The Fundamental Difference

I returned today from the AP Calculus reading. Seeing all those exams (304,490 more or less) is great and seeing all the different solutions is quite an experience. I would like to discuss several parts of the problems that all relate to the Fundamental Theorem of Calculus (FTC)

Let's start with 2009 AB2/BC2 part c. Students were given that

They were asked to find



You could almost hear them thinking, "First I have to find w(t), then evaluate it at the two points and subtract. I can do that." And in fact this works. After far too many lines of computation by hand (omitted here) students (who were good at this sort of thing) came up with the correct answer:



But wait. Isn't



and can't you calculate that on your calculator? Do you really need to know w(t)?

Then let's look at 2009 AB3 part (b). Students asked to explain the meaning of



using correct units in the context of the problem. The integrand was given as the cost to produce a portion of cable that is x meters from the beginning of the cable in dollars per meter.

Explain the meaning of a definite integral has been asked often on the AP Calculus exams. Three things are required in the explanation: (1) what it represents, (2) its units and (3) an accounting for the limits of integration.

There were many different approaches, with many convoluted sentences. The most common mistake may have been forgetting the units (dollars) or giving the wrong units (dollars per meter). Answers like, "The cost in dollars of producing the last 5 meters of a 30 meter cable," and "the cost in dollars of producing the part of a cable between 25 and 30 meters from the end" earned the point.

Students have trouble with all that of course. So my suggestion is to think of the FTC:



Now it should be quite clear that "The definite integral represents the difference in dollars in the cost of producing a cable of length 30 meters and a cable of length 25 meters." is the simplest, most straightforward answer. And it works even if we don't know or can't find f(x).

In 2009 AB1/BC1 part (b) students were given a velocity expression v(t) in miles per minute (a graph actually) and asked to explain the meaning of



The absolute value of velocity is speed and the integral of speed is the distance traveled. So letting p(t) = the distance traveled.



So the meaning is easily seen to be the "distanced traveled in miles from t = 0 to t = 12."

Finally, in 2009 AB6 a few students tried to justify an absolute maximum of a function f given the graph and equation of f ' on the closed interval [-4, 4] as explained next. Alas, few if any, were successful with this approach, but I liked the idea.

The absolute maximum occurred at a point where x = M (the actual value of M was given). Students could have used the Candidates' Test and reasoned this way:



The inequalities in the first and third lines above are true since from the graph



Perhaps the reason I liked this is that I never quite thought of using the FTC in any of these ways until I realized what students were doing.

Even if you don't know or don't what to bother computing the antiderivatives thinking of them this way may help your students better understand the FTC as something more than a way to evaluate definite integrals.