Recent content by Ackbach

And how do you know that the calculator answer is correct? How do you know you didn't fat-finger a number, or hit the wrong operation? As John Wheeler was fond of saying, "Never make a calculation until you know the answer." He meant that you need to have a second, completely independent method...

Agreed, but I'm not seeing where you're going with this. The proof technique above gives you a $\delta$ that is assuredly dependent on $x,$ even if it's implicit.

I think you're entirely missing the point. The point is that I would advocate that the Trachtenberg System be taught in elementary school! That is what Switzerland does, with very good results. The book to which I have linked is admittedly not an elementary-level text. Someone should write that...

Svein: Actually, there are loads of kinds of limits in various contexts. The context of this post is the pointwise limit $$\lim_{x\to a}f(x).$$

Update: you can obtain a free version of the Wolfram Development Engine. It's basically Mathematica without the notebook. You execute code on your own machine (so no timeouts). It's great for one-offs.

Different people have different ideas on what is clear, what is straight-forward, etc. Anyone should feel free to use their own derivation in place of mine.

I have re-framed the OP to focus on what was the main event, in my mind. The main event is the lab, and trying to reduce experimental error. My apologies if that wasn't clear. I'm not interested in continuing any discussions about my method of solving the theoretical problem. I would argue it's...

The derivation is useful in the lab. That is, it is useful for the derivation to have been accomplished so that, in the lab, you get far less experimental error. The method of deriving the result is basically not the point of the OP at all. I derive it so that the student has confidence in the...

Well, that has nothing to do with what I was trying to say, and is what I would call a strawman. If you're in algebra-based physics, you learn algebra-based methods. If you're in calculus-based physics, you can use calculus-based methods. I'm certainly not advocating forcing students to use...

The main point of the original post was that in a lab situation, rolling balls down inclines is a great lab. But you need to have the right theory or you're going to get too much experimental error - too much difference between theory and experiment. The main point of the OP was not that...

True, but it doesn't generalize very well. You could think of the problem geometrically as you've done here, or in a calculus manner. The calculus approach of solving a second-order linear ODE is much more generalizable.

Integrating a constant twice is not what I would call a "trick". I would call that a standard theorem that calculus students must learn. From the context, I'm guessing that the "SUVAT" equations are the kinematic equations for constant acceleration? I've never seen that abbreviation before. You...

Look, this is really not worth quibbling about. If you want to claim that you don't actually have to compute an antiderivative like $$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad\text{for } n\not=-1,$$ fine. But in effect you must integrate one way or the other, because that is how you solve the...

There actually are calculus books (Apostol comes to mind) that focus on integration before differentiation. I never learned from them. But pbuk is actually not just using the chain rule in post #9: it's the chain rule in the middle of a derivation to set up an integral. pbuk still has to...

Well, sure. Any trick can become part of the curriculum somewhere. Not every trick is taught everywhere. I, for one, was never taught to solve any ODE the way pbuk does in post #9. I was taught other, equivalent methods. I do not view my education as lacking simply because I wasn't taught that...