Advanced integration techniques, sequences, series, and polar coordinates
← Back to MathematicsMaster sophisticated integration methods for complex functions.
Solve real-world problems using advanced integration applications.
Solve first-order differential equations and their applications.
Explore infinite sequences, series convergence, and series representations.
Understand power series, Taylor series, and function approximations.
Work with parametric curves, calculus with parametric equations.
Master polar coordinate system and calculus in polar form.
Introduction to vectors in 2D and 3D space and vector calculus.
Begin exploration of functions of several variables.
Master sophisticated integration methods for complex functions that go beyond basic antiderivatives.
Master the integration by parts formula and strategy, including repeated applications and tabular integration.
Integrate products of trigonometric functions using identities and substitution techniques.
Use trigonometric substitutions to evaluate integrals involving square roots and quadratic expressions.
Decompose rational functions into partial fractions for easier integration.
Evaluate integrals with infinite limits or discontinuous integrands using limit processes.
Develop systematic approaches for choosing the best integration technique for any given integral.
Apply integration techniques to solve complex real-world problems involving geometry, physics, and economics.
Calculate the length of curves in Cartesian, parametric, and polar coordinate systems.
Find surface areas of solids of revolution using integration techniques.
Solve complex volume problems using multiple methods and coordinate systems.
Apply integration to work, force, fluid pressure, and center of mass problems.
Learn to solve first-order differential equations and model real-world phenomena.
Solve differential equations by separating variables and integrating both sides.
Use integrating factors to solve linear first-order differential equations.
Identify and solve exact differential equations, including finding integrating factors.
Model population growth, radioactive decay, cooling, and mixing problems with differential equations.
Explore infinite sequences and series, learning convergence tests and applications.
Analyze convergence and divergence of infinite sequences using limits and theorems.
Understand infinite series, partial sums, and basic convergence concepts.
Master integral test, comparison tests, ratio test, root test, and alternating series test.
Distinguish between absolute and conditional convergence and understand their implications.
Master power series, Taylor series, and function approximations using infinite polynomials.
Find radius and interval of convergence for power series using ratio and root tests.
Derive Taylor and Maclaurin series representations for common functions.
Differentiate, integrate, and perform operations on power series term by term.
Use Taylor polynomials for approximation and estimate errors using remainder theorems.
Work with parametric curves and apply calculus techniques to parametrically defined functions.
Graph and analyze curves defined by parametric equations, eliminating parameters when needed.
Find derivatives, second derivatives, and analyze concavity for parametric equations.
Calculate arc length for curves defined parametrically using the parametric arc length formula.
Apply parametric equations to projectile motion and other physics applications.
Master the polar coordinate system and apply calculus techniques to polar functions.
Convert between rectangular and polar coordinates, graph common polar curves.
Find derivatives and slopes of tangent lines for curves in polar coordinates.
Calculate areas enclosed by polar curves and areas between polar curves.
Find arc length and surface area for curves defined in polar coordinates.
Introduction to vector algebra and calculus with vector-valued functions in 2D and 3D space.
Master vector addition, scalar multiplication, dot products, and cross products.
Work with vector-valued functions, their derivatives, and integrals.
Analyze curves in 3D space, finding arc length, curvature, and torsion.
Apply vector calculus to problems involving velocity, acceleration, and motion in 3D.
Begin exploring functions of several variables and partial derivatives.
Understand and visualize functions of two or more variables using level curves and surfaces.
Compute partial derivatives and understand their geometric interpretation as slopes.
Apply the chain rule to composite functions of several variables.
Find directional derivatives and gradients, understanding their relationship to optimization.