This course on Calculus II covers integration methods, series, parametric and polar functions, and vectors. The learning outcomes include mastering integration techniques, understanding series convergence, and applying calculus to parametric and polar functions. The course teaches skills such as integration by parts, trigonometric integrals, Taylor series, and parametric curve analysis. The teaching method includes lectures, examples, derivations, and applications. This course is intended for students who have a basic understanding of calculus and want to deepen their knowledge in advanced calculus topics.
Overview
Syllabus
Welcome to Calculus II.
Intro to Integration By Parts --- Examples: ∫xsinxdx & ∫arctan(x)dx.
Two Tricky Integration By Parts Examples.
Trigonometric Integrals --- ∫sin^n(x)cos^m(x)dx via Pythagorean or Half-Angle Identities.
Intro to Trigonometric Substitution --- Ex: Deriving Area of Circle Formula.
Trig Subs | How To Choose The Substitution & Deal With Indefinite Integrals.
Deconstructing a messy integral | Trig subs & u-subs combined.
Integration by Partial Fractions | Big Idea + First Example.
Partial Fractions | Repeating and irreducible Quadratic Terms.
Choosing what integration methods to use - Part I.
Choosing what integration methods to use - Challenging Example!.
MATH PROF vs TRICKY INTEGRALS.
Improper Integrals: How to Integrate with Infinities, 2 ways!.
Comparison Test for Improper Integrals.
Arclength Formula | Derivation & Ex: Circumference of a Circle.
Area of Surfaces of Revolution | Derivation & Example.
Infinite Surface Area but Finite Volume!?!? *Gabriel's Horn*.
Intro to Sequences.
Limits of Sequences, Limit laws & Function Representations.
The Hierarchy of Big Functions || n^n greater than n! greater than e^n greater than n^100.
Intro to Series: What is 1/2+1/4+1/8+1/16+...?.
Geometric Series | Convergence, Derivation, and Example.
Harmonic Series | It diverges, but insanely slowly!.
Integral Test | Derivation & 1st Example.
Estimating the Remainder of a Series Approximation via the Integral Test.
Comparison Test for Series.
Limit Comparison Test for Series.
Alternating Series Test | Intuition, Statement & Example.
Alternating Series | Estimating the Remainder.
The bizarre world of INFINITE rearrangements // Riemann Series Theorem.
Absolute Convergence vs Conditional Convergence vs Convergence.
Ratio & Root Tests | Geometric Series Generalized.
Choosing Which Convergence Test to Apply to 8 Series.
Solving Inequalities with Absolute Values: Ex |x-2| less than 3.
Power Series & Intervals of Convergence.
Intro to Taylor Series: Approximations on Steroids.
3 Applications of Taylor Series: Integrals, Limits, & Series.
Why Taylor Series actually work: The Taylor Inequality.
Plotting Parametric Curves.
Tangents to Parametric Curves | Multiple tangents at the same point!!.
Area under a Parametric Curve | Formula, Derivation, & Example.
Arclength of Parametric Curves.
Intro to Polar Coordinates.
Sketching Polar Curves.
Areas in Polar Coordinates.
Example: Area Inside Polar Curves.
Why Exponential Growth?? Intro to Separable Differential Equations.
Defining the Natural Logarithm as an Integral?!?!?.
Intro to COMPLEX NUMBERS // Motivation, Algebraic Definition & Fundamental Theorem of Algebra Ep. 1.
The geometric view of COMPLEX NUMBERS.
The Polar Form of COMPLEX NUMBERS // Finding the nth roots of -1.
Taught by
DrTreforBazett
