4032 Fundamental Theorem AP Calculus. Where we have come. Calculus I: Rate of Change Function.
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Transcript of 4032 Fundamental Theorem AP Calculus. Where we have come. Calculus I: Rate of Change Function.
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4032 Fundamental Theorem
AP Calculus
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Where we have come.Calculus I:
Rate of Change Function f f
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x
y
f’
T
T
f
PD
DC
P : f ( 0 ) = 0 + -
∪∩∪2.5 6 8
v(t)
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Where we have come.Calculus II:
Accumulation Function
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x
y
Accumulation: Riemann’s Right V
T
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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
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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
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x
y
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
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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
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)
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( )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
𝑣 (𝑡 )𝑑𝑡 ( )b
ad t 𝑑(𝑏)−𝑑 (𝑎)
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.
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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.
Top-bottom
¿𝑥2
2 | 1−3
¿𝑥3
3− 𝑥| 4
− 1
¿−𝑐𝑜𝑠𝑥| 𝜋2
− 𝜋2
¿12
− −32
2=− 4
¿ ( 643
−4 )−( − 13
+1)523 − 2
3=503
¿− 0+0=0
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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)
𝑏=∆ 𝑥 [− 2,3 ]-0
𝑡𝑜𝑝−𝑏𝑜𝑡𝑡𝑜𝑚𝐴= (𝑥2 −2 𝑥+5 ) ∆ 𝑥
lim𝑛→ ∞
∑ (𝑥2− 2𝑥+5 ) ∆ 𝑥
−2
3
(𝑥2− 2𝑥+5 ) ∆𝑥
∆ 𝑥
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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!
𝑥3
3−𝑥2+5 𝑥| 3
− 2
( 273
−9+15)−( − 83
− 4 − 10)4 53 +
503 =
953
∆ 𝑥
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Method: (Grading)
A). 1.
2.
3.
B) 4.
5.
C). 6.
7.
Graph and rectangle
Base and boundaries
Height (top – bottom) or (right – left) or (big – little)Riemann’s Sum
Definite Integral [must have dx or dy]
antiderivative
answer
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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
0
3
(27 −𝑥3 )𝑑𝑥
27 𝑥− 𝑥4
4 |30(81− 81
4 )− (0− 0 )
2434
𝑏=∆ 𝑥 [ 0,3 ]h=27 −𝑥3 −0𝐴= (27 −𝑥3 ) ∆ 𝑥
lim𝑛→ ∞
∑0
3
(27 −𝑥3 ) ∆𝑥
√
√
√
√
√
√
√
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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
−𝜋
4
𝜋3
( sec (𝑥 ) tan (𝑥 ) )𝑑𝑥
sec (𝑥 ) ⌊𝜋3
−𝜋4
¿¿
(2 − 2√2 )=2√2− 2
√2=
4 − 2√22
𝑏=∆ 𝑥 [− 𝜋4
, 𝜋3 ]
h=sec (𝑥 ) tan (𝑥 )− 0
lim𝑛→ ∞
∑− 𝜋
4
𝜋3
(𝑠𝑒𝑐 (𝑥 ) 𝑡𝑎𝑛 (𝑥 ) ) ∆ 𝑥
√
√
√
√√
√
√
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Last Update:
• 1/20/10
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AntiderivativesLayman’s Description:
2x dx cos( )x dx 2sec ( )x dx
2
1 dxx
1 dxx
Assignment: Worksheet
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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
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Rectangular Approximationsy = (x+5)(x^2-x+7)*.1
Velocity
Time
V = f (t)
Distance Traveled: a)
b)