rates, motion, and "close enough" approximations 🚀
Usually: a word problem with quantities changing over time. Draw a diagram, assign variables, write an equation linking them, differentiate with respect to time t (implicitly), substitute known values.
Classic pattern: expanding balloon (dV/dt linked to dr/dt), ladder sliding (x² + y² = constant), filling cone/tank geometry.
If s(t) is position on a line: v(t) = s′(t) is velocity, a(t) = v′(t) = s″(t) is acceleration.
Speed = |v(t)|. Object at rest when v = 0; direction changes when v crosses 0 (if it does).
This is the first-order Taylor polynomial at x = a — same idea as "use the tangent as a local model."
Write dy = f′(x) dx as an estimate of change in y for a small change dx. Connects to linear approximation: Δy ≈ f′(a) Δx.
next up: graphs, extrema, and optimization 📊