How to get better at calculus?

Question

I'm still more or less at the beginning of my journey to become a mathematician, but I already recognized that there is a huge gap relating to my skills between solving problems in linear algebra and solving problems in calculus. I really like calculus, but solving problems in linear algebra seems way more easy to me then solving problems in calculus, and that bothers me - a lot. Most of the problems in calculus look like this:

"Show the identity of ..."

"Show that the limit of ... is ..."

"Solve ..."

"Show inductive that ..."

All of these tasks have something in common: They include a lot of playing with equations. Linear algebra, on the other hand, requires a (how to say it in English?) "structural approach" or "structural thinking". Its more about the relations between different objects then about writing down $15$ different steps until you showed that the left side is indeed the same as the right side of an equation.

But since it would be way too easy to simply accept the fact that I'm not that good at calculus, I want to get better at it. But where do I have to start? Does anyone have useful hints for me?

Edit:

Since this was asked in the comments, I'll give an example of a task I would have my problems with. In this case, it's simply about showing that the limit of the $p$-norm is equal to the right side.

$$\displaystyle\lim_{p\to\infty}\lVert \textbf{x}\rVert_{p}=\lVert \textbf{x}\rVert_{\infty}$$

What exactly is It that you struggle with? You'll get more specific answers if you are more specific about your problem. Is it applying known properties and formulas? It is formulating a correct proof? Finding examples? ... — Bib-lost, Jun 08 '16 at 09:34
Could you perhaps give a concrete example of a theorem or exercise you find hard? — Bib-lost, Jun 08 '16 at 09:35
Hard to tell. I guess the hardest part is indeed showing that a certain equation holds, and we get a lot of these tasks. I really struggle with recognizing different potential ways of manipulating a certain expression such that I make my way to the other side of the equation. I always try to find shift myself through these tasks by identifying a certain structur (like I do in linear algebra), but most of the time I only succeed by Trial & Error, and this simply doesn't seem ideal for me. (if I solve it at all) — Julian, Jun 08 '16 at 09:38
Not everyone will agree with me in this., but I think the best way of learning the 'tricks' of a certain field of mathematics is by studying the methods used in previous exercises and theoretic proofs. Even if you don't have to study proofs for the exam, understand how they work and why a certain method is used. You'll adapt to these methods eventually. — Bib-lost, Jun 08 '16 at 09:42
Do you simply want to get better at calculus or want to understand it? — , Jun 08 '16 at 10:07
I guess it comes down to getting better at the algebraic manipulations I need to succeed at the task we are normally given. I don't have much trouble with understanding the topics itself. — Julian, Jun 08 '16 at 10:16

score 3 · Answer 1 · edited Jun 08 '16 at 10:04

3

I have a few general ideas that might help you develop your skills and improve:

Study the structure of proofs carefully (especially well-written ones!) This will help you understand how to write a proper proof. Pay attention to the logic of the proof, and how each declaration follows from previously established results (whether they be theorems or things demonstrated already in the course of the proof).
The key concepts in analysis are usually buried in the proofs, so try to see past the equations and on to the ideas that are being represented (for example, the derivative of a function at a point is merely the slope of the tangent line of said function at said point). Pictures are extremely helpful here, but take care that the pictures you make actually represent the ideas being discussed.
Topology is very worth learning as it is the modern foundation for analysis as a whole. Several concepts are much easier to understand from a topological viewpoint than from an analysis perspective. (For me, I found the topological definition of continuity to be both simple and easy to understand and apply, to the point that I promptly decided to forget the usual definition seen in Calculus classes because it was unnecessarily complicated and entirely replaceable by the topological definition, another example of topology helping analysis understanding is compactness. It's hard to really appreciate the reason for it's bizarre looking definition until you actually do proofs that use compactness, then you start to understand why the property is actually important.)
My last bit of advice is rather specific: Practice doing arguments/proofs with inequalities and seek to develop and intuition for it. It's immensely helpful with certain things you do often in analysis (like proving limits and converge), so much so the my Advanced Calculus professor lamented that the curriculum leading to Advanced Calculus doesn't spend more time on it so that we would be better prepared (this after many people in the class complained that they had no intuition for how to manipulate the inequalities in our problems in order to get something we could actually use to prove what we needed to prove).

edited Jun 08 '16 at 10:04

answered Jun 08 '16 at 10:03

Justin Benfield

3,992

I must say that I don't have much trouble understanding proofs - I have my ways to make the content itself understable for me, up to the point that I find most of the topics quite intuitive. But when it comes down to "simple" algebraic manipulations (is this the right term?), I feel quite lost. This might result from the fact that I was horrible at math when I was at school - since most of the time, mathematics at school is nothing else than doing and learning those algebraic manipulations. – Julian Jun 08 '16 at 10:14
@Julian: People that I have tutored with that difficulty usually have it because they haven't yet learned/gotten used to/experienced with thinking in forms rather than dealing with the exact expression in front of them. For instance, I presume you recall how to solve $ax^2+bx+c=0$? I claim it is easy to use that knowledge to solve $ax^4+bx^2+c=0$. Can you see why? The reason is that this latter equation can essentially be expressed as the former: let $u=x^2$, then $ax^4+bx^2+c=0$ can be re-expressed as $au^2+bu+c=0$ which is easy to solve for $u$, after substituting, solving for $x$ is easy. – Justin Benfield Jun 08 '16 at 10:23
Yes, and that is exactly what I'm trying to do: "thinking in forms". This doesn't seem to be a problem in linear algebra, but it obviously does in calculus. Its just that I don't get so much ideas when working on those tasks, you know? I just "don't feel it". When I work on a problem in linear algebra, I often see directly where this is heading. (which doesn't mean that I solve them all or easily of course) But in terms of calculus, its always just "jumping in and hoping for the best". – Julian Jun 08 '16 at 10:32
Are you having issues figuring out what it is you actually need to show? Or is it in the manipulations to get you there? – Justin Benfield Jun 08 '16 at 10:43
My biggest issues are definitely related to the manipulations that get me there / somewhere. – Julian Jun 08 '16 at 10:45
So for the example problem added to OP, you can intuitively see why the $p$-norm converges to the infinity norm (e.g. picture unit circles defined by those norms). So the challenge is: How to take that intuition and turn it into an actual proof? – Justin Benfield Jun 08 '16 at 11:04
Since I already looked up the proof anyway, I might also describe where I would have my problems here: http://fs5.directupload.net/images/160608/jsphxv67.jpg So it's obviously easy to follow the definition of the $p$-norm, and it's also easy to recognize that the sum converges to 1 once you did the necessary manipulations, but I would definitely struggle at making the step in the middle. I would just go like "Mhh, I have to manipulate the sum somehow, but I simply don't see how to do this." An easy manipulation like the one in the middle simply wouldn't arise in my mind. – Julian Jun 08 '16 at 11:13
I have a sketch of the proof in my head, but it makes no use of that rather clever rewrite. The shortest and most symbolic proofs may look nice, but they are often hard to understand and even harder to come up with, and often there are much easier to understand (or derive on one's own) ways of proving something (though such ways are also more verbose). I think we sometimes over-value brevity and undervalue clarity in proofs. Brevity is a luxury, clarity is essential. – Justin Benfield Jun 08 '16 at 11:32
Although this is true, this specific step isn't new to me - I've seen it rather often in different proofs. Therefore, one would assume that I would be able to come up with it myself, but it would be really hard for me to do so. – Julian Jun 08 '16 at 12:05
Another possible factor might be that I don't try to memorize steps like these. I read them, I understand them, but soon enough, I simply forgot about them, and therefore I'm not able to recall them in a situation where I'd need them. Maybe I should also focus on something like this. – Julian Jun 08 '16 at 12:11
My point with the previous comment is that you really shouldn't worry too much about whether or not you'd have come up with that creative form of 1 that they did. As long as you can convert your intuition into a properly rigorous and correct proof, you will probably get by just fine. Perhaps I should show you how I went about proving this, considering that I had to look up the definition of $p$-norm and infinity norm at first because I haven't actually done anything with them before? – Justin Benfield Jun 08 '16 at 12:13
Sure, why not? :-) – Julian Jun 08 '16 at 12:15

score 2 · Answer 2 · answered Jun 08 '16 at 10:00

The best suggestion to make is: use us!

Calculus requires more "mechanism" than linear algebra. Even $(x+h)^n-x^n$ is already quite mechanical in that sense. A lot of the business of working your way through calculus is acquiring an instinct of which bits of mechanism to keep, and which to discard. You will, at the beginning, end up with 12-line proofs that could have been half the length; and equally with 4-line answers that end up proving that $0=0$.

It does get better.

To help the process along, I suggest that you pick something specific that you aren't happy with. You may feel you have had to guess more than you should; or that you have taken a circuitous route to a result that should have been easier. Then write it up properly as a question on here. You probably know by now that "Do my homework!" and "Check this proof!" questions are not welcome, but if you point out "here I had to guess, and I shouldn't have had to", or "it doesn't seem right that all these manipulations are required, are they really?", then people reading your question will get inspired, and want to help.

Thank you! I already learned about the quality of this board, I'll definitely gonna used it more often. :-) — Julian, Jun 08 '16 at 10:19
Tell you what,.pick up one or two good or well written books from the library shelves... by good I mean what you can yourself understand, enjoy following with own instinct. — Narasimham, Jun 08 '16 at 10:23
@Narasimham - that's a good point. We sometimes forget that mathematics is a principally literary activity. Assessing other people's styles is an important part of developing one's own. — Martin Kochanski, Jun 08 '16 at 11:07

score 1 · Answer 3 · answered Jun 08 '16 at 12:35

As a follow up to the comments in my previous answer, here is my approach to proving the proposition given as an example by the OP.

I started by taking a look at the definitions of $p$-norm and infinity norm, the images of the unit circles on the wikipedia page for them clued me in to an intuitive picture of why the proposition was true (as $p$ got large, the unit circle started looking more and more like the square that you get with the infinity norm).

Noting that the infinity norm was the max of the magnitudes of the components of $x$, I realized what I needed to show was that the $p$-norm would, for sufficiently large $p$ be dominated by the largest component (specifically, the proportion of $\frac{|x_j|^p}{|x_i|^p}$ for the relevant $i$ and each other $j\in \{1,...,n\}$ would tend to $0$). Which is true because $|x|^p$ is strictly monotone for $x\neq 0$ (strictly decreasing/increasing).

As a result, in the limit, the sum approaches the $(\operatorname{max}\{|x_i|:i\in\{ 1,...,n\}\}^p)^{1/p}$, with the exponent becoming 1 and thus the result is that the $p$-norm indeed converges to the infinity norm (I've swept some details under the rug here, but I think it's pretty clear how this proof works, and that is it actually correct). Notice how I was able to develop that proof off of intuition in a pretty straightforward manner, only using a few things I knew.

thexiv · Answer 4 · 2020-12-30T20:13:52.987

I had to do a crash course in Calculus. I was doing some stuff and honestly, it was the trig that was messing me up the whole time. I don't know $e$ and I don't pretend to. But what I did instead was sort of miraculous. $$y = | \text{baseline} - \text{Y-axis} |$$ $$t = \text{length of a to b} $$ $$x=2$$ $$m = \frac{sec(x)}{2}$$ $$n = | y - t |$$ $$P = (m*2) + (n*2)$$ $$L_n = \frac{P}{2} + \frac{n}{2}$$

If you add the $L_n$ answers together, you get an integrand of the entire curve.

If you average them using the $V = L_0 + L_1 + L_2 + ...$ and dividing by the largest integer, $G = n-1$ then you'll have the integral of the equation: $VG$

Finally, if you the differential of a Gaussian equation, you do the first algorithm I made, and replace $x$ with the answer to it. The derivative will be divided into the first answer. This is the differential. And obviously you can keep going in derivatives.

How to get better at calculus?

4 Answers4