exact accumulation and
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Exact Accumulation and . AP Calculus. A). Sigma Notation. REM:. Ex. REM: Sum of the first 100 numbers. 1+2+3+…….+98+99+100. Summation Formulas. Finite Series. EX 1. Evaluate using the formulas:. EX 2. Evaluate using the formulas:. Infinite Series. REM: Limits at Infinity. - PowerPoint PPT PresentationTRANSCRIPT
Exact Accumulation and
AP Calculus
A). Sigma Notation
REM:
6
3 3 3 3 3 3 3
1
1 2 3 4 5 6i
i
6
3
1i
i
7
3
2 1i
i
Ex.
Summation Formulas
1
n
i
c cn
2
1
12 2
n
i
n n ni n or
3 22
1
1 2 1 2 3
6 6
n
i
n n n n n ni or
22 4 3 23
1
1 2
4 4
n
i
n n n n ni or
REM: Sum of the first 100 numbers. 1+2+3+…….+98+99+100
Finite Series
EX 1. Evaluate using the formulas:
15
1
2 3i
i
EX 2. Evaluate using the formulas:
15
2
1
2i
i
Infinite Series
EX 3. Evaluate: 2
31
13
n
ni
Lim in
REM: Limits at Infinity
NOTICE: Inside:
i is the variable.
Outside: n is the variable
23
1
43 2
n
ni
Lim i xn
EX 4. Evaluate:
AREA
DEFN: The Area (accumulation) defined by a region in the plane is:
1
n
i in
i
A Lim f c x
1i i ix c x
We will use the RIGHT ENDPOINT and REGULAR PARTITIONS (easier!) b a
xn
if x
THEOREM: The Limit ( as ) of the Under Approximation and the Over Approximation (and therefore any Riemann’s Sum) are equal.
n
a b
a
x
( )if x
ith
ix
ith
Rect.
( )if x1st
Rect.2nd
Rect.
ix
Area: Exact AccumulationFind the Area (accumulation) determined by the region bounded
by and the x - axis on the interval [ 0, 2].2y x
Area: Exact AccumulationFind the Area (accumulation) determined by the region bounded
by and the x - axis on the interval [ 0, 3].24y x x
x
y
Area: Exact Accumulation Find the Area (accumulation) determined by the region bounded
by and the x - axis on the interval [ 1, 2].2( ) 4f x x
Area: Exact AccumulationFind the Area (accumulation) determined by the region bounded
by and the x - axis on the interval [ 2, 4].24y x x
x
y
Last Update:
• 1/15/10
• Assignment: Worksheet