# How to get a more conceptual understanding of calculus

I am currently a junior in high school, and am at the beginning of my second semester of AP calculus (beginning of the bc portion). While I am able to scrape by with As in the class, I have trouble really understanding the class. It is taught in a very "memorize this formula, do that process, and don't ask questions" kind of way, and I am hopeless at simply memorizing formulas and processes in that way, such that I work more slowly and make more small mistakes than others.

From what I've heard when asking how to remedy this problem, I need to understand in a more "conceptual" way, so that I can figure out the solutions of my own accord rather than just memorizing stuff. Essentially, my question is, is there a more conceptual way of understanding calculus?

I've been able to do exceptionally well in a physics class in this way, mainly because the math there is far more applied to real situations, and I'm great with that kind of intuition. I was able to ignore much of what the teacher said beyond the basics, and was able to find my own way to reach the solution without being methodical in that way. My problem is that I don't know how to apply that intuition to calculus, which is a huge handicap for me as I desperately want to go into astrophysics as a career, which obviously utilizes calculus almost exclusively. I desperately want to have the sort of edge in calculus that I had in physics, especially because you really have to be the best of the best if you want to go into a scientific career.

I've tried working with some friends who are better at the class than I, but almost all of them simply reiterate the same methods in processes that the teacher has taught, without explaining why or how to draw that conclusion on an intuitive level. Most websites that I've found either do the same thing or give only hints of the concepts themselves. In particular, I've attempted to make small excursions into Paul's Notes, but ultimately end up getting bogged down in the trivialties and have to spend as much time understanding the math there as I do in a lecture. In addition, other sites start with beginning concepts that I have long since learned, forcing me to sort out the things I already was taught. I'm further wondering, where exactly can I go or what can I do to get a better, more conceptual understanding of calculus?

I've been spending a lot of time lately trying to figure out how to accomplish this understanding, and given my aforementioned experience in physics, I think I have the ability to do so. I don't know if I'm thinking through this all wrong or if calculus really is a completely new form of math that I simply won't be able to grasp. At the rate I'm going, I'm beginning to question my ability to get a respectable grade on the AP exam, and I'm willing to put a lot of work into becoming better, if possible.

• There's pretty much two concepts in one variable calculus: "the derivative is the instantaneous rate of change" and "the definite integral is the (signed) area under a curve". (Standard curricula throw in an oddball chapter about sequences and series, too, but that's still most of it right there.) There isn't all that much more to it, except perhaps in terms of intuitively understanding solution techniques. (For example, looking at $\int x e^{x^2} dx$ and being able to come up with the substitution $u=x^2$.) – Ian Feb 20 '17 at 6:48
• See this question. – Billy Rubina Feb 20 '17 at 6:55
• I wrote these notes that show how to very quickly and easily derive some of the key facts of calculus, using intuition rather than rigor: Quick calculus. – littleO Feb 20 '17 at 6:58
• To be honest, I never "understood" calculus until I took real analysis where we proved all of the theorems covered in a typical calculus course (limits, sequences, continuity, convergence, differentiation, and Riemann integration). Gaining a solid understanding of these concepts is not easy, and will require a lot of study and practice (especially in learning how to understand and write proofs). – Math1000 Feb 20 '17 at 7:30
• There are various calculus books which focus on proofs, for example Hardy's Course of Pure Mathematics (old editions freely downloadable), Spivak, Lang. I agree with @Math1000 that it's hard to understand calculus except in terms of proofs. – ForgotALot Feb 20 '17 at 7:39

## 1 Answer

I was in the same position as you, not too long ago. There are two ways (in my opinion) that intuition can be furthered.

One is in the sense of real analysis, which looks at continuity, differentiation, and integration from truly basic perspective. This is where you get deep intuition (in my opinion), because it reduces everything down to understanding distances between points and functions, which are very basic geometric objects one can easily visualize There are essentially no prerequisites beyond what you already know in terms of calculus. The book Introduction to Analysis by Rosenlicht might a good way to get this understanding, if you think this "atomic" outlook is the way to go.

Unfortunately, real analysis is rather extraordinarily tedious to read, and it is easy to be bogged down in definitions and minutiae. Also, you do not have much time before the exams to battle through an analysis book, in all likelihood.

So my suggestion instead is to do some interesting and difficult practice problems that do not involve rote application of formulas. For me, I was very motivated to learn the proofs for some of the most beautiful formulas I could find. Using only high school level single variable calculus (often via Taylor series, which you should know for the exam anyway), it is possible to show: \begin{align*} e^{{ix}} &= \cos x+i\sin x \\ \displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}} &= \frac{\pi^2}{6}\\ \int\limits_{-\infty }^{+\infty }e^{-x^{2}}\,\mathrm {d} x &= {\sqrt {\pi }} \end{align*} although some might dispute the rigour of the proofs. See here, here (Euler's approach), and here. I guarantee if you can understand those proofs from beginning to end, you will be much further along. I found the beauty of these quite motivating.

My other suggestion is to look at things geometrically and visually. Consider plane and space curves, and their derivatives even. Use WolframAlpha to visualize some examples. Plot say helices and helicoids with their tangent vectors and planes at various points to see why derivatives immediately create linear aproximations to functions. Geometric and intuitive understanding will probably be more useful for you to do physics than analysis anyway (until you reach quantum and GR at least).

Lastly, some advice: there is no way around memorization of some formulas. Even as you get farther in math, you will need to memorize theorems. Nobody wants to (or can) derive every theorem from scratch during an exam! So just buckle down and do it :)

Maybe also look at the AP physics C curriculum (which you may already be taking). If I remember correctly, it uses single variable calculus to solve problems, which I found very pleasant for learning. In fact, very basic astrophysics textbooks are likely already accessible to you! They will use low-level single variable calculus and provide some useful motivation. For instance, Astronomy: A physical perspective by Kutner is on that level I think!