how hard is calculus?

[Shubee]

Originally Posted by Qfwfq

Hmmm, not quite, it's often a matter of finding a method. Unlike what Jay says, sometimes even finding the integral of a function, without looking it up, can be by no means trivial.

The process is so mechanical that there are computer programs that can find the integral of many functions. How many integration methods do you know? There are only a few techniques for finding integrals in lower division calculus, such as change of variables and integration by parts.

[Qfwfq]

That doesn't show what you said, that "the rest is just algebraic manipulation". Neither does it disprove what I said. Even substitution and by parts are not always simple to apply, tricks may be needed and some of these are not algebraic manipulation (especially when even the functions involved are not algebraic). A cleverly written algorithm might incorporate all known tricks and computers are good at trying all possibilities without loosing track or forgetting some of them, so that doesn't mean that it's easy.

[Shubee]

Originally Posted by Qfwfq

That doesn't show what you said, that "the rest is just algebraic manipulation".

Sure it does, in the sense that the answer is arrived at mechanically in a finite number of steps by following a specific algorithm.

Originally Posted by Qfwfq

Neither does it disprove what I said. Even substitution and by parts are not always simple to apply, tricks may be needed and some of these are not algebraic manipulation (especially when even the functions involved are not algebraic). A cleverly written algorithm might incorporate all known tricks and computers are good at trying all possibilities without loosing track or forgetting some of them, so that doesn't mean that it's easy.

I don't recall having to integrate anything in lower division calculus that was outside of the usual routine process. Perhaps you need to find an example from baby calculus where doing an integral requires more than obvious substitution or integration by parts.

[Pyrotex]

I minored in math in college -- and majored in physics. Fortunately, my high school algebra and trig grades were high enough that I did not have to take college algebra. This was all in the USA, so unless the curiculums have changed drastically, this should apply.

Calculus 1 was Analytical Geometry and intro to Differentials. The geometry part was the hardest, I think, but no harder than trig in high school. Differentials were easy after I understood what they were all about. Using geometry to explain this stuff made it all "visual" and easy.

Calculus 2 was advanced Differentials and intro to Integration. Again, geometery was used to explain everything, and we learned that integration was the "converse" (or reverse) of differentiation. This was probably the easiest of all four semesters. I aced the final without even trying.

Calculus 3 saw the introduction of infinite series and other tricks. This became not so easy because I could not visualize what was going on. More memory work, and the problems got trickier. But I made an A.

Calculus 4 saw the introduction of lots of things like partial derivatives, Calculus on vectors, Calculus on surfaces, gradients, infinitesimals, residual error. Some of these topics were no harder than Calc-2, some of them were very difficult, but overall, Calc-4 was only a little harder than Calc-3. Calc-4 was the last of my "easy" math classes. From there and on into graduate school, math required lots of sweat and total dedication.

[Qfwfq]

Originally Posted by Shubee

Perhaps you need to find an example from baby calculus where doing an integral requires more than obvious substitution or integration by parts.

No, I don't need to. Perhaps you need to get the point of what people are saying instead of insisting.

By what Pyro says (my courses were elsewhere and very different) Calculus 1-2-3-4, which php is asking about, seem fairly basic but not without problems that aren't straightforward and trivial. It is misleading to insist on what you are saying; if he asks, he should be told what he's in for. Encouragement is fine, great and important but insisting that it's a cinch when many will find it difficult, or even beyond their ability, is another matter.

[Shubee]

Originally Posted by Qfwfq

Encouragement is fine, great and important but insisting that it's a cinch when many will find it difficult, or even beyond their ability, is another matter.

I find it difficult to understand how someone could be proficient in high school algebra yet find "baby calculus" beyond their ability. I believe that beginning calculus requires the same kind of thinking as high school algebra. Topics like abstract algebra and point set topology are so different however that those fields do require learning a whole new way of thinking.

[Qfwfq]

Do you find it difficult to understand that not everybody will find the same things easy or difficult in the same way? It also depends on good teacher's help, especially for the trickier parts that require the right approach to problem solving and really can't be considered mechanical. Even in algebra, sets of 2nd degree equations can be like this. So can many trig equations. This does not imply some of the things you insist on and I would be wary making exaggerated claims.

Now my problem in here is that php is asking about US course standards which I don't know (and neither whether it's what you mean by "baby calculus"). I can see from Pyro's post that it's simple enough calculus but by the time you get to 4 you really can't say everybody will equally find it as easy as high school algebra. There is no theorem by which anyone who easily gets through the one won't stumble on the other.

So, it seems php is inquiring about pretty basic calculus and apparently given in rather spoon-fed courses (unlike mine), before Pyro's post I was reckoning on a wider range of possible difficulty. Still I think the only serious way to help php is for those familiar with these standards to give him an idea of what he's in for, without equating cats and leopards.

[Shubee]

Originally Posted by Qfwfq

Still I think the only serious way to help php is for those familiar with these standards to give him an idea of what he's in for, without equating cats and leopards.

Equating cats and leopards. That's a perfect analogy. The leopard will be more difficult to handle, but in trying to visualize a leopard after seeing a cat will be no great surprise.

[Qfwfq]

Suppose your kid just loves cats and has always taken excellent care of them, at the zoo he's insisting to be let into the leopard's cage 'cause he says it's really just a big cat. Would you agree to let him in, thinking he's in for no great surprise?

[Shubee]

Originally Posted by Qfwfq

Suppose your kid just loves cats and has always taken excellent care of them, at the zoo he's insisting to be let into the leopard's cage 'cause he says it's really just a big cat. Would you agree to let him in, thinking he's in for no great surprise?

I already agreed that a leopard is more difficult to manipulate. But that is true of everyone. My point was that trying to understand calculus after algebra is like trying to visualize a leopard after seeing a cat. Consequently, your analogy of putting a naive child in a cage with a leopard doesn't hold. Students that are competent in algebra aren't naive. They are very qualified to handle baby calculus.