Calculus 1 — Limits & Derivatives
Limits computed numerically, the difference quotient becoming the derivative, and tangent lines — watch h shrink.
Calculus answers one question: how fast is it changing right now? "Now" is an instant — zero time, zero distance — and dividing by zero is illegal. The way out is the limit: sneak up on the instant and watch where the numbers are heading.
- 01 Limits — sneaking up on a value sin(x)/x at x = 0 is 0/0 — meaningless. But near 0 it is perfectly well-behaved. Make x small and watch it commit to 1:
- 02 The derivative — a limit of slopes Average speed over an interval is easy: Δf/Δx. The derivative is what happens when the interval shrinks to nothing — the instantaneous slope. Watch the average…
- 03 The tangent line The derivative buys you the best straight-line approximation at a point: y = f(a) + f′(a)(x − a). At a = 3 that is y = 9 + 6(x − 3) — plotted against the…
- 04 Using it: optimization Maximums and minimums live where the slope is zero. A farmer has 100 m of fence for a rectangular pen against a barn (no fence needed on that side): area A(w)…
next course: Calculus 2 — Integrals & Series →