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Fundamental Theorem

Fundamental Theorem. AP Calculus. Where we have come. Calculus I: Rate of Change Function. f’. f. T. P D. D C. T. Where we have come. Calculus II: Accumulation Function. Accumulation: Riemann’s Right . V. T. Using the Accumulation Model, the Definite Integral represents

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Fundamental Theorem

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  1. Fundamental Theorem AP Calculus

  2. Where we have come. Calculus I: Rate of Change Function

  3. f’ f T P D D C T

  4. Where we have come. Calculus II: Accumulation Function

  5. Accumulation: Riemann’s Right V T

  6. Using the Accumulation Model, the Definite Integral represents NET ACCUMULATION -- combining both gains and losses Accumulation (2) D V 8 8 6 5 3 T -3 -3 -4 T REM: Rate * Time = Distance

  7. Accumulation: Exact Accumulation V f ( x i) T x

  8. Where we have come. Calculus I: Rate of Change Function Calculus II: Accumulation Function Using DISTANCE model f’ = velocity f = Position Σ v(t) Δt = Distance traveled

  9. Distance Model: How Far have I Gone? V T Distance Traveled: a) b)

  10. B). The Fundamental Theorem DEFN: THE DEFINITE INTEGRAL If f is defined on the closed interval [a,b] and exists , then

  11. B). The Fundamental Theorem The Definition of the Definite Integral shows the set-up. Your work must include a Riemann’s sum! (for a representative rectangle)

  12. The Fundamental Theorem of Calculus (Part A) If or F is an antiderivative of f, then

  13. The Fundamental Theorem of Calculus shows how to solve the problem! Your work must include an anti-derivative! REM: The Definite Integral is a NUMBER -- the Net Accumulation of Area or Distance -- It may be positive, negative, or zero.

  14. Evaluate each Definite Integral using the FTC. 1) Practice: 2). 3). The FTC give the METHOD TO SOLVE Definite Integrals.

  15. Example: SET UP Find the NET Accumulation represented by the region between the graph and the x - axis on the interval [-2,3]. REQUIRED: Your work must include a Riemann’s sum! (for a representative rectangle)

  16. Example: Work Find the NET Accumulation represented by the region between the graph and the x - axis on the interval [-2,3]. REQUIRED: Your work must include an antiderivative!

  17. Method: (Grading) A). 1. 2. B). 3. 4. C). 5. D). 6. 7.

  18. Example: Find the NET Accumulation represented by the region between the graph and the x - axis on the interval .

  19. Example: Find the NET Accumulation represented by the region between the graph and the x - axis on the interval .

  20. Last Update: • 1/20/10

  21. Assignment: Worksheet Antiderivatives Layman’s Description:

  22. Using the Accumulation Model, the Definite Integral represents NET ACCUMULATION -- combining both gains and losses Accumulating Distance (2) D V 4 T T REM: Rate * Time = Distance

  23. Rectangular Approximations V = f (t) Velocity Time Distance Traveled: a) b)

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