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Accumulation & Functions Defined by Integrals. Lin McMullin. Accumulation & Functions Defined by Integrals. Or Thoughts on . My Favorite Equation!.

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Accumulation

&

Functions Defined by Integrals

Lin McMullin


Accumulation functions defined by integrals
Accumulation & Functions Defined by Integrals

  • Or Thoughts on

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….”[1] 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:








AP Example from 1997 BC 89

If f is an antiderivative of such that f (1) = 0

Then f (4) =


AP Example from 2008 AB 7

  • A particle moves along the x-axis with velocity given by

  • for time . If the particle is at the position

  • x = 2 at time t = 0, what is the position of the particle

  • at time t = 1?


AP Example from 2008 AB 87

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?


AP Example from 2008 AB 81

  • If G(x) is an antiderivative for f (x) and G(2) = -7,

  • then G(4) =

  • (A) f ´(4) (B) -7 + f ´(4) (C)

  • (D) (E)




2009 AB 6

M

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)

M


f (0) = 5

Find f (-4)

M


f (0) = 5

Find f (-4)

M


Find the x-coordinate of the absolute maximum value and justify your answer.

M = 3ln(5/3)

M


and since f ´(x) ≥ 0 on [-4, M ] it follows that f (M) > f (-4).

M


and since on [M, 4] £ 0 it follows that f (M) > f (4)

M


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.

M


Lin McMullin

E-mail: [email protected]

Blog: TeachingCalculus.wordpress.com

Website: www.LinMcMullin.net

Click on AP Calculus


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