Starting off with Derivatives?

[Ivan92]

Hey there. So during break, I'm going to try and refreshen up what I have learned over the summer on Calculus for next semester. I downloaded the M.I.T video lecture series for Calculus I with David Jerison. The lecture series is good, however I have a question. In his first lecture, he starts off by teaching what a derivative is. Most books and notes I have seen start off with the idea of limits in Calculus. The professor though starts off with derivatives. Even my friend, who has taken Calculus says that one should learn limits then derivatives. What do you guys think? Are there other lectures that I could refer too? Thanks in advance.

 

[symbolipoint]

Your friend is logical in how he sees the way to approach studying derivatives. The derivative IS a limit. The ratio of two values of a function at two corresponding independant variable values to the difference of those independant variable values as those two values approach zero, is a limit. Studying limits before studying derivatives is justified. What gets difficult and what may be pushed off to the future is the epsilon-delta proofs for limits.

 

[TylerH]

Of course limits should be taught(or learned) before derivatives. The derivative, or more generally, the differential is, by definition, a limit. Remember the difference quotient? Although you should consider the possibility of Prof. Jerison expecting his students to learn limits before they come to his class. They are covered in the last unit of my PreCal book.

Although they are important to understand, they are not essential to solving derivatives. There are sets of rules for the transformations of equations that serve the same ends as the difference quotient, but allow you to be much more productive. Some of those rules are: "power rule", "product rule", "chain rule", and "quotient rule". It would serve you well to go into Calc knowing them.

As for learning limits, if you can even watch those lectures without your head hurting, you'll gain a shallow understanding very easily. There are innumerable resources on the internet for teaching limits. themathpage.com has an exceptional article on limits... in it's Calc section. It is my opinion that it takes longer, even for very intelligent people, to deeply grasp the concept. I'm not sure I'm at that point yet. You'll do great.

 

[slider142]

I've seen courses taught with derivatives, then integrals and finally limits and series are left for the end, as they are thought to be too abstract for starting material (colleges that emphasize engineering and computer science).
As a mathematician, I prefer the courses that teach from the fundamental limits and series approach first, then use theorems to build derivatives and integrals. However, I've seen many people view limits and series with very vague eyes; they may have been taught in the opposite, unnatural way.

 

[Ivan92]

Ahh...Very well. For my Pre-Cal class, we have not gone over limits. In fact we have not gone over many things that I have learned in my Pre-Cal class in high school. Though he does go over limits, but in the 2nd lecture following the first lecture about the derivative. So far, with what I have seen, it seems pretty easy. Though I do understand that it will get harder. Well, I guess I have to go over limits first then I can start the lecture series. Thanks for the answers. :)

 

[TylerH]

So you remember you PreCal stuff? That will help you immensely with the harder limits. If there's nothing weird going on, it's perfectly okay to just plug it in and you're done. But when you have a division by 0, that's were things get slightly more challenging. How about finding the points of discontinuity, remember that? Say you have lim x-> 0 x^2/x, you can tell that there isn't an asymptote, because there's not a constant over an x, and that there is a point of discontinuity at x=0. So to solve it, you cancel out to get x^2/x=x, then plug in 0 for x and you get 0. So you get $$\lim_{x\to0}x^2/x=0$$ EDIT again: It SHOULD say =0, but it won't.

A way you can think about it is that, you can get infinitely close to 0, thus, even though it is undefined at 0, the limit is 0.

EDIT: We haven't covered them yet in my HS PreCal, but if the teacher sticks to what he told me earlier, at the end of the year(a few weeks from now) we will get a chance to choose between our last unit being Calc or Stats.