# Can we skip Newton's Method?

I am teaching an introductory calculus course for high school juniors and seniors. It is not formally described as an AP Calculus course, but it is supposed to map roughly onto Calculus AB.

The students needed more review of algebra, trigonometry, and precalculus than one might hope, and as a result, we are quite far behind, and risk not getting to integrals and the fundamental theorem of calculus, which are supposed to be the high point of this course. So I am looking for things to cut along the way.

The most obvious candidate seems to be Newton's method. There is a section on it before the section on antiderivatives. It seems to be something of a side note.

Am I going to regret cutting this? Does it serve some pedagogical purpose for introducing anti-derivatives or integrals? Is there something else that's going to depend on this method when they get to Calculus II?

• Is this course one semester or a whole year? Is Calculus II a course at your high school, or is the concern about college matriculation? Sep 28 '20 at 14:55
• Is this an XY-problem? "The students needed more review of algebra, trigonometry, and pre-calculus than one might hope" is problematic. Students taking calculus need to have these prerequisites down pat. Is this the right class for these students, and are these the right students for this class? Sep 28 '20 at 15:17
• @shoover Probably, if we were talking about designing a class for students without a strong background. However, at this point, the students are there, whether they should be or not. Sep 28 '20 at 18:38
• Is the course for juniors or for seniors, then? That's a massive difference in terms of personal and mathematical maturity.
– Nij
Sep 30 '20 at 3:46
• Life is prioritization. Definitely cut that topic, especially in a class that is already stripped down from conventional Calc BC (traditional two semester college calc.)
– user14746
Oct 10 '20 at 16:18

If time constraints are so dire that you risk not getting to cover integrals and the fundamental theorem of calculus, then I'd cut Newton's method (and probably much more) since I don't see how you could pass the test without knowing those former topics.

But, if you cut Newton's method from the teaching, wouldn't it still come up in practice AP tests that the students do assuming you'd have time to do it? If I did cut it, then I would have to definitely highlight it when it does come up on practice materials.

What I might do is discuss the organization of the material in how you could streamline the course so that you definitely cover the most important things and then will come back to the "add-ons" (like Newtwon's method) if you have time. Or, you could make it a self-study unit?

• Why is that on the placement exams now? I didn't get to Newton's method until second semester college calculus, and I tested into College calculus knowing only algebra and boneheaded trig. Sep 28 '20 at 22:11
• @Joshua: It seems that Newton's method is no longer on the AP calculus test (see "CF Unit 5", comment 4.8 in the rightmost column), and the results of this google search. As for why certain topics are tested and others are not, see this blog post by Lin McMullin. Sep 29 '20 at 10:58
• One option I think we did in one of my classes (many years ago) was this: The teacher told us "We're not covering X, Y, and Z, because we don't have time. They won't come up on the multiple-choice section. If they come up as an option on the essay section, pick the other option." I believe this class was trying to teach to two different standardized tests simultaneously. Oct 5 '20 at 18:28

AP classes have become much more common these days, and at many schools the result has been that very few students actually pass the AP exam with a grade that would allow them to skip the course in college. The trouble is that even if 90% of your students fall in this category, you also have a duty to serve the other 10%. Those students are going to be shortchanged if they miss out on this standard topic of freshman calculus.

I would suggest that you just offer an optional 1-hour zoom session on a weekend for those students who want to have a little bit of instruction on this.

If this is a high school class with 5 contact hours per week for 18 weeks, then I don't really understand how you can literally run out of time for all the standard topics of a freshman calculus course. An AP calculus course is a college course. That means that your students need to be working at the level of a college student, meaning that they don't need to be spoon-fed every topic and then hand-held on practicing it in excruciating detail.

• The OP said that this is not an AP class, but "is supposed to map roughly onto Calculus AB". On the other hand, removing a topic, originally listed in the class syllabus, from the class is indeed shortchanging the students who are qualified for it. This is a broader problem in American education, where there is no national curricula, and every state, district, school or even teacher come up with their own courses that roughly map into "algebra" or "geometry" or "history", reinventing the wheel again and again. A student never knows what he will be learning next year or in three years. Sep 28 '20 at 17:35

Reasons to study Newton's method:
-It's an application of derivatives
-It is a good example of numerical methods
-It can help strengthen understanding of relationship between derivative and tangent line
-It's likely going to be on the AP test

I've put these roughly in order of how important they are to your situation; as nice as it is to get greater conceptual understanding, what benefits your student from a practical point of view is passing the test. And if you use AP tests in your course as practice tests, you'll have to go through them and mark any questions on Newton's method as optional.

You should consider making a list of the things you're cutting, finding self-study resources on those topics, and sending an email to students and/or parents explaining that the course isn't fully matching the AP test, and if they want to take the test at the end of the course, they should study these topics on their own.

I recently encountered -

$$3^{x-1}+2^{x}=5$$

I spent too much time trying to use logs to isolate X and got nowhere. I then rearranged to look at it as

$$y=3^{x-1}+2^{x}-5$$

and that's when I realized that this was a classic case of the need for Newton's Method. I'd say that I'd strongly advise against underestimating the value of this technique. (FWIW - After using this method for the problem I shared, I searched here to find Q&A on this. Ironic to see this as the question that resulted)

Many traditional calculus classes completely omit all mention of numerical calculation and approximation (in any concrete sense). It is certainly viable to cut Newton's method and it might make sense if students can't properly manipulate logarithms and trigonometric functions.

On the other hand, numerical calculation and approximation introduce lots of interesting and useful ideas (notice I did not say anything about their direct utility). Newton's method is instructive in this sense. Moreover it is one of the few methods accessible at the elementary level that is actually useful and used in practice (Simpson's rule and, later, Runge-Kutta are the other two that come to mind). Teaching it might help to explain how calculus is used for something other than passing exams.

One way of motivating Newton's method is: follow the tangent line until it intersects the horizontal axis. This makes it an instructive application related to derivatives and the tangent line approximation. Often the tangent line approximation (the linear Taylor approximation) is presented as if it were important with little or no application that serves to justify in the student's mind the claimed importance.

Perhaps teaching approximation techniques can help eliminate a bit of the magic thinking associated with calculator use. Students generally have no idea how any digital approximation is actually calculated nor that what devices return are approximations. How in the world did anyone calculate square roots before there were handheld calculators? How do calculators do it now?

It should not be that hard to teach Newton's method in a short, self-contained way. Students typically like algorithms. Both the bisection method and Newton's method are easy to teach and easy to learn. Both are easily presented in a step by step way. It should be easier to teach Newton's method to mediocre uninterested students than it is to teach the chain rule or integration by parts. What can be a bit complicated about teaching Newton's method is that sometimes it fails. However, it may be good for students to see this too.

It is not formally described as an AP Calculus course, but it is supposed to map roughly onto Calculus AB.

It sounds like there isn't a clear definition for the level of the course or the content that needs to be covered. It may be beneficial to poll the students to find out how many are planning on skipping Calculus II in college and moving straight to Calculus III (multi-dimensional applications) or other upper level courses. If you find out that most of them will be taking Calculus II, then they will eventually learn integrals, while any applications of derivatives such as Newtons method will likely be missed. However, If you find out that most plan to skip Calculus II, then they will miss the basics of integrals, which is critical for the more advanced courses. This also applies to other Calculus II topics such as sequences, series, and summations that will likely be skipped. Keep in mind that they will still be missing Newton's method and any other Calculus I content which will be skip.

Also poll students on their field of study and career path. Any students who are interested in a career in programming and computer science would greatly benefit from numerical methods applications such as Newton's method. This is especially true for schools that do not offer computer science courses. In those cases, students rely on mathematics courses to get that experience.

• how many are planning on skipping Calculus B in college --- I believe you are misinterpreting the naming. In ap-calculus, the 'A' in AB represents certain precalculus material mostly no longer part of the test (but the name AB remains due to historical inertia) and the 'B' represents a typical 1st semester elementary calculus course (limits, derivatives, conclude with Fundamental Theorem of Calculus and some basic integration by simple substitutions). (continued) Sep 29 '20 at 16:03
• The BC course, which is intended to be taken by stronger students instead of the AB course and not after the AB course, represents a typical first two semesters of elementary calculus: includes some integration techniques, some applications of derivatives, Taylor series, and at least at one time if not still, polar coordinates, arc length, and some other topics, but NOT partial derivatives or multiple integrals. Also, I suspect fewer than 10% of the students in the course will take math beyond multivariable calculus and linear algebra, and perhaps only 20%-30% will take multivariable calculus. Sep 29 '20 at 16:09
• @Dave L Renfro thanks for the clarification about The ABC naming convention. I was assuming it corresponded to the course numbering and their main topic, namely Calculus I (derivatives), Calculus II (integrals), and Calculus III (multi-dimensional applications), which is the naming system used in my area. I had heard of classes that combine Calculus I and II or Calculus II and III, particularly for application to specific fields like business. Sep 29 '20 at 17:24
• The naming is a bit confusing, and the tests have changed a lot since the 1970s and 1980s (and earlier) when a fair amount of precalculus material (possibly even things like the rational root theorem) was included. Much of this kind of stuff started going out of style, and also it was decided to focus more on what college calculus classes actually covered (which was not precalculus math), and additionally the use of calculators (and later, graphing calculators) also had a huge impact. Multivariable calculus, however, was never included. Sep 29 '20 at 18:24
• FYI, during 2001-2005 I taught on several occasions business calculus 1 and business calculus 2, which in two semesters covered a water-downed version of "regular" differential, integral, and multivariable calculus (no theory, no trig used in anything, no sequences and series except rudiments of Taylor expansions; logarithmic and exponential functions heavily used; "business" max/min problems emphasized). Sep 29 '20 at 18:28