6035 functions defined by the definite integral ab calculus
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6035 Functions Defined by the Definite Integral
AB Calculus
Accumulation Functions
Given
Then A (x) is the Accumulation function. The points on A(x) reflects the amount under the curve f (t).
2
bA x f t dt
Net Area: Net Distance Net Money:
BIG PICTURE:
Functions Defined by the Definite Integral
f (t)
2( ) 1f t t
2
0
( ) ( 1)x
A x t dt Also can work with negative accumulation.
A (-1) =
A (-2) =
Functions Defined by the Definite Integral
f (t)
2( ) 1f t t 2
0
( ) ( 1)x
A x t dt A (0) =
A (1) =
A (2) =
A(3) =
TI-89 Graph then F-5 Math #7
TI-83 2nd Calc #7
x
y
x
y
x
y
0
12
4
3
14
3
x
y
Functions Defined by the Definite Integral
f (t)2( ) 1f t t 2
0
( ) ( 1)x
A x t dt Also can work with negative accumulation.
A (-1) =
A (-2) =
x
y
x
y
x
y
4
3
14
3
Functions Defined by the Definite Integralf (t) A (x)
2( ) 1f t t
2
0
( ) ( 1)x
A x t dt
A (x) points indicate the quantity of accumulation under f (t).
x
y
Verify: Write the equation A(x)
2
0
( ) ( 1)x
A x t dt
A (0) = , A (1) = , A (2) = ,
A(3) = , A (-1)= , A (-2) =
=
3
( ) 03
xA x x
=
Writing the Equations: Initial Values = Particular Solutions
2
0
( ) ( 1)x
A x t dt
2
2
( ) ( 1)x
A x t dt
2
1
( ) ( 1)x
A x t dt
What do -2, 0, and 1 represent?
REM: The Antiderivative finds…
Writing the Equations: Initial Values = Particular Solutions
Example:3
3
( 4 )x
t t dt
2
cos 2
x
t dt
20 (3 1)
x tdt
t
Initial Value Problems : 1st Fundamental Theorem
Think: I have $200.00 and deposit $20.00 a week for 4 weeks.
My brother has $350.00 and deposits $20.00 a week for 4 weeks.
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
b
a
b
a
b
a
f t dt F b F a
F b F a f t dt
F b F a f t dt
Words:
or
Initial Value Problems (concept)
If A (0) = 4 , Find A (7)
b
0
A(x)= f(t)dtIf
7
0
7
0
7
0
A(7) - A(0) = f(t)dt
(7) A(0) + f(t)dt
(7) 4 + f(t)dt
A
A
7
0
f(t)dt
7
0
f(t)dt4 +
4
Initial Value Problems
If A (5) = 6 , Find A (8)
82
5
A(x)= (t +1)dtIf
Accumulation Functions
Suppose f (1) = 10. Find f (3)
Suppose f (0) = 0. Find f (1), f (2), f (3)
The graph the derivative, f / ,is given.
Accumulation Functions
The graph of a function, f , is shown.
a. Evaluate
b. Determine the average value
of the function on the
interval [ 1 , 7 ].
c. If F( 1) = -2 find F ( 7).
7
1( )f x dx
d. Determine the answers to parts a, b and c if the graph is translated two units up.
AP type
Last Update:
• 01/30/10
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