Fundamental Theorem
23
Fundamental Theorem AP Calculus
description
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 - PowerPoint PPT Presentation
Transcript of Fundamental Theorem
Fundamental Theorem
AP Calculus
Where we have come.Calculus I:
Rate of Change Function f f
x
y
f’
T
T
f
PD
DC
Where we have come.Calculus II:
Accumulation Function
x
y
Accumulation: Riemann’s Right V
T
Accumulation (2)Using the Accumulation Model, the Definite Integral representsNET ACCUMULATION -- combining both gains and losses
V
T
D
T
REM: Rate * Time = Distance
x
y
5 8 8 63
-3-4
-3
x
y
Accumulation: Exact AccumulationV
T
1
8 8 8 x ( )
8 8lim
i i
n
n i
i ix f x fn n n
ifn n
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
f f
x
y
Distance Model: How Far have I Gone?V
T
Distance Traveled: a)
b)
B). The Fundamental Theorem
DEFN: THE DEFINITE INTEGRAL
If f is defined on the closed interval [a,b] and
exists , then0 1
lim ( )n
i ix i
f c x
0 1
lim ( ) ( )bn
i ixi a
f c x f x dx
B). The Fundamental Theorem
0 1
lim ( )n
i ix i
f c x
0 1
lim ( ) ( )bn
i ix i a
f c x f x dx
The Definition of the Definite Integral shows the set-up.
Your work must include a Riemann’s sum! (for a representative rectangle)
( )b
b
aa
f x dx F x
The Fundamental Theorem of Calculus (Part A)
If or F is an antiderivative of f,
then
F b F a
( ) ( )F x f x
( )
=
bb
aa
f x dx F x
F b F a
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) 1
3xdx
2).
4 2
1( 1)x dx
3).2
2sin( )x dx
The FTC give the METHOD TO SOLVE Definite Integrals.
Example: SET UPFind the NET Accumulation represented by the region between
the graph and the x - axis on the
interval [-2,3].
2( ) 2 5f x x x
REQUIRED:
Your work must include a Riemann’s sum! (for a representative rectangle)
Example: WorkFind the NET Accumulation represented by the region between
the graph and the x - axis on the
interval [-2,3].
2( ) 2 5f x x x
REQUIRED:
Your work must include an antiderivative!
Method: (Grading)
A). 1.
2.
B). 3.
4.
C). 5.
D). 6.
7.
Example:Find the NET Accumulation represented by the region between
the graph and the x - axis on the
interval .
3( ) 27f x x 0,3
Example:Find the NET Accumulation represented by the region between
the graph and the x - axis on the
interval .
( ) sec( ) tan( )f x x x
,4 3
Last Update:
• 1/20/10
AntiderivativesLayman’s Description:
2x dx cos( )x dx 2sec ( )x dx
2
1 dxx
1 dxx
Assignment: Worksheet
Accumulating Distance (2)Using the Accumulation Model, the Definite Integral representsNET ACCUMULATION -- combining both gains and losses
V
T
D
T
REM: Rate * Time = Distance
4
Rectangular Approximationsy = (x+5)(x^2-x+7)*.1
Velocity
Time
V = f (t)
Distance Traveled: a)
b)
