**Calculus - Single Variable**

**Calculus - Part 1: Limits and Derivatives**

Limits 1a - Definitions and Concepts

Limits 1b - Delta-Epsilon Formulation

Limits 1d - Polynomials and Rational Functions

Limits 1e - Compositions and the Squeeze Theorem

Limits 1f - Trigonometric Functions

Continuity 1a - Definition and Basic Concepts

Continuity 1b - Polynomial/Rational Functions and The Extreme Value Theorem

Fast Solution of Inequality Using Continuity

Horizontal Tangent Lines to a Polynomial

Derivative of sin(x) and cos(x)

Power Rule for Rational Exponents

Example of Chain Rule 1 - Basic examples

Example of Chain Rule 2 - Approximation with Tangent Line

Example of Chain Rule 3 - Trig Functions

Example of Chain Rule 4 - Triple Chain Rule

Graphs and Higher Order Derivatives

Implicit Differentiation 1 - Definition and Basic Concepts

Implicit Differentiation 2 - Basic Example

Implicit Differentiation 3 - Approximation with Tangent Line

Implicit Differentiation 4 - Example with Trig Functions

Implicit Differentiation 5 - Higher Derivatives

Example of Extreme Value Theorem 1

Example of Extreme Value Theorem 2

Example of Extreme Value Theorem 3

Increasing/Decreasing and Derivatives 1

Increasing/Decreasing and Derivatives 2

Example of Increasing/Decreasing 1

Example of Increasing/Decreasing 2

Example of Increasing/Decreasing 3

First/Second Derivative Test for f(x) = x^4 - 12x^3

First/Second Derivative Test for f(x) = sin(x)

First/Second Derivative Test for f(x) = x^2 - 6x^{4/3}

Concavity and the Second Derivative

Concavity for f(x) = (x^2 - 36)/(x-2)

Concavity for f(x) = |x^2 - 4x - 12|

Example of Limit at Infinity 1

Example of Limit at Infinity 2

Example of Limit at Infinity 3

Checklist for Sketching Functions

Graph of f(x) = sin(x) + cos(x)

Graph of f(x) = sin(x)/(1+cos(x))

Graph of f(x) = x^{4/3} - 8x^{2/3}

Optimization - Maximizing Profit

**Calculus - Part 2: Basic Integration**

Antiderivative of a Polynomial

Antiderivative of (x-1)(x-2)/sqrt(x^3)

Antiderivative of sin(x)/[1-sin^2(x)]

Solving Differential Equations

Antiderivative of a Piecewise-Defined Function

Overview of Rectangular Approximation of Area

Rectangular Approximation of Area

Overview of Summation Formulas

General Method for Integer Sum Formula

Definition of Definite Integral

Definite Integral as Area 1 - Using the Graph

Definite Integral as Area 2 - Breaking Up the Region

Definite Integral as Area 3 - Area Below the x-axis

The First Fundamental Theorem of Calculus

The Mean Value Theorem for Integrals

Example of Mean Value Theorem for Integrals

The Second Fundamental Theorem of Calculus

Example of 2nd Fundamental Theorem of Calculus 1

Example of 2nd Fundamental Theorem of Calculus 2

Integration By Substitution: Antiderivatives

**Calculus - Part 3: Log, Exp, and Inverse Trig Integrals**

Definition of ln(x) Using Integration

Properties of ln(x) Using Integration

Derivative of ln(x) as a Slope

The Chain Rule for ln(x) and ln|x|

Derivative of f(x) = ln(ln(x^2 + 1))

Implicit Differentiation with ln(x)

Antiderivative of tan(x) Using ln(x)

Antiderivative of ln(x^4)/x (HD Version)

Antiderivative involving 1/x 1

Antiderivative involving 1/x 2

Log Integral for (x^2+5x+1)/(x^2+1)

Trig Antiderivatives involving 1/x

Inverse Functions: ln(x) and exp(x)

Inverse Function for f(x) = 2 e^{x-2}

No Inverse Function for f(x)=ln(x)-exp(x)

Derivative of an Inverse Function

Properties of Derivative of exp(x)

Examples of Derivatives with e^x

Implicit Differentiation with e^x

Tangent Line to x^2 e^x at x=1

Definite Integral of exp(-3x+2) (HD Version)

Definite Integral of (e^x - 1)(e^x - 2)/e^2x (HD Version)

Derivatives of log_b(x) and b^x

Tangent Lines to log_b(x) and b^x

Indefinite Integrals with b^x and log_b(x)

Definite Integrals with b^x and log_b(x)

Evaluating Inverse Trig Functions 1

Evaluating Inverse Trig Expressions 2

Evaluating Inverse Trig Expressions 3

Derivatives of Inverse Trig Functions

Graph of f(x) = tan^{-1}(1-x^2)

Tangent Line to f(x) = xsin^{-1}(x/2)

Integrals with Inverse Trig Functions 1

Integrals with Inverse Trig Functions 2

Integrals with Inverse Trig Functions 3

**Calculus - Part 4: Applied Integration**

Example of Volume Using Cross Sectional Area

The Disk/Washer Method for Volume 1

The Disk/Washer Method for Volume 2

The Disk/Washer Method for Volume 3

Arc Length Along Parabola 1: Base Case

Arc Length Along Parabola 2: Sinh Formula

Arc Length Along Parabola 3: Log Formula

Area of a Surface of Revolution

**Calculus - Part 5: Advanced Integration Techniques**

Integration by Parts 4 - Antiderivative of e^(3x)cos(4x) (Double IBP)

Integration By Parts 5 - Antiderivative for e^{3x}cos(4x) (Fast Solution)

Fast Antiderivative of x^2 exp(3x)

Reduction Formula for x^n exp(ax)

Integrals with cos^m(x) sin^n(x)

Trig Substitution 1 - Basic Inverse Trig Integrals

Trig Substitution 2 - Integral for (1+x^2)^{5/2}

Trig Substitution 3 - Integral of x^2/sqrt(1-4x^2)

Trig Substitution 4 - Integral of sqrt(e^{2x} - 1)

Trig Substitution Integral of sqrt(4-36x^2)/x^2

Integration with Partial Fractions 1

Integration with Partial Fractions 2

Integration with Partial Fractions 3

Integration with Partial Fractions 4

Integration with Partial Fractions 5

Partial Fraction Integral with Linear and Quadratic Factors

Partial Fraction Integral of x^2/(4x^-36)^2

Trig Sub Integral of x^2/(4x^2-36)^2

Growth of Functions at Infinity

Improper integral of exp(-1/x)/x^2

Mean of the Exponential Distribution

**Calculus - Part 6: Sequences and Series**

Examples of Recursive Sequences

Infinite Series 1b - Geometric Series

The Integral Test for Series 1a

The Integral Test for Series 1b

The Integral Test for Series 2

Estimating Sums with the Integral Test

Direct Comparison Test for Series 1

Divergence of Series for 1/ln(n)

Limit Comparison Test for Series 1

Limit Comparison Test for Series 2

Rational Function Test for Series

Alternating Series 1b - Estimating the Remainder

Root Test for Series Sum (1-1/n^2)^{n^3}

Motivating Taylor Polynomials 1

Motivating Taylor Polynomials 2

Approximating with Maclaurin Polynomials

Fast Maclaurin Polynomial for Rational Function

Approximating with Taylor Polynomials

Taylor's Theorem for Remainders

Taylor's Theorem : Remainder for 1/(1-x)

Example of Interval of Convergence Using Ratio Test

Derivative/Antiderivative of a Power Series 1a

Derivative/Antiderivative of a Power Series 1b

Derivative/Antiderivative of a Power Series 1c

Increasing the Interval of Convergence

Constructing Power Series from Functions 1a

Constructing Power Series from Functions 1b

Constructing Power Series from Functions 1c

The Taylor Series for f(x) = ln(x) at x = 1

The Maclaurin Series for f(x) = 1/(1-x)^2

The Maclaurin Series for f(x) = e^x

The Maclaurin Series for sin(x), cos(x), and tan(x)