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# Lin McMullin - PowerPoint PPT Presentation

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

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&

Functions Defined by Integrals

Lin McMullin

• 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:

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

Then f (4) =

• 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?

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?

• 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)

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