4035 Functions Defined by the Definite Integral

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Presentation transcript:

4035 Functions Defined by the Definite Integral BC Calculus

Accumulation Functions BIG PICTURE: Given Then A (x) is the Accumulation function. The points on A(x) reflects the amount under the curve f (t). Net Area: Net Distance Net Money:

Functions Defined by the Definite Integral f (t) Also can work with negative accumulation. A (-1) = A (-2) =

Functions Defined by the Definite Integral f (t) A (0) = A (1) = A (2) = A(3) = 12 TI-89 Graph then F-5 Math #7 TI-83 2nd Calc #7

Functions Defined by the Definite Integral f (t) Also can work with negative accumulation. A (-1) = A (-2) =

Functions Defined by the Definite Integral f (t) A (x) A (x) points indicate the quantity of accumulation under f (t).

Verify: Write the equation A(x) = = A (0) = , A (1) = , A (2) = , A(3) = , A (-1)= , A (-2) =

Writing the Equations: Initial Values = Particular Solutions REM: The Antiderivative finds… What do -2, 0, and 1 represent?

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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. or Words:

Initial Value Problems (concept) If If A (0) = 4 , Find A (7) 4 + 4

Initial Value Problems If If A (5) = 6 , Find A (8)

Accumulation Functions The graph the derivative, f / ,is given. Suppose f (1) = 10. Find f (3) Suppose f (0) = 0. Find f (1), f (2), f (3)

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). d. Determine the answers to parts a, b and c if the graph is translated two units up.

AP type

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