Would like to do a follow-up on the real meaning of dx.
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In derivatives
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In integrals
The core idea is that dx is the “size of our model”.
- How big is the gap between samples?
We get an estimate for the change / total accumulation based on a model size of “dx”. What is the estimate when we have no gap between measurements?
- Think about a smoothly changing function. A line. If you’re at “x = 3”, what is the gap to the next decimal number? It seems that the line is filled-in, the neighbors are touching, right? (What is the actual gap between digits?) Setting dx = 0 lets us model what happens when have no gap.
So, we use a real gap to estimate what it’s like if there’s no gap.
We humans build a supercomputer, which estimates what the planet would be like without people. But we need people to make the prediction about having no people!
We need dx to make the prediction about what happens when there is no dx.
dx gives us a model. We ask the model what happens when dx=0. Limits are practice in asking the model questions (what happens when dx=2? When dx=3?)
Example:
- Predict change when dx=3, going from $2^2$ to $5^2$ .
Forget about slope, area under the curve, etc. Those are specific interpretations. What is the core message?
dx: what is the gap between pixels?
- Idea… pixels being a fence, or pixels being a sample (dithered image). Individual dot. What is the dithering gap between samples? (Then we get a model and let it go to zero.)
References: