Mathematics Extension

Description

Course presentation. Explanation of the teaching guide. 1. Single-variable integration. Meaning of the definite integral. Fundamental theorem of calculus. 1.1 Indefinite integrals: Immediate integration and integration by parts. Integration of rational and trigonometric functions. Integration of some irrational functions. 1.2 Definite integrals: Areas between curves. Volume of a solid of revolution. Other applications: function average, arc length. 1.3 Numerical integration: Trapezoidal and Simpson's algorithms (simple and composite formulas). Error estimation. 2. Double integrals. Double integrals over rectangles. Fubini's theorem. Iterated integrals. Double integrals over general regions. Applications: surface area, density, and probability. 3. Ordinary differential equations. Geometric interpretation of solutions, integral curves, and vector fields. 3.1 Some exact solution methods: Separation of variables, homogeneous equations, exact equations. Linear equations. 3.2 Numerical methods for solving differential equations: Euler and Heun methods. Runge-Kutta method. 4. Partial differential equations. First-order linear PDEs. Characteristic equation. Characteristic curves. Fourier transform. Separation of variables. Wave and heat equations. Laplace transform. 5. Interpolation and methods to approximate functions. Lagrange interpolating polynomial.