Calculus

Calculus studies change, approximation and accumulation. The strand assumes familiar differentiation, substitution and integration by parts and builds from integration and differential equations to series and functions of several variables.

Chapters

1. Integration techniques: choosing a method from the form of the integrand. Substitutions change the variable, trigonometric identities change the expression and reduction formulae relate a family of integrals.
2. Ordinary differential equations: first-order methods such as separation and integrating factors, second-order constant-coefficient equations, when solutions exist and are unique, and numerical solutions with error control.
3. Taylor series: Taylor polynomials and remainder bounds, then sequences, series and convergence. A Taylor series represents its function only when the remainder tends to zero.
4. Functions of several variables: gradients as level-set normals and linear models, the Hessian test at interior stationary points where it applies, boundary checks for global extrema and Lagrange multipliers for equality constraints.
5. Double integrals: accumulation over planar regions using slices and changes of variables, applied to area, mass and probability.

Links to Algebra

• Polynomial spaces: Taylor polynomials live in the polynomial vector spaces of Algebra.
• Kernels: the solutions of a homogeneous linear ODE form the kernel of a linear differential operator; every forced solution is one particular solution plus an element of that kernel.
• Characteristic roots: complex characteristic roots describe exponential growth, decay and oscillation, read through polar form. Coupled equations use the eigenvectors of linear ODE systems.

See Core for how the two strands fit together.