) Derivative of the Inverse The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals. (b,a) f(x) slope @ a = 3 Derivative of an Inverse Function: Given f is a differentiable one-to-one function and f -1is the inverse of f . If b belongs to the domain of f -1and f /(f(x) ≠ 0 , then f -1(b) exists and

  11. (a,b) Derivative of the Inverse The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals. (b,a) Derivative of an Inverse Function: If is the derivative of f, Then is the derivative of f -1(b) CAUTION: Pay attention to the plug in value!!!

  12. (4,16) ILLUSTRATION: (16,4) Find the derivative of f -1at (16,4) a) Find the Inverse. b) Use the formula.

  13. Find the derivative of the Inverse at the given point, (b,a). EX: (-1,6) Theorem:

  14. Inverse Functions REMEMBER: The x in the inverse (S) is the y in the original (f) If S(x) = f -1 (x), then S / (3) = If S(x) = f -1 (x), then S / (10) = 3 is the y value 10 is the y value

  15. Last Update • 1/8/14 • Assignment: Worksheet 91

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