Accumulation & Functions Defined by Integrals. Lin McMullin. Accumulation & Functions Defined by Integrals. Or Thoughts on . My Favorite Equation!.
Functions Defined by Integrals
My Favorite Equation!
The goals of the AP Calculus program include the statement, “Students should understand the definite integral … as the net accumulation of change….” The Topical Outline includes the topic the “definite integral of the rate of change of a quantity over an interval interpreted as the [net] change of the quantity over the interval:
If f is an antiderivative of such that f (1) = 0
Then f (4) =
An object traveling in a straight line has position
x(t) at time t. If the initial position is x(0) = 2 and
the velocity of the object is , what is
the position of the object at t = 3?
The x-intercepts are x = - 2 and x= 3ln(5/3) = M
With the initial condition f (0) = 5
f (0) = 5
Find f (4)
Find the x-coordinate of the absolute maximum value and justify your answer.
M = 3ln(5/3)
and since f ´(x) ≥ 0 on [-4, M ] it follows that f (M) > f (-4).
and since on [M, 4] £ 0 it follows that f (M) > f (4)
Since M is the only critical number in the interval [-4, 4] and f (M) > f (-4) and f (M) > f (4), x = M is the location of the absolute maximum value by the Candidates’ Test.