The art of doing mathematics is finding that special case that contains all the germs of generality.
David hilbert
Why we care about something (dependent variable) change due to a small change in independent change? I think it because a lot of cases in reality, the new state includes previous states. We only care what the change will lead to. Therefore the only current state decide the next state. We are trying to predict the future from this small change. (This is only my thoughts. )
Note: your textbook starts with limit concept which is formal definition of derivative. This video starts with some intuitive concept and in Chapter 7 it brings out the formal definition easily.
Chapter 1
- Area of circle = Pi R x R. Divide and conquer and Approximation
- Hard problem => sum of many small values
- smaller dr to better approximations of the original problem; how dr is related to R.
- Approximation ~ original problem (hard problem)
- “Integral” of X*X. dA = X*X *dX.
- Derivative of A
Chapter 2. The paradox of the derivative
- s(t) and v(t)
- ds/dt (t) = rise/run
- dt -> 0. is not “infinitely small”, is not 0. And ds/dt is the slope of the tangent to the curve at certain point. Instantaneous change = best constant approximation around a point.
- The Art of Problem Solving