Let's start with this example of walking from Point A to Point B and back.
1) What does it even mean to talk about derivatives and differentials in one dimension? The derivative is supposed to reflect the ratio of changes between two dimensions.
2) How can df = 1dx but Δf = x_B - x_A? My understanding from this video is that df and Δf are effectively synonymous, differing only in fine numerical precision, but the Wikipedia example seems to treat them very differently.
3) How would the physical scenario change if df = 1dx was replaced with df = 3dx? For example, would this triple the speed of the round trip?
4) Is there a difference between x_B - x_A (the expression assigned to Δf) and B - A?
5) What quantity does f actually represent?
6) In the total distance case, why is the exact differential dx used instead of the inexact differential δx?
Consider now a new example. Say we start at Point C on the real number line, move right and left haphazardly for a while, and end at point D.
7) For net distance, the exact differential dx represents a tiny movement along the x-axis (as described from the very beginning of Calculus). For total distance, however, I don't understand what the inexact differential δx actually represents, if it even appears. Does it have the same geometric interpretation as dx?
8) Same for δf; what is its geometric interpretation in this case?
1) What does it even mean to talk about derivatives and differentials in one dimension? The derivative is supposed to reflect the ratio of changes between two dimensions.
2) How can df = 1dx but Δf = x_B - x_A? My understanding from this video is that df and Δf are effectively synonymous, differing only in fine numerical precision, but the Wikipedia example seems to treat them very differently.
3) How would the physical scenario change if df = 1dx was replaced with df = 3dx? For example, would this triple the speed of the round trip?
4) Is there a difference between x_B - x_A (the expression assigned to Δf) and B - A?
5) What quantity does f actually represent?
6) In the total distance case, why is the exact differential dx used instead of the inexact differential δx?
Consider now a new example. Say we start at Point C on the real number line, move right and left haphazardly for a while, and end at point D.
7) For net distance, the exact differential dx represents a tiny movement along the x-axis (as described from the very beginning of Calculus). For total distance, however, I don't understand what the inexact differential δx actually represents, if it even appears. Does it have the same geometric interpretation as dx?
8) Same for δf; what is its geometric interpretation in this case?
