Calculus Sequences
First and second year Calculus courses
Distinct Real Roots
Constant Coefficients with Distinct Real Roots
Complex Conjugate Roots
Constant Coefficients with Complex Conjugate Roots
Repeated Roots
Constant Coefficients with Repeated Roots
Reduction of Order
Recovering the full solution set given one homogeneous solution
Cauchy-Euler Equations
The Monomial Analogue of the Constant Coefficients Equations
Exponential Functions
Finding the right constant multiple
Polynomial Functions
Using either Iterative Refinement or Undetermined Coefficients
Sine and Cosine Functions
Using either Iterative Refinement or Undetermined Coefficients
Exponential Functions with Conflicting Roots
The desired exonential is alredy a homogeneous solution
Variation of Parameters
A Determinant and Integration Heavy Method for finding Particular Solutions
Extra Variation of Parameters Problems (external reference)
Warning: These problems can be very tedious. Practice these sparingly if at all.
What Are Eigenvectors?
Getting a better understanding
How Do We Use Eigenvectors?
Using eigen-axes to model matrices
Computing Eigen-coordinates
A very short guide
Computing Eigenvectors
How to Compute Eigenvalues and Eigenvectors of a Matrix
DE Connection
Using eigenvectors in differential equations
Review Worksheet
A comprehensive pdf with the definition through diagonalization and applications
Multi-variable Limits
Considering more approaches than just left and right
Partial Derivatives
Why do we hold y constant?
Lagrange Multipliers
Level sets and gradient alignment
Gradient Descent and Area
How to Improve Riemann Sum Estimates
Fundamental Principles of Area and Volume
Building shapes with prisms and cones
Volume of a Pyramid
Slices, Riemann Sums, and 3D Objects
Double Integrals
How can we integrate surfaces with slices?
Line Integrals
How should we think about integrating curves?
Understanding Surface and Flux Integrals
Building intuition for how both are computed with a (sometimes signed) scaling factor
Dot Products and Approximations
Connecting dot products to approximation methods
Sine and Cosine Properties
Geometric Properties of Sine and Cosine following from the previous section
Computing Fourier Coefficients
Geometric Properties of Sine and Cosine following from the previous section
Wave Equation Example
Seeing what a solution does and how it connects to the initial conditions
Lecture Notes by Topic
A custom reader for lecture notes
Area Under the Graph
Riemann Sums, Signed Area, and Visualizing FTC
Area Practice
Randomized Practice Problems: Signed Area and Average Height, Riemann Sums, Integrals, and Word Problems
Midterm 1 Practice Problems
Derivatives, Integrals, Word Problems
Sine Waves
Matching Sine Waves with the right constants: A, K, ϕ, and B
Taylor Coefficient Search
Finding the best fitting polynomial
Taylor Series
Visualizing Taylor Series
Analysis and Approximation Practice
Product Rule, Second Derivative Test, Sine Waves, and Linear Approximations (for Orders of Smallness/Taylor Series)
DE Practice
Randomized Practice Problems: Anti-Derivatives, Exponential Growth and Decay, Logistics
Slope Field Games
Get a better understanding of Slope Fields and Euler's Method with games
Slope Field Practice
Randomized Practice Problems: Solutions, Isoclines, Long-term Behavior, Euler's Method
Understanding Partial Derivatives
Why do we hold y constant?
Multivariable Practice
Randomized Practice Problems: Partial Derivatives, Tangent Planes, and Optimization
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