Unit 16: The derivative as a local stretch
About this unit
Near a point, a function acts like multiplying small changes by f′(a), so the derivative is how much it locally stretches, squishes or flips the number line.
By the end, you can: See a function as moving points from one number line to another, read f′(a) as how much it stretches tiny intervals near a, and use that for the chain rule and error estimates.
Modules
5 modulesLearn
- Functions move pointsArticle · Treat f as sending each input to an output on a second number line; a tiny interval near a comes out about |f′(a)| times as long, flipped when f′(a) < 0.Not started
- Stretches multiplyArticle · Composing functions stretches twice, so the factors multiply: the chain rule. Small measurement errors are stretched the same way: Δy ≈ f′(a)·Δx.Not started
- Find the half-size spotExperiment · Drag along y = x² to the point where it shrinks tiny intervals to half their length (slope 0.5, at x = 0.25).Not started
- Stretch factors in briefReview · f′(a) as stretch factor: size for stretch or squish, sign for flip, factors multiply, errors scale by |f′(a)|.Not started