How hard is calculus?

[php111]

hi,

 

How hard is pre calculus through analytic geometry & Calculus 4? Analytic Geometry combined with calculus 1, 2 ,3 and 4? I am taking taking pre algebra at a community college but mathematics is my major and some time within the years i would need to take calculus.

[Jay-qu]

You want to do a maths major and seem worried about how hard calculus is?

 

Its not really a matter of how hard.. it depends on how much and how deep you get into it. Integration and differentiation are easy concepts and not to hard to perform - when you get to some ODE's and PDE's (ordinary and partial differential equations) in multiple dimensions, then we are starting to get hard. Here we take that stuff in 2nd year. But I guess it would be possible to do a math major without touching to much calculus, surly it cant be avoided entirely though.

[php111]

You want to do a maths major and seem worried about how hard calculus is?

 

Its not really a matter of how hard.. it depends on how much and how deep you get into it. Integration and differentiation are easy concepts and not to hard to perform - when you get to some ODE's and PDE's (ordinary and partial differential equations) in multiple dimensions, then we are starting to get hard. Here we take that stuff in 2nd year. But I guess it would be possible to do a math major without touching to much calculus, surly it cant be avoided entirely though.

 

Ok so calculus 1 is easy? and calculus 2 starts to get hard? What is required in part 3 and 4? What is required in Differential Equations with Linear Algebra?

[Rade]

Ok so calculus 1 is easy? and calculus 2 starts to get hard? What is required in part 3 and 4? What is required in Differential Equations with Linear Algebra?

I think what is required is that you do all the homework problems, then some more.

[freeztar]

Ok so calculus 1 is easy?

That depends...How is Pre-Calc working for you?

and calculus 2 starts to get hard?

Every step up will become more difficult intrinsically as it depends on prior knowledge.

What is required in part 3 and 4? What is required in Differential Equations with Linear Algebra?

 

These are great questions for the teachers (at your school) of those subjects. Show interest early and you will benefit. :hihi:

[Jay-qu]

Ok so calculus 1 is easy? and calculus 2 starts to get hard? What is required in part 3 and 4? What is required in Differential Equations with Linear Algebra?

I cant really give you an answer to that question because its going to depend on the place you study - here in Australia we do things a bit different :hihi:

[Shubee]

How hard is pre calculus through analytic geometry & Calculus 4?

I thought that calculus was just as easy as high school algebra. There are only a few fundamental theorems to learn (for elementary calculus) and the rest is just algebraic manipulation.

[Qfwfq]

...and the rest is just algebraic manipulation.

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. Try for instance:

 

[math]\int_0^\infty dx\,\frac{e^{-ax}-e^{-bx}}{x}[/math]

 

or even the Gaussian:

 

[math]\int_0^\infty dx\,e^{-\frac{x^2}{\sigma^2}}[/math]

[Michaelangelica]

Too Hard.

And what is the point anyway?

[Jay-qu]

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. Try for instance:

 

[math]\int_0^\infty dx\,\frac{e^{-ax}-e^{-bx}}{x}[/math]

 

or even the Gaussian:

 

[math]\int_0^\infty dx\,e^{-\frac{x^2}{\sigma^2}}[/math]

:hihi: right you are Q, the concept of whats going of isnt to hard to understand, while actually doing it proves much harder and in the case of some integrations (and differentiations?) are impossible to perform analytically.

[Shubee]

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]

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.

 

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]

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]

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.