Dive into advanced calculus concepts through a comprehensive 7-hour course covering integral calculus, sequences, series, and parametric equations. Master techniques for calculating volumes, arclengths, and areas between curves. Learn integration methods including parts, trig substitution, and improper integrals. Explore convergence tests for sequences and series, and delve into power series and Taylor series. Conclude with an introduction to parametric equations and polar coordinates. Enhance problem-solving skills with proofs and real-world applications throughout the course.
Overview
Syllabus
Area Between Curves.
Volumes of Solids of Revolution.
Volumes Using Cross-Sections.
Arclength.
Work as an Integral.
Average Value of a Function.
Proof of the Mean Value Theorem for Integrals.
Integration by Parts.
Table Method For Integration By Parts by Morgan Goetz.
Trig Identities.
Proof of the Angle Sum Formulas.
Integrals Involving Odd Powers of Sine and Cosine.
Integrals Involving Even Powers of Sine and Cosine.
Special Trig Integrals.
Integration Using Trig Substitution.
Integrals of Rational Functions.
Improper Integrals - Type 1.
Improper Integrals - Type 2.
The Comparison Theorem for Integrals.
Sequences - Definitions and Notation.
Series Definitions.
Sequences - More Definitions.
Monotonic and Bounded Sequences Extra.
L'Hospital's Rule.
L'Hospital's Rule on Other Indeterminate Forms.
Convergence of Sequences.
Geometric Series.
Koch’s snowflake problem.
The Integral Test.
Comparison Test for Series.
The Limit Comparison Test.
Proof of the Limit Comparison Test.
Absolute Convergence.
The Ratio Test.
Proof of the Ratio Test.
Series Convergence Test Strategy.
Taylor Series Introduction.
Power Series.
Convergence of Power Series.
Power Series Interval of Convergence Example.
Proofs of Facts about Convergence of Power Series.
Power Series as Functions.
Representing Functions with Power Series.
Using Taylor Series to find Sums of Series.
Taylor Series Theory and Remainder.
Parametric Equations.
Slopes of Parametric Curves.
Area under a Parametric Curve.
Arclength of Parametric Curves.
Polar Coordinates.
Taught by
Linda Green
