1. 4032 Fundamental Theorem AP Calculus

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

3. f’ T T f PDPD D C P : f ( 0 ) = 0 +- 2.5 6 8 v(t)

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

5. Accumulation: Riemann’s Right V T

6. Accumulation (2) Using the Accumulation Model, the Definite Integral represents NET ACCUMULATION -- combining both gains and losses V T D T REM: Rate * Time = Distance 5 886 3 -3 -4 -3

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

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) If I go 5 mph for one hour and 25mph for 3 hours what is the total distance traveled? Ending position-beginning position

10. B). The Fundamental Theorem DEFN: THE DEFINITE INTEGRAL If f is defined on the closed interval [a,b] and exists, then Height base Rate time The Definition of the Definite Integral shows the set-up. Your work must include a Riemann’s sum! (for a representative rectangle)

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 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.

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

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

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. 3. B)4. 5. C).6. 7. Graph and rectangle Height (top – bottom) or (right – left) or (big – little) Riemann’s Sum Definite Integral [must have dx or dy] antiderivative answer

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. Antiderivatives Layman’s Description: Assignment: Worksheet

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

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