20072008 AP Calculus AB Electronic Project

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2007-2008 AP Calculus ABElectronic Project

• Created by Larissa Kruesi
• Period 2

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Question 1

• Lightning McQueen is moving along the x-axis of
  the speedway. His motion is represented by the
  equation v(t) 3sint1 on the closed interval
  0,10. Let v(0)1.
• Find Lightning McQueens acceleration at the
  given time
• b 3.
• Answer?

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Answer 1a

• Solution
• Take the first derivative of the equation v(t)
  3sint1
• v(t)3cost.
• Then plug in 3 for t.
• Lightning McQueen is accelerating at 1.42 m/s

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Question 1

• Tinkerbell is flying along the x-axis over
  Neverland. Her motion is represented by the
  equation v(t) 3sint1 on the closed interval
  0,10. Let v(0)1.
• b) Is her speed increasing or decreasing at t
  3?
• Answer?

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Answer 1b

• Solution
• At t3 seconds the velocity is positive and the
  acceleration is positive.
• When v(t)0 and a(t)0 the speed 0.
• b) Tinkerbells speed is increasing.

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Question 1

• Lightining McQueen is speedin along the x-axis,
  with his motion represented by the equation
  v(t) 3sint1 on the closed interval 0,10. Let
  v(0)1.
• c) When does he change directions?
• Answer ?

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Answer 1c

• Solution The particle changes directions
  whenever it hits the x-axis.
• c) Lightning McQueen changes directions at
• x3.48
• x5.94
• x9.76

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Question 1

• Tinkerbell is flying along the x-axis over
  Neverland, with her motion represented by the
  equation v(t) 3sint1 on the closed interval
  0,10. Let v(0)1.
• d) Find the total distance Tinkerbell traveled
  over the given time interval 0,10.
• Answer?

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Answer 1d

• Solution
• Use the formula
• 3sint1
• 3sin(10)1-3sin(0)1 15.52 meters

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Question 1

• Dumbo is flying along the x-axis, with his motion
  represented by the equation v(t) 3sint1 on the
  closed interval 0,10. Let v(0)1.
• e) Find Dumbos position at t8 seconds.
• Answer ?

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Answer 1e

• Solution
• 1 v(t) dt 1 3sint 1
• V(0) 1, so you know that Dumbo started at 1.
  Then find the area on the interval (0,8), and add
  1.
• e) 12.44 meters

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Question 1

• Pongo is moving along the x-axis while trying to
  avoid Cruella DeVil. His motion is represented by
  the equation v(t) 3sint1 on the closed
  interval 0,10. Let v(0)1.
• f) When is Pongo farthest to the left?
• When is Pongo farthest to the right?
• Answer ?

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Answer 1f

• Solution
• Pongo is the farthest to the right when the
  velocity is a maximum. He is the farthest to the
  left when the velocity is a minimum.
• f) Pongo is farthest to the left at
• x5.9
• Pongo is farthest to the right at
• x9.76

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• Find g(a).
• Answer ?

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Answer 2a

• Solution Find the area of the triangle from
  (0,1).
• g(1)
• ½ (20)
• a) g(1) 1

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• b) Find g(a).
• Answer ?

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Answer 2b

• Solution g(a)f(a)
• Use the graph to find the y-coordinate of f(1).
  At x1, y4.
• b) g(1) f(1) 4

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• c) Find g(1).
• Answer ?

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Answer 2c

• Solution g(a) f(a)
• f(1) (0-2) / (2-0)
• c) g(1) f(1) -1

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• d) Find the average rate of change of g on the
  interval 0 x 4.
• Answer ?

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Answer 2d

• Solution The interval is 0 x 4. The average
  rate of change
• g(4)-g(0) ¼
• 4-0
• d) ¼ (( ½ (3)(5)) ( ¼ (35))) 19/8

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• e) On what intervals is g decreasing? Justify
  your answer.
• Answer ?

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Answer 2e

• Solution
• The function is decreasing when
• gf(x) (1.8, 2.2).
• e) Decreasing on (1.8, 2.2)

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• f) On what intervals is g increasing? Justify
  your answer.
• Answer ?

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Answer 2f

• Solution
• The function is increasing when
• g f(x) 0. The slope of the graph of f is
  positive on the intervals (0,1.8) and (2.2,4).
• f) Increasing on (0,1.8) and (2.2, 4)

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• g) On what intervals is g concave up? Justify
  your answer.
• Answer ?

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Answer 2g

• Solution
• The graph of f is concave up when
• g f(x)0.
• g) Concave up (0,1) and (2,4)

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• h) On what intervals is g concave down? Justify
  your answer.
• Answer ?

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Answer 2h

• Solution
• The graph of f is concave down when g f(x)
  
• h) Concave down (1,2)

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• i) Find the absolute minimum of g on a given
  closed interval. Justify your answer.
• Answer ?

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Answer 2i

• Solution
• An absolute minimum occurs when the graph of f
  changes from negative to positive. Use a sign
  chart to find the absolute minimum.
• (hint the equation of the second line is
  y-5x9 and the equation of the third line is
  y5x-11)
• i) There is an absolute minimum at x 2.2

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Question 2

• 2.Let the function g(x)
• when 0 x 4. The graph of f is given below.
• j) Find the absolute maximum of g on a given
  closed interval. Justify your answer.
• Answer ?

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Answer 2j

• Solution
• An absolute maximum occurs when the graph of f
  changes from positive to negative. Use a sign
  chart to find the absolute maximum.
• (hint the equation of the second line is y
  -5x9 and the equation of the third line is y
  5x-11)
• j) There is an absolute maximum at x 1.8

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Works Cited

• Question 1
• 2002 Form B 3
• 2003 Form B 4
• 2007 Form B 2
• 2002 Form B 4
• Question 2
• 2003 Form B 5
• 2006 Form B 2
• 2007 Form B 4