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From Secant to Tangent

Secants collapsing into a local slope · BC Calculus · greeting

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Owl · Open a trail map of elevation versus distance. Pick a point at mile 1.4 where the path is curving. Draw a chord from that point to mile 1.9 — that's a secant, and its slope is an average climb rate. Now slide the second point closer: 1.7, 1.55, 1.42… Each new chord is still a secant, but the second foot is creeping toward the first. If you zoom hard enough on a smooth bit of trail, the squiggle starts to look like a straight segment. The slope of that "almost line" is the idea of a tangent slope. We're inventing a local slope by limiting a family of secants. What do you expect those secant slopes to do as the second point approaches the first?

Owl: Open a trail map of elevation versus distance. Pick a point at mile 1.4 where the path is curving. Draw a chord from that point to mile 1.9 — that's a secant, and its slope is an average climb rate. Now slide the second point closer: 1.7, 1.55, 1.42… Each new chord is still a secant, but the second foot is creeping toward the first. If you zoom hard enough on a smooth bit of trail, the squiggle starts to look like a straight segment. The slope of that "almost line" is the idea of a tangent slope. We're inventing a local slope by limiting a family of secants. What do you expect those secant slopes to do as the second point approaches the first?

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