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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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Presentation Transcript
Accumulation

&

Functions Defined by Integrals

Lin McMullin

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)
A quick look at some free-response questions
• 2000 AB 4
• 2008 AB2 / BC2 (d)
• 2008 AB 3 (c)
• 2008 AB4 / BC 4 (a)
A quick look at some free-response questions
• AB 1 (a, c, d)
• AB 2 (c)
• 2010 AB 3 (a,d)
• 2010 AB 5 (a)
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

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