My experience with Calculus II is that it is a hodge-podge of several topics unrelated topics: integration, numerical series, Taylor series... There is no "unifying concept".
Most students hate and struggle in a calculus 2 course more so than they do in calculus 1 and 3 or so I read. My guess is that calculus 2 goes deeper into integration than calculus 1. Students who understand calculus 2 integration well will not struggle with double and triple integration, right? Anyway, what is your view? Why do so many students struggle in a calculus 2 course? I am learning calculus on my own with the help of an online tutor. We are going through the calculus 2 textbook one section at a time and it is challenging. I wonder what calculus 3 will be like.
At my university, about 36% of the student fail calculus 1, and only 10% of the students fail calculus 2; We try to get prepared most students very well with calculus 1 in order to not struggle as much as they did with the concepts in calculus 2. The best way to learn math is by understanding very well the concepts.
in my school we just have differential calculus, then integral calculus, then differential equations and vector analysis, then advanced engineering mathematics (Laplace transforms, Fourier transforms, eigenvalues, partial differential equations)
I believe that the root of the problem goes far back to high school. Students are not learning to learn. High school students are not probably learning the basics of the math needed before stepping in calculus 1. Thus, calculus 2 becomes a complete nightmare. I must say that the summation and series lessons in calculus 2 are interestingly complicated.
I believe that the root of the problem goes far back to high school. Students are not learning to learn. High school students are not probably learning the mathematics needed before stepping in calculus 1. Thus, calculus 2 becomes a complete nightmare. I must say that the summation and series lessons in calculus 2 are interestingly complicated.
Actually I find sometimes as a problem is the students depending only on what is coming out in class.. and not comparing to anything else.. there are books and forums to read, one should not be limited to a what comes from a single teacher.
Let me pose a simple question:
What is the next word in this sequence:
Line, Triangle, Square, ____________
A. Circle
B. Pentagon
C. Triangle
D. None of the above
Spoiler:
The trouble, as I see it, as that students are not trained to think for themselves. Many such students come to these forums "looking for the answer" to a problem. What if the problem doesn't HAVE an answer? To be blunt, schools turn out a lot of trained monkeys, and very few creative thinkers. Part of the problem is that schools do not teach, so much as "train future citizens". Enjoyable activities are frowned upon, actual exploration is discouraged, you are given tasks (with the dreary name of "home-work") in preparation of your future life as a wage-earner laboring under someone else's discretion. You are expected to learn everything at the same rate as everyone else, and your standing in this mock society is determined by "grades" on "standardized tests".
Schools are doing away with anything artistic (my GOD, what's the point in teaching MUSIC, or useless things like "art"), information about the past is generally white-washed and sanitized, recreational activities are kept to a minimum. Provocative reading materials are often banned, and it is generally felt that students cannot be trusted with the constitutional freedoms guaranteed by law. You are expected not to derive too much enjoyment from life, not to be too disruptive; to learn how to be punctual and efficient (later, they will fine you money, or fire you outright for tardiness at your future life), and above all, to follow the rules and instructions given to you (there are substantial punishments for refusing-you may be expelled, subjected to performing labor tasks against your will, confinement, or, worst of all: they will make you stay an extra year or two).
I find it miraculous anybody ever learns anything at all, in such a environment.
When students finds themselves in college, many of these restrictions are mysteriously lifted. Human nature being what it is, there are always those who do not do any more than is mandatory. They flounder, having received "instruction" before, and are now in dire need of "learning how to learn". I don't generally fault teachers for this-I fault the school administrations, who see people in an inhumane fashion. I fault the parents for not demanding (and be willing to pay for) better. And I fault the students themselves, for being blind to the fact that they are being sold a bill of goods. Many people need to wake up.
This planet has real problems, problems that may demand the best our minds can produce, to ensure our species' continued survival. This business as usual attitude has got to stop. You do not have to look very far to see our social systems are broken, outmoded, and ultimately self-destructive.
So teachers: if you want your students to get more out of class, you're going to have to try harder, if a student fails because you didn't teach them, then what are you doing?
So students: if you want an easy ride, do something else. Nothing ventured, nothing gained. You can't count on the world being kind to you, it generally won't be. Mathematics is especially unforgiving, the subject itself doesn't care if you succeed or not. It's up to you.
Everyone else: if you're not trying to change this, you're accepting it. Enjoy.
(and a heartfelt congratulations to those who ARE trying to "make things better"-you know who you are, and it is a noble endeavor, with little reward).
Deveno, so what does the answer to the first question imply? That there's no link between anything, unless we view it in a context; and so that means our choice of context predetermines the link?
Like I could think of the shapes as having 2, 3, 4 sharp points respectively and choose a pentagon (well, I guess that wouldn't be totally correct (or at all), because a line has no thickness, but just to illustrate the point; the example could have been different and so could the set of common properties, anyways).
In a way, I feel like there could be an infinite amount of contexts so that we could find one for any link. I'm not sure about this one, though. But if this is so, it should be necessary to specify why are we considering this (well, in this case) sequence.
Right?
Often, when such a question is posed, the underlying assumption of the answerer is there has to be a pattern. When one is given a home-work question, for example, one assumes it IS solvable (and perhaps, the answer is even in the back of the book). Real life isn't like that.
We are taught to solve problems mechanically. That is not thinking, it's cooking from a recipe.
I remember a story about a geometry class, and on the first day, they didn't talk about geometry at all. Instead, they talked about giving an award for "student of the year". Now the thing is, they immediately ran into trouble: who should be eligible for this award? Could someone who transferred in in the last month win it? What about someone who was handicapped and did their homework by correspondence? Did they have to attend the entire year?
It's not so much that these questions aren't answerable, it's to underscore the importance of defining terms. If we want to think clearly, we have to be clear on what it is we are thinking ABOUT. In mathematics, we are free to make definitions willy-nilly, but are these definitions useful? Do they offer insight? Do they facilitate understanding, or impede it?
The idea was, with my question, was to make clear that learning really ISN'T an orderly re-cataloging of "what the greats before us learned". You should question-what is going on, here? Indeed, context IS important, some things are only true within a certain narrow scope. Many continuous functions are differentiable, but by no means are ALL continuous functions differentiable. We cannot always assume "the nice case".
In a certain sense, Calculus I is "practice", you're just getting your feet wet. In Calculus II, you get thrown in the water, and told to swim. The answer just might BE "hippopotamus".
Deveno,
Is calculus 3 harder than calculus 2?
We know that calculus 1 is just PRACTICE and calculus 2 more involved. But what about multivariable calculus? Does calculus go beyond calculus 3? I never took calculus 3. I am curious.
Im not sure how calc I vs II vs III is defined at different universities, however with respect mutivariate calculus i fell it takes a little bit of digesting to develop an intuitive sense for what you are doing (if you want to get beyond just applying formulas). The joint densities vs. marginal densities vs. conditional densities and relationships between them require some reflection to develop an intuitive feel. I find drawing pictures is essential to understanding the correct domains and ranges for integration on mutivariate problems. In general for calculus drawing pictures can really help you "see" what you are doing with the formulas.