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The Car from 2005 AB/BC5
This is the first in a series of Featured Questions that will build on problems chosen from the 2005 AP Calculus AB and BC Examinations, both the operational examinations and Form B. We begin the series with an extension of the common problem AB/BC5 that dealt with a car traveling on a straight road. The car's velocity was presented graphically in the problem, and the graph in question is reproduced below.
The problem was difficult for many students because the velocity function was not differentiable at the t values 4 and 16, and hence the acceleration of the car was not defined at these values of t. In the new part (H) below, we consider one way in which the velocity function can be modified so that the revised function is differentiable.
The official scoring guidelines for the original four parts of this Question of the Month are available on AP Central. The solutions for these four parts given in this Question of the Month are comparable to the official ones, but we include an alternate solution to part (D).
AB/BC5
A car is traveling on a straight road. For 
seconds, the car's velocity v(t) is modeled by the piecewise-linear function defined in the graph above.
(A) Find 
. Using correct units, explain the meaning of 
.
(B) For each of 
, find the value or explain why it does not exist. Indicate units of measure.
(C) Let a(t) be the car's acceleration at time t, in meters per second per second. For 
, write a piecewise-defined function for a(t).
(D) Find the average rate of change of v over the interval 
. Does the Mean Value Theorem guarantee a value of c, for 
, such that v'(c) is equal to this average rate of change? Why or why not?
(E) Find the distance, in meters, traveled by the car for t = 0, 4, 8, 12, 16, 20, and 24 seconds. Present your answers in tabular form. You should be able to do these calculations in your head and should not have to use a calculator.
(F) Write a piecewise-defined function that gives the distance traveled by the car as a function of t on the interval [0,24].
Now we consider how the original velocity function might have been modified so that it would be differentiable on its entire domain. We will assume that the velocity can only be modified on the interval [4,16]. Thus we need a function that is differentiable on [4,16] and satisfies the
following conditions:
and 
. The last two conditions ensure the left- and right-hand derivatives of v are equal at t = 4 and 16 so that the function v will be differentiable at these two values of t.
The simplest such function would be quadratic. Convince yourself that a quadratic function,
v2(t) , that satisfies v2(4) = 20 and v'2(4) = 5 must be of the form A(t-4)2 + 5(t-4)+20, where A is a real constant. Part (G) shows that this approach cannot succeed.
(G) Show that a quadratic function of the form v2(t) = A(t-4)2 + 5(t-4) + 20 cannot satisfy both
v2(16)= 20 and v'2(16) = -5/2.
The next simplest function on [4,16] would be a cubic. Convince yourself that a cubic function,
v3(t), that satisfies v'3(4) = 5 and v3(4) = 20 must be of the form
A(t-4)3 + B(t-4)2+5t-4)+20,
where A and B are real constants.
(H) Find A and B so that if v3(t) is given by A(t-4)3 + B(t-4)2 + 5(t-4) + 20, on [4,16], then
v3(16) = 20 and v'3(16) = -5/2. [Note that the constants A and B are uniquely defined.]
It is instructive to graph the new velocity function, using the values of A and B from part (H), that is defined by
.
A graph is given following the answer to part (H). A natural question to ask is, how does changing the velocity function change the behavior of the car? In particular, answer the following questions, assuming that the velocity function has been modified as in part (H).
(I) With the velocity function v3(T) , for how many values of t on the interval [4,16] is the car's acceleration 0? What is the maximum velocity of the car? What is the total distance traveled by the car as t goes from 0 to 24 seconds? [Compare your value for the total distance with the corresponding value obtained either from the table in part (E) or by using t = 24 in part (F).]
We next consider a special case of a more general problem. First go back to the original piecewise-linear velocity function, v(t), and suppose you want the car to travel an extra 80 meters by modifying the velocity function for t > 16. In particular, you will allow the car to travel until time t = T (potentially greater than 24), but on the interval [16,T] the velocity must be linear, v(T) must be 0, and the velocity must be continuous at t = 16. You should be able to do the next part of this question in your head. [All that is needed is that the triangle whose vertices are (16,20), (24,0), and (T,0) has 80 as its area.]
(J) Show that the value of T is 32.
With this new piecewise-linear velocity function, v1(t) , the total distance traveled by the car is 360 + 80 = 440 meters. However, this new velocity function fails to be differentiable at t = 4 and 16, just like the original piecewise-linear velocity function. Suppose that we want the new velocity function to be differentiable on [0,T] and decide to accomplish this by modifying its values on the interval [4,16] as we did earlier. Just as in part (G), we cannot replace the constant 20 with a quadratic and obtain differentiability, but as in part (H), there is a (new) cubic function that will do the job. In part (K), you are asked to find this cubic function, and in part (L), you are asked to compute the corresponding maximum velocity of the car and the total distance traveled by the car. Before you do so, you might want to predict how these latter two values should compare with the corresponding values you found in part (I).
(K) Find the unique cubic function of the form
on [4,16] such that v4(16) = 20 and
.
Suppose that we use the cubic from part (K) to "smooth" the piecewise-linear velocity function
by replacing the 20 in the definition of (v(t) with the cubic. We denote the resulting smooth velocity function by v4(t).
(L) For this new smooth velocity function, find (1) the maximum velocity of the car and (2) the total distance traveled by the car as t goes from 0 to 32 seconds. [Compare your value for the total distance with the 440 meters traveled by the car using the piecewise-linear velocity function. v1(t).]
Complete the question before viewing the answers and explanation!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every reading since then and served as a Question Leader at the 2005 Reading. In 2003, he completed a four-year term on the AP Calculus Development Committee.
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