4032 Fundamental Theorem AP Calculus

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

f’ T T f PDPD D C P : f ( 0 ) = v(t)

Where we have come. Calculus II: Accumulation Function

Accumulation: Riemann’s Right V T

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

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

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

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

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)

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)

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.

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!

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

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)

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!

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

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

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

Last Update: 1/20/10

Antiderivatives Layman’s Description: Assignment: Worksheet

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

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