Calculus in the Wild & Study Capstone

Where this shows up

• Physics: velocity/acceleration, work as ∫ F dx, centre of mass.
• Economics: marginal cost as a derivative; consumer surplus as an integral.
• ML / data: gradients for optimisation; loss surfaces.
• Biology: population models dy/dt = ky(1 − y/K).
• Engineering: signals, control systems, heat and diffusion PDEs (next layer).

A durable study loop

For each new technique: (1) state the pattern in one sentence, (2) do three graded examples, (3) invent one word problem that needs it, (4) explain the why without notes.

• Limits → definition of derivative → rules → applications.
• Antiderivatives → FTC → techniques → area/volume.
• Keep a personal "error log" of algebra slips — calculus fails most often on algebra, not on the new idea.

Capstone mini-project

Pick a real quantity you care about (battery drain, running pace, monthly savings). Model it with a simple function, compute average vs instantaneous rate on an interval, and compute net change with a definite integral of the rate. Write five sentences interpreting the numbers.