Calculus MOC
Calculus studies change and accumulation. The core idea is a limiting process: shrink a local change until it becomes instantaneous, or add many tiny contributions until they become a total.
Foundations
- Limits and continuity - the language underneath derivatives and integrals.
- Derivative - instantaneous rate of change, tangent slope, velocity from position.
- Differentiation rules - power, product, quotient, chain, exponential, and logarithmic rules.
- Integral - accumulation, area, displacement from velocity, Riemann sums.
- Fundamental theorem of calculus - differentiation and integration are inverse processes.
- Integration by parts - product-rule integration technique.
Bridge topics
- Taylor series - local polynomial approximation and error control.
- Ordinary differential equation - equations where the unknown is a function.
- Vector Calculus index - gradients, divergence, curl, and multivariable extensions.
Physics links
- Velocity: derivative of position with respect to time.
- Work: integral of force through displacement.
- Action: time integral of the Lagrangian, central to variational mechanics.
- Lagrangian mechanics: uses partial derivatives and stationary-action conditions.
- Power: rate of doing work or transferring Energy.
- Cosmology distance methods: integrals and logarithmic measures in cosmology.
Common study loop
- Draw the quantity and its rate of change.
- Decide whether the problem asks for a derivative, an integral, or an equation involving derivatives.
- Check units; derivatives divide units, integrals multiply by the differential unit.
- Interpret the answer, not just the algebra.