AP Math Free Response Question and Solution

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AP Math Free Response Question and Solution

• BC Calculus
• Year 2002
• Problem II. A 3

By Wes Montierth
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The figure below shows the path traveled by a
roller coaster car over the time interval 0 t
18 seconds. The position of the car at time t
seconds can be modeled parametrically by
y
x
3
m 1.794

• Find the slope of the path at time t 2.
• Show the computations that lead to your answer.

Slope is the derivative of y with respect to x,
which parametrically results in
2
1.794
2
2
2
2
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• Find the acceleration vector of the car at
• the time when the cars horizontal position is x
  140.

First , it is important to know what the value of
t is when x 140.
Use the solver in your calculator over the
interval 0,18 to find t.
The value of t is 13.64708 seconds.
Now, identify the acceleration vector as the
derivative, with respect to t, of the velocity
vector.
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Use der2, or some other method, to get the second
derivative on both x(t) and y(t), and evaluate
at t 13.64708.
der2(10t 4sin t, t, 13.64708) -3.529
der2((20- t)(1- cos t), t , 13.64708) 1.226
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A note about the velocity and acceleration
vectors.
The velocity vector is in the direction of travel.
The acceleration vector is at right angles
(orthogonal) to the velocity vector, toward the
center of the circle having slope and concavity
in common with the curve at that point.
140
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• Find the time t at which the car is at its
  maximum
• height and find the speed in m/sec. of the car
• at this time.

To maximize the height, find a graph of y as a
function of x (in function mode, with x
representing time) and find the maximum height
over the time interval 0,18.
Height in meters
t 3.024
Time in seconds
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• Find the time t at which the car is at its
  maximum
• height and find the speed in m/sec. of the car
• at this time.

t 3.024 seconds
Speed is the absolute value of velocity.
0
meters per second
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y(t) 0 when cosine equals 1? 0, 2p, 4p, etc.
y(t) 0 at t 20 Outside the range

• For 0 lt t lt 18, there are two times at which the
  car
• is at ground level (y 0). Find these two
  times.

Examination of y(t) reveals that the zeros must
occur at even multiples of p , that is 0, 2 p,
and 4 p (6 p gt18) and at 20, which is also
greater than 18. So the two values of t on
the open interval (0,18) are 2 p and 4 p .
2p
4p
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• For 0 lt t lt 18, there are two times at which the
  car
• is at ground level (y0). Find these two times
  and
• write an expression that gives the average
  speed
• in m/sec, of the car between these times.
• Do not evaluate the expression.

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• For 0 lt t lt 18, there are two times at which the
  car
• is at ground level (y 0). Find these two
  times and
• write an expression that gives the average
  speed
• in m/sec, of the car between these times.
• Do not evaluate the expression.

Oops, we evaluated it!
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Hope you enjoyed the ride!
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Credits

• The function graphic was imported from the
  mathematical software DERIVE, Windows version 1.
• The roller coaster graphic was imported from the
  Microsoft clip art file.
• The actual AP question came from the COLLEGE
  BOARD.
• It is intended that this Power Point Presentation
  be used for non-profit educational purposes only!