<p>Is it the Lack of algebra skills, trig topics or is it just too complex to understand? I always hear people complaining about it.</p>
<p>Really, it’s all about not making mistakes in your algebra.
Do that and you’ll be golden.</p>
<p>Most people get through Calculus I, Limits and Derivatives, without too much difficulty but Calculus II, Integration and Infinite Series, tends to be a merciless weeder course that ends the careers of many aspiring engineers. I think it is because unlike all previous courses, you can not just memorize a load of formulas, functions and derivations and stick in the numbers. While it can be incredibly tedious you can always find the derivative of a continuous function by applying the various formulas such as the product rule, quotient rule and chain rule etc.</p>
<p>On the other hand, while there are some general guidelines on how to approach certain kinds of problems, there are no formulas that will always work in finding the indefinite integral of a function. Possibly for the first time, the student has to employ ingenuity, imagination and original thinking to come up with a way to solve a Math problem. You tend to run into the same issues in evaluating Infinite Series problems which is also included in Calculus II. Many people who had been very good Math students through high school, Precalculus and Calculus I suddenly find that many of the methods that always worked to get good grades in Math in the past are of little value to them in solving problems involving integrals and infinite series. The ones that survive are the ones who can adapt to relying on one’s own ingenuity to solve problems with Integrals and Infinite Series.</p>
<p>The good news is that once you get past Calculus II, the worst is generally over. My oldest son, who is a Geology major, got a C in Calculus II but the next semester got a B+ in Calculus III, Multi-variable Calculus.</p>
<p>You may or may not have to do these, but to this day, I still don’t understand Epsilon Delta proofs.</p>
<p>Think of epsilon-deltas as defining sets in your domain and image. Say we have a function f: X → Y.
For every open set B ⊂ Y in the image, there’s an open set in the domain A ⊂ X such that f(A) ⊂ B. That is, every point in A is mapped to a point in B. </p>
<p>Now think of a standard picture of a discontinuity, which is a “gap” at a single point. Call this point y and let y = f(x). Then if we take an open set B centered at y which is smaller than the width of that gap, then the definition of continuity is false. For every open set containing x, there will be points which are mapped to one side of the gap as well as points which are mapped to the other side of the gap. </p>
<p>The “for all epsilon there exists delta” means that there cannot be any such gaps, because if there was one, we just choose an epsilon less than the height of the gap. Epsilon and delta just define a notion of size for your open sets.</p>
<p>“Epsilon-delta” proofs are just a way of formalizing this notion given a concrete function.</p>
<p>People hit a wall in calculus for one of two reasons, and the first reason is by far the more common one:</p>
<p>1: Weren’t mathematically prepared algebra and/or trig-wise. Can you simplify
(1-x)/(1-x^2) easily, like almost without thinking about it? Or at least, are you comfortable with the math it takes to simplify? Meaning:</p>
<p>(a^2-b^2)=(a-b)(a+b) so you can simplify the above to</p>
<p>(1-x)/((1-x)(1+x))=1/(1+x)</p>
<p>Can you quickly see that
1/sqrt(1-(sinx)^2) is the same as secx? IOW, can you see that because
(sinx)^2+(cosx)^2=1, then 1-(sinx)^2=(cosx)^2 and sqrt(1-(sinx)^2)=cosx, so
1/cosx=secx? Are you cozy with this? If not, you need more mathematical preparation/practice.</p>
<p>2: Just don’t get what the derivative, and/or limit, and/or integral, and/or slope is all about. They don’t get the analytic geometry part of things. They try to understand everything formally and formula-wise instead of seeing the bigger picture. I can explain the concept of the derivative and the integral to somebody with almost no math training purely through pictures and some explanations. I know this is true because when I was tutoring remedial math I had fun telling people that I could explain the two big concepts in calculus to them. And they would get it after I explained it all with my pictures and whatnot. But it’s like getting the purely conceptual version of quantum mechanics or relativity like they do on those science shows with their fancy graphics–you wouldn’t know how to do anything with any of your knowledge because you need the mathematical background to go farther than merely getting the concepts.</p>
<p>Heh, at my school the first-year computer science sequence relies entirely on mathematical definitions for EVERYTHING that are in the same style as the epsilon-delta definition of the limit, only about strings and trees and pointers and such. It’s to force you to understand everything formally. Imagine a calculus sequence where they teach epsilon-delta-style definitions for <em>every</em> concept and <em>every</em> problem you do, eep.</p>
<p>^^ it’s called real analysis.</p>
<p>What is? I never got the difference 'tween real analysis and calculus, could you explain your comment?</p>
<p>Real analysis is the study of sets of real numbers and functions on said sets. You could think of calculus as a subset of real analysis. </p>
<p>Real analysis can prove calculus. Using real analysis to prove calculus is typically the goal of undergrad real analysis courses.</p>
<p>Nothing really. It gets hard when the integrals are really complex. Only thing I had trouble grasping conceptually were Taylor Series.</p>
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<p>That is how my freshman calculus course was taught. It was a grueling, beast of a class, but I came out so much better for it.</p>
<p>Algebra is the hard part; calculus is easy.</p>
<p>My experience is that all “calculus” texts and courses will teach the how to derive most of the derivative and antiderivative relations from first principles, is this not the norm? Then again, I freely admit I almost never paid attention in calculus class as I had already taught myself the material months in advance.</p>
<p>I don’t think much of a calculus class where you don’t learn where the derivative rules come from, or where you don’t get a proof of why the integral is the area under the curve. Has anybody taken a calculus class where they were just given those things without proof?</p>
<p>In my highschool AP Calculus class, I was given that type of instruction. We were told what the answers were and how to get the answers. Never told WHY they were done that way and proven. They were given as is. That frustrated me a lot, especially after spending a summer learning through the MIT OCW lectures. High school math classes were always like that. I hated high school for it. I do not feel high school has prepared me enough. The sad part is that my school was considered one of the best in our county…</p>
<p>Most of the other students’ responses, when I explained the intuition and proofs of the topics they were learning, were to stop confusing them with extra information… Even the teacher said for me to shut up and teach when I get my degree instead of trying to teach right now.</p>
<p>In a regular calculus course, the integral is introduced by adding rectangles of small width together, and letting the width grow smaller.</p>
<p>In real analysis, it’s introduced by defining lower/upper step functions which bound the function from below/above respectively. The integral exists if and only if the limit as the width of the steps goes to 0 of the lower step function equals that of the upper step function.</p>
<p>They’re describing the same thing, but one is just physical intuition and the other is mathematically rigorous.</p>
<p>The real gem of real analysis comes from studying things like sequences of functions, abstract measures on spaces, etc. In those places, you can’t really use physical intuition to figure out what’s going to happen, and this is where the mathematical machinery is actually needed. Calculus is a very nice and clean area to introduce the machinery of real analysis to students, and that’s why single variable calculus is usually covered in the first semester of a real analysis course.</p>
<p>That sounds like the squeeze theorem proof of the integral that I saw, which I always thought was the best one. Or the inner product/dot product thing…</p>
<p>Calculus is a bit like puzzles so I think it’s pretty fun. THe limits, derivatives, and integrals are pretty easy if you have a good teacher but I hate anything dealing with geometry or shapes in general (Riemman’s sums, related rates with shapes, volume of a cross section). That stuff isn’t too bad either, but I just hate it because for me it’s not even fun, its just work. After you take Calculus and if you don’t know much about the higher math classes, you will think that you can rule the world with the ability to create functions and will bug your teacher to teach you how to discover and make functions like me =p. Integrals are the best in my opinion, they can tell you so much (if you have a function that is).</p>
<p>My initial issue with calculus was that it seemed like a whole different type of math and like a new language. It didn’t seem like it was based on anything I ever did before. But once I understood what limits, derivatives, and integration was, I was good to go.</p>