Deeper

Lab · c10.s02.L1

Secant to tangent

The slope of a curve at a point is the one number the secant slopes approach as the gap between the two points shrinks.

Question: What is the slope of a curved thing?

Secant to tangentA navy parabola y = x squared has a fixed point at a = 1. A dashed russet secant runs through it and a second point a gap h to the right. As h halves from 1 to 0.000244140625, the secant pivots about the fixed point and settles onto the gold tangent, slope 2. Readouts show the secant slope falling 3, 2.5, 2.25, 2.125, its distance from 2 halving. Moving the point to 2 settles the process on 4.f(x) = x^2curvesecanttangent04016aha = 1.0 · h = 1.0secant slope 3.0 · tangent 2.0 · distance 1.0
Re-simulated Re-simulated: the secant slopes are recomputed exactly from f(x) = x^2 by data/scripts/c10.s02.L1.py, and the lab displays only the numbers in data/c10.s02.L1.json. The first frames hold the point at a = 1.0 and halve the gap from h = 1.0 to h = 0.000244140625, so the secant slope runs 3.0, 2.5, 2.25, 2.125 toward the tangent slope 2.0; later frames move the point to 0.5, 2.0 and 3.0 to show the limit belongs to the point.
Lab controls
1

Left/Down and Right/Up move one position; Page Down/Page Up move four; Home goes to 0.5, End to 3.0

0.5

Arrow keys move through the gaps; Home goes to 1, End to the smallest gap

700 ms per frame

Arrow keys

Frame 1 of 12: Two points, one secantThe fixed point sits at a = 1.0 on y = x^2, and a second point sits a gap h = 1.0 to its right. The dashed secant through the two points has slope 3.0; the dotted tangent at a = 1.0 has slope 2.0.

Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.

Text description of this lab

A navy parabola y = x squared has a fixed point at a = 1. A dashed russet secant runs through it and a second point a gap h to the right. As h halves from 1 to 0.000244140625, the secant pivots about the fixed point and settles onto the gold tangent, slope 2. Readouts show the secant slope falling 3, 2.5, 2.25, 2.125, its distance from 2 halving. Moving the point to 2 settles the process on 4.

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Caption status: re-simulated. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.