The perennial question of AP Calculus teachers is this, what to do after the exam. I first taught AP Calculus in the 1980s when I took over the class from a teacher who had a heart attack. His answer was, “ Do nothing. ” It is hard to believe now, but standard practice back then was to treat the AP exam as the final exam, and to end class. It was certainly nice for me; but while the students enjoyed the free time, letting them out a month early did not exactly further their learning.

Over the years I have tried several ways of extending the year, usually by studying calculus topics not covered earlier. In other words, I did more of the same. This did not inspire seniors who were tired, had other commitments, and were already accepted in college. They were polite, but not turned on.

Then I discovered software for visualizing three-dimensional surfaces, and found my answer. Ever since, I end the year by introducing the class to multivariable calculus. This was a subject I struggled with in college, because I could not visualize what was happening. But with good software to help me see what the books had been talking about, multivariable calculus began to make sense to me. I know that only some of my students will continue with multivariable, but for those students these last weeks provide a grounding on which they can build Calculus 2. And for the rest of the class, well, the software is terrific fun, so they just enjoy the experience of extending what they have done with differentiation and integration into functions of two or more variables.

The calendar is always our master, so in some years I have had nearly a full month of classes, and been able to take the students through partial differentiation and multiple integrals. More typically, I only have two weeks before our final exams. This is just enough time for a basic introduction to graphing and a taste of either differentiation or integration. This year it will be integrals again, finding volumes of solids that we could not even describe a month earlier.

I hope that you all have found your answer, course continuations that work for you. If not, perhaps I will give you some new ideas. In any case, I would love to hear your ideas.

Next week is my last week of classes before final exams, and so this is my next-to-last blog of the year. I hope they have been helpful to you and that I’ll see you one more time, next week.

A Look Back—Week 28

Monday, May 7: Review

There were many last-minute questions before the exam. I even had one student see me for extra help three times during the day. I told her, and I told the entire class, that they needed to relax. They may not trust their skills yet, but I know that they were ready for the exam.

I told them not to study calculus Tuesday night. The best thing they could do for themselves was to get a good night’s sleep and have a good breakfast in the morning. This is the one day of the year when my homework is to not study calculus!

Wednesday, May 9 The AP Exam!

Since our class usually meets at 8:00, there was no class today. Not exactly what I would call a day off!

Thursday, May 10: Begin Multivariable Calculus Introduction

Today we began with the idea of a spatial coordinate system. I use the intersections of the walls of the classroom as the coordinate axes. The ideas of graphing are so well developed by now that this is easy. In a few minutes the class is describing regions such as |x| = 4, the volume enclosed by |x| = 5, |y| = 9, |z| - 2.

Then we moved on to the question of what x² + y2 = 9 graphs as in space. The class decided it was an infinite cylinder whose cross section is the (ordinary) circle x² + y² = 9. Tonight they will work on a number of similar questions requiring visualizing surfaces in space and the interpretations of equations from a three-dimensional perspective.

Friday, May 11: Multivariable Calculus Introduction . . . Again

We did not get to the lab today as I had planned. I forgot that calculus is but one of many AP exams for my students. Yesterday’s class was diminished by competing exams. So today we covered the same ground. This gave the students who were present the opportunity to be “experts” and teach the absentees.

After the review, we developed the distance formula in space. This let me ask a host of new questions. Is (3, 4, 3) inside, on, or outside the sphere x² + y² + z² = 25, for example? This evoked an interesting question from a student, “How can (3, 4, 3) be outside a sphere of radius 5 when none of the coordinates are larger than 5?

Since we were not in the lab, I took the opportunity to introduce the class to the software they will be using next week, DPGraph. (This is a wonderful piece of software written by David Parker; I recommend it highly.)

We used DPGraph and the smart board to create various surfaces, turn them so that we could look from a different perspective, magnify them, etc. The class is going to have lots of fun playing with this next week!

Saturday, May 12: The AP Exam Redux

The AP Exam was released Friday, so I was finally able to give my students their green and blue sheets back. Now I know why they were so despondent after the exam. I think this may have been the most difficult exam I have seen in over twenty years. It is not that any one question was difficult, but most of the questions included parts that were not obvious, so every question pushed at their understanding. There was no question that I thought unfair, but the combination of one challenge after another, with no place to relax in comfort, must have taken a psychological toll. I’ll bet that the statistics on this exam will show a fatigue factor creeping in.

So I commiserated with my students and reminded them that if they found it difficult then everyone found it difficult. It just means the cut points for a 3, 4, 5 will be lower than last year. I don’t know if that made anyone feel any better. I like it when my students are upbeat and ready for a challenge; it is painful to see them feeling like failures. But this too will pass.

So we spent today going over a few of the free-response questions. Question 3 was the first one that everyone wanted to discuss. While they knew the Intermediate Value Theorem and the Mean Value Theorem, the questions were embedded in an unusual context, where h(x) was defined as a composition of functions defined by a table. I think the context made the problem more difficult for them than it otherwise would have been.

We did a few other portions of problems, but I wanted to reserve some time for questions on the homework. I’ll be interested to see how they do when they get to the lab on Monday.

A Look Ahead—Week 29

Monday, May 14: Graphing in the Lab with DPGraph

This is the class I had planned for last Friday. The class will work in pairs and use DPGraph to help them answer questions about three-dimensional surfaces.

Wednesday, May 16: Catching Up
         
The class will be puzzled by some of the questions from the computer lab, so this will be a time to pull together the threads that we developed there. The key idea that I am pursuing (since I only have a few more classes) is the idea of finding volumes.

With that in mind, after the homework Q & A we will develop solutions to the following two questions. First, find the volume enclosed by the surfaces y = x² and y = x³, and the planes z = 2 and z = 10. This is not difficult, once you realize that this is just the volume of a prism with an unusual base. Alternatively, this can be seen as a volume formed by building rectangular slabs on the area between the two power functions. The second question is more challenging.

The plane y = 1 divides the sphere x² + (y – 2)² + z² = 26 into two spherical caps. Find the ratio of their volumes. We will use DPGraph to help visualize what is happening. Essentially, we need to find the volume of a spherical cap centered on the y-axis. My challenge is to help the class see that this is just the volume of a solid of revolution.

Thursday, May 17: More Graphing in the Lab with DPGraph

We will return to the lab today for more hands-on work. The culminating question in the lab is to find the volume between the xy-plane and the paraboloid z = x² + 2y² – 2. This is not a solid of revolution. The trick is to see this as a volume formed by slabs, either elliptical cross sections parallel to the xy-plane or parabolic cross sections parallel to one of the other coordinate planes. The slabs are not the easy slabs we teach in AP Calculus (triangles, semicircles, etc). But the cross sections do form slabs, whose volumes can be computed by integration. The result is that the desired volume is evaluated at an iterated integral.

Friday, May 18:

The class will certainly have struggled with the iterated integral problem. Today I will do several similar problems so that they can see the method in action. We do not have time to fully develop the idea of multiple integrals, but at least they will see that the methods they have already learned can be used to compute new types of volumes.

We are finally here, the week of the exam. For my students this is a time of high anxiety, but for me it is a time to relax, to recognize that I have done everything that I can, that my students are well prepared.

But that does not mean it is time for complacency. Every AP teacher has to meet the problem of what to do after the exam. For many years, my answer has been to introduce multivariable calculus. I have several reasons for this unusual choice. At least I think it is unusual.

First, I had trouble with multivariable calculus when I took it (ages ago) in college, so I can easily anticipate that some of my students will have similar difficulties when they take Calc 2 in college. It seems to me that anything I can do to set the foundation for the course will serve my students well. Of course, not all my students will continue in calculus, so this alone is not sufficient reason for putting my class through these three-dimensional hoops.

A second reason is that we now have wonderful graphing tools available, tools that might have transformed my experience of multivariable had they been available to me back then. My intuition was that students would enjoy working with software in the computer lab. Now that I have done this for many years, I can report that I was right. Students love working with the software.

The third reason, however, is the kicker for me. Much of multivariable calculus is a simple extension of the ideas we have been developing all year long. Partial differentiation, and multiple integrals, for example, can be approached with very little more than ordinary differentiation and integration. Most students get the idea of a section of a function. For example, an x-section of z = f(x, y) = x² – 2y³ + 7 is an ordinary single-variable function z = g(y) = c² – 2y³ + 7. Once we have a single-variable function, we can find its derivative, but this ordinary derivative g'(y) is a partial derivative of the multivariable function z.

This gives students the opportunity to apply their knowledge of calculus in a new situation. So I expect to have some fun in the next few weeks. I hope that you will join me here.

A Look Back—Week 27

 Wednesday, May 2: 2004 Free Response Part A

There were lots of questions from the midterm, as I had hoped. This encouraged students to ask questions based upon questions they had difficulty with. This is generally more productive since it is already pointed in the right direction. We spent part of the class working on the free-response questions. The class will finish these for homework.

Thursday, May 3: 2004 Free Response Part B

There were several questions on part A. The traffic question (1) raised several questions. Part (a) asked for the number of cars passing through an intersection in 30 minutes. My students had the correct integral but were unsure about how to handle the answer. In this case, the problem specified “the nearest number of cars,” so the answer choice was clear. But without that instruction, the value of the integral should still be rounded to an integer since we can never have a fractional number of cars.

The last two parts of this question also needed to be clarified for some students. The question asks for both the average value of a given function, and the average rate of change of that function. The vocabulary is similar and so easy to confuse.

The last part of question 3 was challenging for many students. The problem gave a formula for the velocity of a particle moving along the y-axis (v(t) = 1 – tan–1(e–t)) and an initial value v(0) = –1. Part (d) asks whether the particle is moving toward or away from the origin at t = 2. It is easy to compute v(2) to determine whether or not the particle is moving up or down the axis, but to find out whether it is moving toward the origin requires finding its position at t = 2, and this requires the Second Fundamental Theorem of Calculus.

Friday, May 4: 2005 Free Response A and B

Questions 4 and 5 each raised issues for my students. Question 4 presented an implicit function, and one part of the question asked for the second derivative at a particular point. We had done several problems like this earlier in the year, but my students were inclined to do more work than required. The question gave the derivative as
.
My students differentiated to get
.
They then multiplied out and simplified; some even substituted for dy/dx! These students were chagrined to see the simplest solution which substitutes for x, y, and dy/dx at this point.

Problem 5 gave the graph of a function and then defined its antiderivative using an integral. My students were confused that two parts of the question asked for values on the open interval (–5, 4) and one asked for values on the closed interval [–5, 4]. This led to a good discussion about function properties and where they may occur.

Part (b) asked for relative maxima on the open interval. In this case, the question could have specified the closed interval since the endpoints may be relative maxima. Part (d), however, asked for a point of inflection. Since a point of inflection is a location where a graph changes concavity, an endpoint can never be a point of inflection, so this specification was unnecessary. Finally, the third part of the question asked for the absolute minimum of a function, and since a continuous function must have an absolute minimum on a closed interval, it made sense to specify the closed interval.

A Look Ahead—Week 28

Monday, May 7: Review

I didn’t give the class homework over the weekend, and I don’t plan on assigning any tonight. At this point they know what they know. What they don’t know they are not going to learn through cramming. The best they can do is relax, get a good night’s sleep, have a good breakfast, and walk in prepared to do what they can do.

So today will be a relaxed class, our last before the exam (since we don’t meet on Tuesdays).
There will be questions, I am sure, and we will answer them. This is not the time to push.

Wednesday, May 9The AP Exam!

Thursday, May 10:Begin Multivariable Calculus Introduction

Today we will consider graphing in space. Most students have had a small introduction to this in Algebra 2, but I don’t mind starting from scratch. At this point, students are ready for the topic, they can visualize, and they have a better understanding of how to interpret equations and inequalities.

We will define regions in space using inequalities, and derive the distance formula in space.

Friday, May 11: Begin Multivariable Calculus Introduction—in the lab

This is one of my favorite classes. We go into the lab and use DPGraph to draw three-dimensional surfaces. This is a powerful graphing program which makes it easy to visualize several surfaces and their intersections in space. The surfaces can be defined implicitly, so any equation in three variables is fair game.

DPGraph is a dynamic program which allows students to turn the surfaces and animate intersections with planes. This makes it a wonderful teaching tool as well as an exciting “toy.” Students love playing with this, trying out strange equations and seeing the complex surfaces they form.

This is play with a purpose. We’ll take these visualizations and use them to find volumes in space. Some of these volumes are ones students have already found as solids of revolution, but others will need new methods. These will be the springboards for introducing multiple integration.

Saturday, May 12: Questions

Today we will return to the classroom. There will be many questions, some from the homework and others from the experimentation in the lab. After working through the questions, we will introduce some of the vocabulary of multivariable calculus. In particular, we will talk about analyzing a function by examining sections of the surface formed by holding fixed one variable.

The exam is just around the corner.

Last weekend was our Spring Parent’s Weekend. Traditionally, this extravaganza ends with a long weekend. We get both Monday and Tuesday off and return to classes on Wednesday. While this is a lovely and much needed respite, it clashes against the early placement of the AP Calculus exam the following Wednesday. Worse, since classes meet four times each week, and since my class doesn’t meet on Tuesdays, I have just four more classes before the exam.

It can’t be helped, and there is always some point that is just “four classes before the exam” for everyone. My heavy review schedule for the past two weeks is partly a response to this scheduling nightmare. The truth is, I think my students are as prepared as they are going to be.

I don’t really think that they are going to learn a lot of calculus that has eluded them for the past seven months. They may make some headway on the parts of calculus they don’t know, but the real purpose of the review is to help them to be confident on the large part of calculus that they do know. If they can enter the exam with confidence, and access what they know well, I am confident they will do fine next week. For me, that is the sole purpose of the review: to help them feel at ease in the exam situation.

And so my advice to all of you, my unknown colleagues, is, “Relax.” You have done what you can, and your students have learned what they can. Anxiously memorizing another antiderivative formula isn’t going to bring the big payoff. If you can help your students feel comfortable with what they currently know so that they can enter the exam room with a feeling of calm rather than panic, then you will have helped them do the very best that they can.

By all means, help the class work through a few more past exams (that’s what I am doing!) but remember to tell them not to sweat it. They really know what they know, they just need to believe that is enough.

See you next week!

A Look Back—Week 27

Monday, April 23: Free Response 2006

Last Friday I gave the class a quiz on area and volumes. I was pleased that the class generally did well. I also gave them an initial value problem to solve, since they did not do well on that earlier quiz. I was pleased that most students seemed to have “put it together.” However, there were two common errors that did appear in solving

.

The first error surprised me when I saw it, and surprised me even more when I saw it repeated on several papers. Some students “separated variables” by writing this as y dy = x² dx. I am quite certain that these students know the correct algebra, so I can only see this strange lapse as an error caused by time pressure.

The second error is more substantive, but also algebraic. Most students correctly separated variables and antidifferentiated to get
.
The function was supposed to pass through (0, –2) and these students used this equation to get C = ln(2) and then incorrectly solved for y:
.
A second incorrect solution reveals difficulty working with absolute values. These students took the logarithm equation and produced

and either chose the positive solution or left both solutions.

Both types of error reveal a misunderstanding of the idea that a solution is a continuous function passing through (0, –2). The first solution does not pass through the given point, while the second solution is not a function.

For homework I assigned the 2006 free-response questions.

Wednesday, April 25: Multiple Choice 2003

Today we went over some of the 2006 questions. Many students were thrown by part C of question 2. The question gave the graph and formula of a complex function representing the rate in cars/hour turning left at a certain traffic light. The question asks if there is any two-hour interval in which the total number of left-turning cars exceeds a certain value. The question is puzzling since there doesn’t seem to be any way to get at every two-hour interval.

The trick is to realize that you only need to find a single two-hour interval in which the number of cars is larger than the specified amount, and from the graph [13, 15] is a natural choice since it is (visually) the interval with the largest rate of turnings.

My students also had difficulty with problem 4, where the velocity of a rocket is given in a numerical table. Part A asks for the rocket’s average acceleration. This problem is difficult because it puts a simple concept, average rate of change, in an unfamiliar context. Had the problem asked for average velocity of a distance function, there would have been no confusion; but asking for average acceleration required students to recognize acceleration as the rate of change of velocity.

My students had no problem with part B, which asked for the interpretation of an integral of velocity, but a few were puzzled by part C. Once they saw that this question was just asking for the solution of a differential equation, however, the solution was easy.

For homework I gave the class the multiple-choice question from the 2003 exam.

Thursday, April 26: Questions?

With 45 multiple-choice questions, nearly everybody was puzzled by some question. So we spent the class going over as many of these multiple-choice questions as we could.

For homework, I gave the class their December midterm exams and asked them to correct them. This came as a surprise for most students, but it seems to me that the best way to focus the review is on problems that are hard, not easy. I hoped that they would discover that many of their December errors were, in retrospect, simple errors to correct.

Friday 27 April: Questions from the Midterm

We went over several problems from the midterm. I think the idea of using this old test was a good one. Those students who did very well on the exam could either focus on some other type of problem or on some other subject. (After all, most of these students are taking one or two other AP classes.)

For the students who had difficulty with the exam, they had the opportunity to revisit the problems they found challenging. Hopefully, they will not make the same mistake on these problem types on the AP exam or on their final.

A Look Ahead—Week 27

Wednesday, May 2: 2004 Free Response

There are many problems from last week that we did not get to go over. I’m sure the class will have no problem finding questions to ask in this first day back from our long weekend.

Thursday, May 3: 2005 Free Response

Yesterday, I gave the class the 2004 free-response questions for homework. We will focus on those problems today.

Friday, May 4: Review

Today we will work on the 2005 free-response problems the class had for homework yesterday.

I really enjoy these last weeks before the AP exam. The pressure of covering new material is over. Instead, we can really focus on student questions. All of them.

Classes nearly always have a core of highly vocal students, the ones who lead the discussions. They also have a number of students who prefer to stay silent. There are many reasons for this, but the dynamic occurs in almost every class. What I love about these last weeks is that everybody is asking questions.

Nobody “feels stupid.” Almost everybody is willing to ask “dumb questions,” because they really want to know the answers. And since everybody is asking, no one feels out of place.

The result is that everybody is fully engaged with the material, and we are really using every minute of class time to good advantage. Still, it will be nice to relax with my class on May 10, put the exam behind us, and move on to new work.

But for now it is review and then more review. I only have eight more classes before the exam, the result of a four-classes-per-week schedule and a Spring Parent’s Weekend that will cost two class days. Can’t be helped, but it does put on the pressure.

So let’s get to it. I hope that you and your classes are also deep into review. See you next week.

A Look Back—Week 26

Monday, April 16: 2003 Free Response

I was away at a workshop. It is always more work to prepare for a class that I am not teaching than one that I am. But the class and I survived.

Wednesday, April 18: Go over 1997 Multiple-Choice Questions

It is fascinating to notice the questions that arise when we look back over the year. The approaching exam seems to give permission to students to ask the questions they have been embarrassed to ask before. It is also interesting to see how much forgetting really occurs, even with very strong students.

I really appreciate the students who are willing to ask (finally!) why the second derivative gives information about concavity. So there were questions about the exams, but there were also questions about the definition of limit, the definition of derivative, the Second Fundamental Theorem, and many more.

Thursday, April 19: Go over Multiple-Choice Questions & 2003 Free Response

Some students were frustrated at the number of questions they did wrong. This was a wonderful opportunity to review test-taking strategies and to explain how the AP exam is graded. One student observed that the “hard questions” (based on the national averages published by the College Board) were not all at the end. This gave me the opportunity to present the strategy of scanning questions and skipping the hardest ones. The idea is simple: Make sure you answer the questions you think are easy.

After finishing the questions that seem easy, go back through the questions, attempting the ones that seem a little more challenging. By moving through the test multiple times, students are less likely to get into time pressure. The second strategy is related to the first: Save the hardest questions for last so that you have time for all the easier questions.

To those students who were worried about the number they got wrong, I pointed out that the AP exam is not graded like a classroom test, where a “5” is an “A” and requires 90% accuracy. In fact, students can get a 3 with little more than half the test done correctly. Not that I am encouraging my students to be satisfied with less than their best effort, but they should not feel frustrated if there are questions they cannot do.

Another student was frustrated by her careless mistakes. She understood the calculus ideas in the questions she got wrong, but made careless algebraic and arithmetic errors. To her and the students like her, I suggested the nonintuitive strategy: Slow down. This is a hard one for most students, especially on a timed test, but the idea is to work at a comfortable pace, especially for those anxious students who, under time pressure, tend to make careless errors. For them, slowing down is a winning strategy. They may not get to complete all the questions, but they are likely to be more accurate on the ones they do complete.

Friday, April 20: Area and Volume Test (with IVP Question)

I hate giving up class time for testing, but it is a necessary evil.

A Look Ahead—Week 27

Monday, April 23:Free Response 2006

We will be practicing these questions until the day before the exam. This is a winning strategy. I am convinced that the more time we devote to practice, the better my students do.

Wednesday, April 25: Multiple Choice 2003

It is particularly important to give students practice with multiple-choice problems. Especially for me, since I do not include multiple-choice questions on my own tests.

Thursday, April 26: Questions?

We will need today to work through the questions we did not get to on Monday and Wednesday. In addition, I have a surprise for my class. I plan on returning their midterm exams and asking them to correct them for homework. It stands to reason that the questions they had the most difficulty with in December are probably the ones they need to review now.

Friday, April 27: Questions from the Midterm

Today is the beginning of our Spring Parent’s Weekend, and I expect to have visitors in class today. Some teachers try to “put on a show” for parents. I have always found the best “show” is a normal class. Most parents cannot follow the content, but they are quite adept at following the logic of an argument and the energy of the class. And they like to see what their children have been doing.

So we will continue to practice, practice, practice.

The calendar tells me that there are still 17 school days until the AP Exam, but I have only 12 classes, the result of a schedule that meets four days a week plus a long weekend following our Spring Parent’s Weekend. So I am heavily into review mode.

I will continue to insert a few new topics in the next dozen days, but the strong emphasis will be on review. I have had my students work on both free-response questions and multiple-choice problems, and this push will continue this week and next.

I have found this extended look at problems from former exams to be invaluable preparation for the exam. My students may lose a few points because I have not had enough class hours to cover them, but they will pick up far more points by being comfortable with the structure of the exam. By the time May 9 rolls around they will be saying, “Oh, yes. Another one of those. Ho hum.”

Or words to that effect.

I hope that your students will be equally blasé; this greatly boosts their confidence and performance levels.

A Look Back—Week 25

Monday, April 9:Volumes of Solids of Revolution, Washers

This class followed on the heels of the Slab Lab and a more formal introduction to volumes. Today we found volumes by slabs, by disks, and by washers. The hardest part for most students is when the axis of revolution is parallel to one of the coordinate axes. In this case the typical problem is expressing the inner and outer radii in terms of x or y.

So we took several simple regions (e.g., the area between y = x² and y = x³) and revolved them about several different axes. A good picture of a typical washer is a great help in finding a formula for the inner and outer radius of the washer. But no matter how many times I draw the picture, no matter how many times I say, “Always draw a typical slab,” some students will march to the beat of their own drum. In this case, the drum is usually out of tune and they are not successful. But we shall see.

Wednesday, April 11:AP Volume Problems

I had planned on using the entire class for students to work on problems from AP exams. However, we spent most of the class going over two problems from the homework, both from the text, so I assigned the AP problems for homework (1996/AB2, 1999/AB2, 2000/AB1, 2001/AB1, 2002/AB1).

First we went over Hughes-Hallett’s 8.2/27. This describes a ship whose hull looks like a parabolic prism; the length of the hull is L and the cross sections have the shape y = ax² to height H. The problem was to find the weight of water displaced.

First we needed to do a little easy physics. The class seemed to understand that the weight of water would be the volume of the hull multiplied by 10,000, the metric weight of a cubic kilogram of water. So this is essentially a volume problem.

I was pleased that the class thought to take horizontal cross sections across the parabolas. This cross section is a rectangle, much easier to work with than the vertical cross sections, which are parabolas. The problem the class had, I think, was in figuring out where the axes were. Once I helped them locate the axes so as to be able to draw the parabola, it became clear that the volume of a typical (horizontal) rectangular slab was the length of the hull L times the width of the hull, and the width was the length of a typical horizontal slice across the parabola, 2x.

Therefore, a typical slab had volume ΔV = 2xLΔy where x = (y/a)^1/2. This led to the integral
.
This is not a difficult integral, and we soon had the total weight of displaced water as
.
I had anticipated that they would have difficulty with problem 29, a problem that combines volumes and related rates. (There was a similar problem on the AP exam a few years ago.) The Hughes-Hallett problem defines a bowl in the shape of y = x⁴ between (1, 1) and (–1, 1) rotated around the y-axis. We are told that when the bowl contains water to a depth of h units, it flows out through a hole in the bottom at a rate (volume/time) proportional to h^1/2 with constant of proportionality 6. Students were asked (a) to show that the water level falls at a constant rate and (b) how long it takes to empty a full bowl.

Part of the problem here was that some students didn’t “get” that this was a related rates problem. Or they got that, but didn’t know how to find the volume of the bowl. The problem here is to relate two different domains of knowledge: volumes and related rates.

With a little help, we identified the given information as
.
As with all related rates problems, this one boils down to finding some equation we can differentiate. In similar problems with more familiar shaped tanks (cylinders, spheres, cones), we differentiated the volume, so we set about finding the volume of this solid of revolution

The class wanted to evaluate the integral as
.
Now differentiate implicitly, to get
.
Combine this with the given derivative for V and we get that dh/dt is constant.

But there is a better way (better meaning faster, and showing more insight into calculus). I pointed out that we first antidifferentiate the integral in order to evaluate it, and then differentiate it in order to get dV/dt. Suddenly the light went on in several places around the room. “The Fundamental Theorem,” I heard.

Yes indeed. Instead, we could take the integral for V and differentiate directly:

by applying the Second Fundamental Theorem of Calculus. Now use the Chain Rule:

to complete the problem.

Thursday, April 12: AP Advice, AP Cautions

I had originally planned on a short IVP quiz today, but I knew there would be questions on the AP problems. So we spent the class on the AP problems they worked on yesterday. I took the opportunity to talk about the structure of the test, and to provide both helpful hints and dire cautions. Most of these hints and cautions I have given many times throughout the year; however, the imminence of the AP exam seemed to sharpen their hearing today.

For example, we solved parts of problem 1 from the AB 2000 exam:

Let R be the region in the first quadrant enclosed by the graphs of
,
y = 1 – cos(x), and the y-axis. (a) Find the area of R, and (b) Find the volume of the solid generated when the region R is revolved about the x-axis.

The class understood how to set up part (a), but some students got stuck trying to find the coordinates of the point of intersection algebraically.

So my first caution to the class was,

Use your calculators appropriately!

Even if the solution was possible using algebra (it isn’t), it is better for them to find the point of intersection with the calculator since this is faster and less apt to produce a careless error. Which brought up my first helpful hint: Sometimes less is more.

My next suggestion was more pragmatic. When you find that the intersection is x = 0.940624418, store it in calculator memory. There are two reasons for this. First, if you store it then you can use it without copying many digits. Secondly, if you store it, you will be working with a nine-digit decimal and produce a more accurate value for the area and volume than if you round x to three decimals. Which brings up helpful hint two:

Don’t round until the last step.

Intermediate rounding can only increase the error, and this may result in a final answer that does not satisfy the College Board’s “three-decimal accuracy” criteria.

The class arrived at the correct integral for part (a):

which brought us again to caution #1. Some students tried to evaluate this integral exactly. I pointed out that this is a good way to use up the 45 minutes of this part of the exam. Instead, since we’ve already entered the two graphs in memory as

and
,
now we just have to enter
,
which will compute the area (where a is the stored x-value of the point of intersection. This raised the next caution:

You must write the integral in mathematical notation. You will not get credit for just writing the (correct) integral but using calculator notation.

By the way, if you are not familiar with these built-in variables (y1, y2, etc.), you should be. You can find them under the VARS button; just choose y-Vars from the menu. These are the same names given to the entries on the graphing screen, and they provide a quick way to have access to them (so that you do not have to retype everything each time you use the function. For example, in part (b) we are revolving this area around the x-axis; the correct integral is

which can be entered as
.

The final caution/hint I gave early in the class, when I returned their homework. In spite of my oft repeated mantra, “ Draw a typical slab! ” most of the homework problems I saw had either no picture, or one so tiny as to be useless. Therefore,

Draw large, clear, well-labeled images to help see the algebraic relationships.

Friday, April 13: AP 2000

I had planned on introducing density to the class, but yesterday’s class convinced me that I should begin AP review. Density can wait. For now, I want to give the class the experience of struggling with a variety of questions. So today the class worked on the 1997 multiple-choice questions, and will finish them for homework.

A Look Ahead—Week 26

Monday, April 16: 2003 Free Response

I spent the day away at a workshop. The class worked on the 2003 free-response questions and will finish them for homework.

Wednesday, April 18: Go over 1997 Multiple-Choice Questions

Between the 1997 multiple-choice questions and the 2003 free-response questions, we are going to have far more questions to go over than time to go over them. Can’t be helped.

Tonight, more free-response questions. We are moving on to 2004. This is a lot of homework, but the push is necessary.

Thursday, April 19: Go over Multiple-Choice Questions & 2003 Free Response

Today we’ll spend more time on the 1997 multiple-choice and 2003 free-response questions.

Friday, April 20: Area and Volume Test (with IVP Question)

Today, for a change of pace, I’ll give the class a quiz on areas and volumes. This is really more free-response practice in disguise. It will also give me the opportunity to retest the IVP question that so many students blew on the quiz last week.