# How to make Calculus II seem motivated, interesting, and useful?

I am due to teach Calculus II in the fall at an American state university. Our calculus sequence is somewhat slow, due to the fact that many of our students come with limited backgrounds. Most of our typical syllabus consists of the following.

1. Techniques of integration. We teach the students to memorize and apply complicated techniques for evaluating obscure integrals --- all of which Wolfram Alpha can do in a fraction of a second.

Why do they work? Well, that requires a lot of mathematical maturity to understand, and for the most part we don't even try. For example, Stewart's calculus book makes no attempt to explain why partial fractions work, not even a handwavy one. Just "A theorem in algebra guarantees that it is always possible to do this," and the rules are presented for memorization, in all of their complicated glory.

2. Sequences and series. This chapter is a huge, long slog in every calculus book I've ever seen. Stewart's book starts with lots of scary theorems ("Every bounded monotonic sequence is convergent", proved with epsilon-delta), so that by the time you write $1 + \frac{1}{2} + \frac{1}{4} + \cdots = 2$ on the board, or even $.9999\dots = 1$, the students are made to feel that it secretly means something complicated and technical which they half-admit they don't really understand.

From there it gets even worse. Typical problem in Stewart: Does $\sum_{n = 1}^{\infty} \frac{n + 5}{\sqrt{n^7 + n^2}}$ converge or diverge? Well, there is a rule for that -- one of very many -- and for the comparison and integral tests you can at least give a fairly convincing simple explanation for why they work. But I shudder to ponder how many of our students understand that the Latin word converge means equals a number -- this fact is obscured because usually there is no good way to evaluate what this number is.

In the meantime -- where do these series come from? The students will have seen few or none, and typically applications are de-emphasized in the book (and also feel a bit artificial or canned). The textbook does not provide the material with which to convince the student that such a series is a natural thing to write down, except if one is looking for convenient fodder for homework or an exam.

Now, I could revise the material to do it "my way", but if I really did it my way, I'd end up teaching something radically different from your average Calculus II course. (And my department expects me to stay fairly close to the book.)

I'm feeling unprepared to make the case to my students that this course is interesting, useful, or important. What can be done with this material?

• There is a genuine problem here. Such courses are most often "weeder" courses, and the kids know this, either explicitly or subliminally. E.g., they sense that they don't genuinely "need" the material (except to survive it to get to their chosen major...). The engineering and science majors know it is antique, etc. In particular, you would not want to find yourself defending the choices made for the syllabus, but neither would you want to criticize it. But, then, too, the kids' motivation to pay attention to "optional" stuff is severely compromised, and they may feel you are misallocating... May 26, 2014 at 13:31
• ... class time which could better be spent helping them prepare to not get weeded out. Conclusion: enrichment additions to such a course have to be rather slight and in-passing, or the students may have a legitimate complaint. Long-range conclusion: such courses are archaic to the point of embarrassment, but it'd take considerable effort to update them. May 26, 2014 at 13:33
• Frank, speak to faculty in other departments that require the course to find out from the syllabus what topics they want their students to know and why. Or create an online survey with lists of topics and ask the faculty to mark those they consider important and allow them to submit written responses too as feedback.
– KCd
May 27, 2014 at 11:59
• @KCd, be careful. Around here (technical university, mostly engineering) the math folks asked the other departments what they would like their students to know. The result is a sequence of four calculus classes, with syllabi long enough to kill a cow. In reality, what the students really need is a more in-depth understanding of some key issues, not a truckload of specific formulas (long forgotten by the time they would come handy) and half-digested ideas (which they would never link with their applications). Feb 12, 2016 at 16:55
• @vonbrand, it was helpful when we met faculty in other departments to know what mattered to them, so we could justify to our own faculty why certain topics should be covered or in fact did not matter to anyone outside of the math department. Just for fun I put "Fourier series" in the online survey we made about calculus topics, even though we don't actually teach that, and it was a very popular choice by the engineering faculty.
– KCd
Feb 12, 2016 at 17:41

While it doesn't motivate much of the dense and technical morass that is Calculus II, I've made it a little more fun (for me anyway) by thinking about the course as trying to understand the infinite. Our version of the course also has improper integrals (clearly about infinity) and power series (a polynomial of infinite degree).

I tend to drop in some facts about infinity throughout the semester.

• +1 Short + sweet! May 26, 2014 at 1:23
• Came here to say the same thing that @ncr did -- the basic idea behind all the concepts in Calc 2 is infinity. That's why students can have such a tough time grasping the subject matter! So make it a course about infinity. And while you're at it, make sure the technical algebra doesn't take over -- cut back on some of the more mechanical integration techniques like some of the trig substitution. You won't miss them. May 26, 2014 at 23:18
• Hmm...I'd consider Calc 1 as a course that is all about understanding the infinite. Calc 2 is just more of the same.
– user507
May 29, 2014 at 19:43

Don't you also do volumes of rotation, and polar, parametric etc?

The volumes and work problems are the most down-to-earth problems in the course I teach.

And I end the infinite series unit with Maclaurin series, and their ability to make integrals of things like e^(-x^2) solvable, which has a solid connection with statistics.

In the techniques of integration unit, I have them do an area project, using Simpson's rule, so they get a sense of how messy real-life applications may be.

I find Calc I much more satisfying, but I enjoyed Calc II this past semester. I am still searching for ways to make it work better for the students.

• I would do $\int e^{-x^2/2}\,dx$ at the beginning, not the end of infinite series. It was soon after all the integration techniques, and many of the students know the 68 95 99.7 rule better than I do. (Not to spend a lot of time on Taylor series, just given them the one for $e^x$ and observe its derivative is itself.) Dec 26, 2017 at 15:34

The real problem is that the course has to serve the needs of math majors, engineering/physical science majors, and life science majors.

For math majors, 100% of the material in a traditional freshman calc course is relevant, and no apologies are required. It doesn't matter that computer algebra systems can do hard integrals; we need to keep on educating human beings who have the skills to write a computer algebra system. These students need to understand the foundations of the subject, which are nontrivial and took hundreds of years for the world's best minds to work out after the invention of the calculus by Newton and Leibniz. For example, people like Euler used to seriously manipulate series like $\sum_{n=0}^\infty (-1)^n n!$, because the notion of convergence wasn't obvious and hadn't been invented. Lagrange tried to build calculus on Taylor series as a foundation, in order to avoid limits and infinitesimals; people at that time didn't understand that not all functions were analytic. For a student who's a math major, the only serious problem with this type of course is that the curriculum typically ignores the rehabilitation of infinitesimals in the 20th century.

For physical science and engineering majors, series, especially Taylor series, are important, practical, easy to motivate, and will be used by them for the rest of their working lives. Methods of integration, on the other hand, are less likely to be useful, although there may be times in their upper-division coursework where their teachers will want to use these ideas without having to apologize, and in contexts, e.g., proofs of general facts, where a CAS isn't the right tool. For example, integration by parts is used in the standard derivation of the Euler-Lagrange equations of motion from the principle of least action; because this is being proved in general, not for some specific example, you need to understand integration by parts in order to understand the proof.

For nearly all life science majors the second semester of calculus is a ridiculous waste of time. It's required of them at some more selective schools because majors like biology are "impacted," so the biology department is looking for a way to weed students out.

There is no good reason why all students who take calculus should take the same two-semester freshman calc course. At the community college where I teach, we offer four different flavors of freshman physics. The math department could do the same. The most probable reason that freshman calc is one-size-fits-all is that other departments want it as a "weeder" course.

• You've elevated Robinson pretty high among the many (e.g. Bishop, Conway, Lawvere) who refounded the calculus. I'm happy leaving all of them out: even for math majors, probabilities and partial derivatives and numerical computations are the serious omissions for me.
– user173
May 27, 2014 at 2:30
• @vonbrand: I added a specific example.
– user507
May 27, 2014 at 16:40
• @nomen: Sorry, I don't understand what you mean by "fractional" techniques. Do you mean "infinitesimal?" Yes, scientists and engineers take a sophomore vector calculus course that discusses multivariate integration. No, differential forms are not normally covered in such a course. Differential forms are neither necessary nor sufficient to cover all of the traditional-style things that scientists and engineers do with infinitesimals.
– user507
May 29, 2014 at 19:40
• @nomen: I'd be interested if you could point me to some information on what you mean by fractional techniques. Differential forms are not invertible. Since differential forms are nilpotent, you can't use them to express ideas like $ds^2=dx^2+dy^2$. That would be an example of what I mean when I say they're not sufficient to do what people have traditionally done with infinitesimals. There's a good discussion here: mathoverflow.net/q/25089/21349
– user507
May 30, 2014 at 1:08
• "The most probable reason that freshman calc is one-size-fits-all is that other departments want it as a "weeder" course." -- and possibly the fear that if an 'engineer-only' course is offered, the engineering department might decide it can teach that course instead of the maths department, and take back the funding that service teaching brings to the latter. Jan 29, 2016 at 23:12

I'd try to get more understanding than rote regurgitation of formulas out of them. As you note, there are computer algebra systems around (I'm a fan of maxima, it even runs on my cellphone) that can handle the routine effortlessly. It would be nice to have the same at hand for everybody, and as an open source one maxima fits that bill. Or insist on some web-based system, like Wolfram alpha.

Instead of having them compute nasty integrals by hand, let the CAS handle that. Do compute all sort of weird volumes of rotation and such, but get symbolic/numeric results (and ask for that in homework). Ask for the set up, and have them attach the commands and output of the CAS, and then state the result. In exams ask for how to set up the computation, how to use e.g. partial fractions on a sum or integral (not "give me the coefficients", but "to compute this, do you split ...").

For sequences and series, I'm afraid there isn't much that can be done. But freeing up time by leaving out the drudgery should help a lot.

Take care to coordinate with colleages working in parallel, check what exactly following courses require. In particular, if some CAS is standard there, or there are "student editions" available through the bookstore, use them.

"Every bounded monotonic sequence is convergent", proved with epsilon-delta

Snore. Seriously, that won't motivate anyone. But if you show me how it applies to real-world problems that I actually want to solve....

I took calculus in University and even got a surprisingly good mark in the class. The one and only thing I can recall is I should use 2πr for circumference - the professor (who was great, and Welsh. That's important here) would mark πd wrong. Why do I remember this point? This is the only thing covered that had applications outside of physics labs, where we used differentials to work out absolute error. In 1985 that was a pencil-grinding slog that crashed the calculator half the time, and no one really cared.

As you pointed out, Wolfram Alpha can do the grunt work instantly and produces pretty graphs. Skim over the mechanics (which I won't need any more than long division), connect input directly to output, and then explain why it works.

• What is wrong with $\pi d$?? May 26, 2014 at 12:10
• "pie-dee", in Welsh, is apparently a rather substantial insult. I (much) later asked a Welsh co-worker what it meant, his reply was "Wot? 'Ew said that to ya? Good way to start a punch-up."
– paul
May 26, 2014 at 14:15

I second Alexander's suggestion to focus on approximation.

I particular:

In applications of integrals, you can make a big deal about the transition from finite approximation to the definite integral. In each case you have some desired quantity (like area, volume, work, mass, etc) which you cannot compute directly. However, you can break the problem into millions of tiny pieces, each piece can be approximated in an "easy" way. The sum of these approximations gives a Riemann sum for an integral. Taking the limit gives an exact answer.

Since most functions do not have elementary antiderivatives, you often cannot get an "exact" answer by using the fundamental theorem of Calculus. So ask your students to bound the error. How many terms do I need in my approximating sum to get to within a desired degree of accuracy of the final answer?

A similar story plays out for infinite series. Often, even if a series is summable you cannot find a nice expression for the exact sum. Bounding the tail end of the series by a geometric series (or by a suitable improper integral) can give you precise error estimates. They learn the skills of comparison test, integral test, alternating series test, etc in the context of approximation.

Use Taylor series to solve problems (like finding solutions to physically relevant Diff EQs) and have them bound the error.

I think all of this can be highly motivated from the perspective of the future engineer (you need to know how precise your answers are from an engineering perspective) or the mathematician (this is very good practice with epsilonics).

I agree it is hard to motivate getting good at techniques of integration. For these, I let them know that the skill itself is not so useful (given that only a limited class of functions yields to such methods), but that the process of learning it will make them very good at algebra. It is really the meta level skills which are valuable here.

From my experience tutoring, what has worked is first presenting a real world problem that needs solved then show how Calculus can be used to solve it. Position Calculus as a tool that can solve real problems and show how to translate problems to math and back again. Math is the tool that makes Science possible. Science without math is simply observation!

It also helps to reinforce that math is exact and a chain of steps that has to be done correctly otherwise it only takes one break in the chain to miss a problem. Don't let one misunderstanding of one formula break the chain! Students have to memorize formulas and understand the steps exactly. It takes practice to perfect understanding; and patience with yourself to go through the perfection process.

I always say they are lucky as every problem presented in class has a correct answer, they just have to find it. Thinking a problem cannot be solved is simply not true, they just need to find the secret on how to solve it.

If the mathematical content is not interesting, then focus more on useful meta-level skills:

1. Clearly state what is known and what one wants to figure out or proof. In case of a proof this means explicitly writing down the assumptions and the claim. In case of word problems this means writing down the known information (and naming variables) and giving a name to what one wants to find out. When done, explicitly write down the solution and check that it is plausible.
2. Justify each and every step of an argument.
3. Translate a word problem or a real world problem into the language of mathematics. Calculus has plenty of applications; use them and make the students write problems from their own fields that use calculus, and then solve the problems posed by the others.
4. Have students work in groups of mixed majors. Teach them how to communicate a problem from their own field to a mathematician, solve the problem together, translate the solution back to the language of applications, and then check it for plausibility.

You can even make this explicit to the students: "We will learn these sorts of meta-skills during this course. But, since practicing them with only the mathematics we know already is not very meaningful, we will also learn some integration techniques along the way."

When moving on to sequences and series, you can make the following analogy: Translating an applied problem to the language of mathematics is pretty similar to turning a theorem, such as every monotone bounded sequence converging, into something one can prove (with epsilonistics).

I have heard of people teaching Calculus II with the theme of approximation. Integrals and Riemann sums (as well as other numerical integration techniques) are approximations of each other, and functions and their Taylor polynomials are also approximations of each other. Integration techniques don't fit so well here, but my opinion is that integration by parts can be taught as approximation with the right picture, and trig substitutions and partial fractions should be left to complex analysis where they both become applications of the Residue Theorem.

For me as an educator it was always about how the material is delivered in the classroom \ lecture hall that made the difference. So perhaps in terms of the broad range of topics you posted, so in the interests of being useful and imteresting could I offer these nuggets...

1. Calculus card matching:

On card (no bigger than a letter envelope write a set of mathematical symbols, say $3x^2, 4x^3, 12x^3, \ldots$ and on other cards write their anti-derivatives (whether you want to write the constant after integration is up to you) .

The purpose of this task is to allow students to visually match the cards by seeing the derivative / integral in their minds without the use of a pencil and paper.

I would often sidestep the need for writing $\frac{d}{dx}$ by giving the students a square of red card that 'fits' in between the two cards to emphasise which card has the derivative on it.

For a slightly advanced technique, a blue card could represent $\frac{d^{2}}{dx^{2}}$ and get the students to match a card with $\frac{x^{6}}{3}$ on it with $10x^{4}$ on it.

Have sets of cards (with red-square 'first deriviatves') between $5-10$ as a lesson Starter or Blue-square (second derivative) as a main bulk of a lesson with $10-15$ cards.

A good one to have is $\ln x$ and $x^{-1}$ and variations that may imply that one is a derviative of another, to test the students' reasoning. random cards as red herrings are also useful, with $e^{-2}$ and $\infty$ on it are also fun to see the students reason with each other where they go, perhaps having a piile for red-herrings off the main bulk of matches.

• Welcome to the site! A quick note on this: if you let students write down common derivatives and integrals as notes to themselves, you will very, very often find things like "$\sin = \cos$" and "$\frac{1}{1+x^2} = \arctan$". I would recommend against skipping notation like $d/dx$ when possible; students at this level often do not have firm knowledge of the equals sign, let alone $d/dx$! Jul 27, 2017 at 14:13
• @ChrisCunningham Yes, that's a good point and very pertinent, I try to bridge the gap with the red / blue squares on indeed we use cards with the derivative operator on them so it reads correctly as $\frac{d}{dx} f(x) = f'(x)$ for some $f$. Thanks! Jul 27, 2017 at 14:17
1. I think trying to get into some sort of philosophical meaning is a mistake.

2. Even worse is to lard in the baby real analysis. You are the only one feeling the lack from that. Not the kids. And making things tougher when the kids are already having issues with prior prep makes no sense at all. It is going the wrong direction.

3. I think you should emphasize tricks and drill and practice. There is a real feeling of mastery from learning the different tricks and how to use them. Look at what Jaime Escalante did in Stand and Deliver.

• I agree with this answer. There is no point in learning rigorous analysis without feeling the need for it.
– user7171
Aug 20, 2017 at 16:22

Try using smooth infinitesimal analysis and point out that if we leave the powers of epsilon in we get finite difference calculus. Also look at this. It seems to me that limit theory only applies exactly to some early results of integral calculus (eg the quadrature of the hyperbola); and apart from that it's an approximation for which nilpotent infinitesimals are equivalent.

• Dear downvoter: counterargument please! May 26, 2014 at 19:53
• I'm not the downvoter here (at least not yet), but: How does smooth infinitesimal analysis fit with the constraint that "my department expects me to stay fairly close to the book"?
– user173
May 26, 2014 at 22:49
• @mistermarko: Your suggestion that "limit theory only applies exactly to some early results of integral calculus" seems, at best, misleading. May 27, 2014 at 6:58

The OP mentioned the issue of $0.999\ldots=1$ in his question. With regard to the problem of making calculus intereting, in my experience, students react with much more interest when one argues that zero followed by an infinite tail of 9s can indeed be a number that falls a little bit short of 1 (by an infinitesimal amount). A relevant paper is that by Ely here. Other relevant papers are here and here.

• This answer is quite terse, I feel. Many would say/think it is just false and thus not reasonable to say this. Now, I know you have something specific in mind, but to make this a useful answer I think this needs to be fleshed out;
– quid
Feb 11, 2016 at 20:26