College Calculus Courses and Topics

1. Differential Calculus of Single Variable Functions

• Limits & Limits Properties
• Limits Laws
• Continuity of f(x)
• Discontinuities
• Derivative of f(x)
• Derivative of f(x)
• Differentiation Rules – Power Rule / Product Rule / Chain Rule / Quotient Rule
• Derivatives of Polynomials and Exponential Functions
• Derivatives of Trigonometric Functions
• Implicit Differentiation
• Derivatives of Logarithmic Functions
• Derivatives of Inverse Functions
• Differentials
• Applications of Differentiation
• Maximum and Minimum Values
• The Mean Value Theorem
• How Derivatives Affect the Shape of a Graph
• Indeterminate Forms
• l’Hospital’s Rule
• Graphing with Calculus
• Optimization Problems
• Related Rates Problems
• Linear Approximations
• Newton’s Method
• and more

2. Integral Calculus of Single Variable Functions

• Antiderivatives
• Integrals
• Areas and Distances
• The Definite Integral
• The Fundamental Theorem of Calculus
• Indefinite Integrals and the Net Change Theorem
• The Substitution Rule
• Applications of Integration
• Areas Between Curves
• Volumes of Rotation
• Volumes by Discs/Washers/Cylindrical Shells
• Work Line Integral
• Mean Value Theorem
• Average Value of a Function
• Techniques of Integration
• Integration by Parts
• Trigonometric Integrals
• Trigonometric Substitution
• Integration of Rational Functions by Partial Fractions
• Strategy for Integration
• Approximate Integration
• Improper Integrals
• Arc Length
• Area of a Surface of Revolution
• and more

3. Calculus of Multivariable Functions

• Parametric Equations
• Arc Length
• Tangents and Areas
• Polar Coordinates
• Conic Sections
• Quadratic Surfaces
• Infinite Series
• Series Convergence Tests
• Taylor Series
• Taylor Polynomials
• Taylor Inequality
• Vector Valued Functions
• Vector Equation of a Line
• Vector Equation of a Plane
• Partial Derivatives
• Extrema of multivariable function
• Chain Rule for Multivariable Function
• Lagrange Multiplier
• Double Integrals
• Triple Integrals
• Coordinate Transformations
• Jacobian of the Transformation
• and more

3. Vector Calculus

• Vector Dot Product
• Vector Cross Product
• Vector Projections
• Fundamental Theorem of Line Integrals
• Line Integrals of scalar functions
• Line Integrals of vector functions
• Vector Fields
• Divergence and Curl of Vector Field
• Surface Integrals of scalar funcitons
• Surface Integrals of scalar fields
• Surface Integrals of vector fields
• Divergence Theorem
• Green’s Theorem
• Stokes’ Theorem
• and more

4. Linear Algebra

• Matrix Algebra
• Inverse Matrix
• Systems of Linear Equations
• Row Reduction and Echelon Forms
• Vector Representation of System of Equations
• Linear Independence
• Linear Transformations
• Leontief Input—Output Model
• Determinants
• Properties of Determinants
• Cramer’s Rule
• Vector Spaces & Subspaces
• Linearly Independent Sets; Bases
• Change of Basis
• Eigenvectors and Eigenvalues
• The Characteristic Equation
• Matrix Diagonalization
• Complex Eigenvalues
• Inner Product, Length, and Orthogonality
• Orthogonal Sets
• Orthogonal Projections
• The Gram-Schmidt Process
• Least-Squares Problems
• Inner Product Spaces
• Diagonalization of Symmetric Matrices
• Quadratic Forms
• The Singular Value Decomposition
• and more

5. Ordinary Differential Equations

• First Order Linear Differential Equations
• Separable Equations
• Integrating Factor
• Exact Equations
• bernoulli equation
• Special Substitution Solutions
• Applications of First Order Equations
• Growth and Decay Problems
• Cooling Equation
• Tank Mixing Equation
• Logistics Equation
• Population Equation
• Preditor/Prey Equation
• Linear Second Order Equations
• Homogeneous Linear Equations
• Characteristic Equation
• Constant Coefficient Homogeneous Equations
• Nonhomgeneous Linear Equations
• The Method of Undetermined Coefficients
• The Method of Undetermined Coefficients
• Reduction of Order
• Variation of Parameters
• Oscillator Equation
• Laplace Transform Solutions
• Laplace Transform
• Inverse  Laplace Transform
• Step Function
• Delta Function
• Convolution IntegralSeries Solutions of Linear Second Order Equations
• Series Solutions of Linear Second Order Equations
• Fourier Series Solutions
• and more

6. Partial Differential Equations

• Boundary Value Problems
• Techniques for Solving Partial DEs
• Heat Equation
• Fourier Series and Integrals
• Wave Equation
• Potential Equation
• Euler Beam
• Higher Dimensions and Other Coordinates
• Laplace Transform
• and more