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Calculus 2

1. Review: Derivatives

Basics of Derivatives

Derivative Rule

Implicit Differentiation

2. Integration

Indefinite Integrals

Definite Integrals

Fundamental Theorem of Calculus (FTC)

3. Applications of Integrals

Basic Applications of Integrals

Area between Curves

Volumes (Cross Sections Method)

Volumes of Revolution (Disks/Washer Method)

Volumes of Revolution (Cylindrical Shells Method)

Arc Length of a Curve

Surface Area of Revolution

4. Integration Techniques

Integration by Parts

Partial Fractions

Mixed Integration Questions

Numerical Integration

5. Improper Integrals

L'Hospital's Rule

Improper Integrals

Comparison Theorem for Improper Integrals

6. Sequences and Series

Basic Definitions

Test of Divergence

Comparison Tests

Absolute Convergence & Alternating Series Test

Which Convergence Test to Use?

7. Power Series

Functions and Power Series

Taylor and Maclaurin Series

8. Parametric & Polar Curves

Parametric Equations

Calculus w/ Parametric Curves

Polar Coordinates

Calculus w/ Polar Coordinates

Conic Sections [Coming Soon!]

9. Differential Equations

Basics of Differential Equations

Separable Differential Equations

Linear Differential Equations

Applications of DE

Second-Order Reducible DEs

Non-Linear DEs [Coming Soon!]

10. Vector Functions

Vector Geometry

Vector Functions

11. Multivariable Functions

Functions of Several Variables

Partial differentiation

Gradient and the directional derivative

Tangent plane & Linear Approximation

Taylor Series in Two Variables [Coming Soon!]

Practice exam questions

12. Bonus: Integral calculus in several variables (Videos Coming Soon)

Double Integration (rectangular)

Double Integration (polar)

Triple Integration (rectangular)

Triple Integration (cylindrical)

Triple Integration (spherical)

Improper Integrals

Applications of multiple integrals

Practice quiz: Multiple integrals

13. Vector Calculus [Coming Soon!]

Linear Algebra

1. Vectors

Basics of Vectors

Vector Operations and Linear Combinations

Vector Properties

Length of a Vector (Vector Norm)

2. Products of Vectors

Dot Product Properties

Angle Between Vectors

Projection (Proj) and Perpendicular (Perp)

Cross Product Properties

Cross Product and Area

Volume of a Parallelepiped (Triple Product)

3. Equations of Lines and Planes

\reals^n

\reals^3

Intersection of Two Lines

Intersection of a Line and a Plane

4. Systems of Linear Equations (SLEs) (Linear Systems)

Basics of Systems of Linear Equations

Augmented Matrix

Reduced Row Echelon Form (RREF)

Reducing a Matrix (Gauss-Jordan Elimination)

Rank of a Matrix

Solving Linear Systems

Geometric Interpretation of Linear Systems

Polynomial Interpolation

5. Matrices

Basics of Matrices

Matrix Multiplication

Matrix Transpose

Matrix Inverse Algorithm

Elementary Matrices

Invertible Matrix Theorem

Solving Matrix Equations

6. Determinants

Basics of Determinants

Determinants and EROs

Determinants of Large Matrices

(Optional) Shortcut for 3x3 Determinants

Determinants and Inverse

Cramer's Rule for Solving Linear Systems

7. Vector Spaces and Subspaces

Linear Independence

Basis and Dimension

Column Space and Null Space (Range and Kernel)

8. Orthogonality

Orthonormal Basis and Gram-Schmidt Process

Orthogonal Complement and Orthogonal Projection (COMING SOON)

Orthogonal Diagonalization of Symmetric Matrices (COMING SOON)

9. Complex Numbers

Basics of Complex Numbers

Powers and Roots

Quadratic Formula

10. Linear Transformations

Linear Transformations

Composition and Inverse

Special Linear Transformations in

\reals^2

11. Eigenvalues and Eigenvectors (Spectral Theory)

Eigenvalues and Eigenvectors

Complex Eigenvalues

Diagonalization

Markov Chains (Random Walks)

Linear Dynamical Systems

12. Differential Equations

Basics of Differential Equations

Systems of Linear Differential Equations

Higher Order Differential Equations

13. Least Squares Approximation

Least Squares Approximation

14. Cryptography (COMING SOON)

Basics of Cryptography

Calculus 1

1. Pre-Calculus (Review)

Interval Notation vs. Set Notation

Operations with Functions

Even and Odd Functions

Transformations of Functions

Dividing Polynomials

Rational Functions

Radical Functions

Piecewise Functions

Absolute Value Functions

Inverse Functions

Exponential Functions

Logarithmic Functions

Trigonometric Functions

Inverse Trigonometric Functions

2. Limits

One-Sided Limits

Infinite Limits

Limits at Infinity

Indeterminate Forms

Computing Limits by Factoring

Computing Limits by Multiplying by the Conjugate

Computing Limits by Getting a Common Denominator

The Fundamental Trig Limit

The Squeeze Theorem

The Intermediate Value Theorem

The Epsilon Delta Definition of a Limit

Exam-Like Practice Questions

5. Applications of Differentiation for Science

Position, Velocity, and Acceleration

Exponential Growth and Decay

Newton’s Law of Cooling

Exam-Like Practice Questions

6. Applications of Differentiation for Business & Econ

Marginal Concepts In Economics

Elasticity of Demand

Compound Interest

Exam-Like Practice Questions

7. Differential Equations

Basics of Differential Equations

Separable Differential Equations

Linear Differential Equations

Applications of DE

Second-Order Reducible DEs

8. Integrals

Antiderivatives & The Indefinite Integrals

Antiderivatives of Trig Functions

Antiderivatives of Inverse Trig Functions

Antiderivatives of Exponential Functions

Antiderivatives of

\frac{1}{x}

Initial Conditions

Approximating Areas & Riemann Sums

The Definite Integral

Properties of Definite Integrals

Computing Definite Integrals Geometrically

The Fundamental Theorem of Calculus - Part 1

The Fundamental Theorem of Calculus - Part 2

Integration by Substitution (U-Substitution)

Exam-Like Practice Questions

9. Applications of Integration

Distance and Displacement

Average Value of a Function

Area Between Curves

Volumes of Revolution by Cross Sections (Washers)

Volumes of Revolution by Cylindrical Shells

Surface Area of Revolution

Exam-Like Practice Problems

10. Applications of Integration for Physical Science

Position, Velocity, and Acceleration

Mass and Density

Moments and Center of Mass

Pressure and Force

Exam-Like Practice Questions

11. Integration Techniques

Integrating Even and Odd Functions

Integration by Parts

Trigonometric Integrals

Trigonometric Substitution

Partial Fractions

The Midpoint Rule

The Trapezoid Rule

Error of Approximations

Exam-Like Practice Questions

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