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AP Calculus BC Chapter 4 – Applications of Derivatives Lesson 4.2 – Mean Value Theorem

AP Calculus BC Chapter 4 – Applications of Derivatives Lesson 4.2 – Mean Value Theorem. Goal: Apply the MVT and find the intervals on which a function is increasing or decreasing. The Mean Value Theorem for Derivatives.

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AP Calculus BC Chapter 4 – Applications of Derivatives Lesson 4.2 – Mean Value Theorem

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  1. AP Calculus BC Chapter 4 – Applications of DerivativesLesson 4.2 – Mean Value Theorem Goal: Apply the MVT and find the intervals on which a function is increasing or decreasing.

  2. The Mean Value Theorem for Derivatives • Mean Value Theorem for Derivatives: If y=f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of it interior (a, b), then there is at least one point c in (a, b) at which • Note: The MVT says that the instantaneous change at some interior point must equal the average change over the entire interval.

  3. Increasing/Decreasing Functions • Increasing Function/Decreasing Function: Let f be a function defined on an interval I and let x1 and x2 be any two points in I. • f increases on I if x1 < x2 f(x1 ) < f(x2 ). • f decreases on I if x1 < x2 f(x1 ) > f(x2 ).

  4. Corollaries to the MVT • Corollary 1 – Increasing and Decreasing Functions: Let f be continuous on [a, b] and differentiable on (a, b). • 1. If f’ > 0 at each point of (a, b), then f increases on [a, b]. • 2. If f’ < 0 at each point of (a, b), then f decreases on [a, b]. Corollary 2 – Functions with f’ = 0 are Constant: IF f’(x) = 0 at each point of an interval I, then there is a constant C for which f(x) = C for all x in I.

  5. MVT Corollary and Antiderivatives • Corollary 3 – Functions with the Same Derivative Differ by a Constant: If f’(x) = g’(x) at each point of an interval I, then there is a constant C such that f(x) = g(x) + C for all x in I. • Antiderivative/Antidifferentiation: A function F(x) is an antiderivative of a function f(x) if F’(x) = f(x) for all x in the domain of f. The process of finding an antiderivative is antidifferentiation.

  6. Assignment and Note: • HW 4.2A: #3-24 ( every 3rd). • Quiz next Friday.

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