Accumulation Functions Defined by Integrals Lin Mc Mullin
Accumulation & Functions Defined by Integrals Lin Mc. Mullin
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: ”
Final Value = Starting Value + Accumulated Change
Final Position = Initial Position + Displacement
The first time you saw this ….
The first time you saw this ….
The first time you saw this ….
The first time you saw this ….
AP Example from 1997 BC 89 If f is an antiderivative of Then f (4) = such that f (1) = 0
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 the position of the object at t = 3? , what is
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) (D) (B) -7 + f ´(4) (E) (C)
A quick look at some free-response questions 2000 AB 4 2008 AB 2 / BC 2 (d) 2008 AB 3 (c) 2008 AB 4 / BC 4 (a)
A quick look at some free-response questions 2010 AB 1 (a, c, d) 2010 AB 2 (c) 2010 AB 3 (a, d) 2010 AB 5 (a)
2009 AB 6 M The x-intercepts are x = - 2 and x = 3 ln(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 = 3 ln(5/3) M
![and since f ´(x) ≥ 0 on [ -4, M ] it follows that](http://slidetodoc.com/presentation_image_h2/d72120a9b9b343eae4695c77aca51e7d/image-23.jpg "and since f ´(x) ≥ 0 on [ -4, M ] it follows that")
and since f ´(x) ≥ 0 on [ -4, M ] it follows that f (M) > f (-4). M
M
![Since M is the only critical number in the interval [-4, 4] and f](http://slidetodoc.com/presentation_image_h2/d72120a9b9b343eae4695c77aca51e7d/image-25.jpg "Since M is the only critical number in the interval [-4, 4] and f")
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 Mc. Mullin E-mail: lnmcmullin@aol. com Blog: Teaching. Calculus. wordpress. com Website: www. Lin. Mc. Mullin. net Click on AP Calculus
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