Question

Just a question about the proof of the quotient rule in Session 10 on the Accompanying Notes (PDF).

It says that "since u(x+delta x) = u(x) + delta u"

So u(x+delta x) is basically the second y-coordinate we're interested in.

Would I be correct in saying that the reason why "u(x+delta x) = u(x) + delta u" is because u(x) is the first y-coordinate PLUS the change in the function "u"?

If I'm wrong, can someone explain why?

Answer 1

Yes, you are correct. Say, for example, u(x) = 2x, if x = 2 and dx = 1, then
u(x+dx) = u(2+1) = 2(2+1) = 6, and
u(x)+du = u(2)+(1) = 2.2 + 2.1 = 4 + 2 = 6.

Answer 2

That's one way of looking at it. More simply, this follows from the way delta u was defined, which in these notes is immediately above the statement you're looking at. By definition,\[\Delta u=u(x+\Delta x)-u(x)\]so it follows that\[u(x+\Delta x)=u(x)+\Delta u\]

Answer 3

Thanks guys. I understood that \[\Delta u = u(x+ \Delta x) - u(x)\] but I wanted to understand it on the graph because it'll help me be more intuitive with what I'm reviewing/learning.

Answer 4

I already took math up to multi-variable calculus but I only learned how to do problems but never understood the why so that's why I'm brushing up on it through OWC :)