Notes for Students
General Notes
Notes for Basic Algebra Classes
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Basic Rules of Algebra
- Functions-Definitions and Notation
- Solving Inequalities
- Multiplying Polynomials
- Factoring Polynomials
- Quadratic Formula
- Functions-Definitions and Notation
- This is a derivation of the Quadratic Formula using Completing the Square.
Notes for Calculus Courses
- Slope of a Tangent Line
- This note uses only algebra and analytic geometry to demonstrate that the slope of the line tangent at (3,9) to the graph of y=x2 is 6.
- Logarithmic and Exponential Functions
- This note describes some of the algebra of logarithmic and exponential functions, but does not go into any Calculus.
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Limits
- Limit Calculations
- This note describes three categories of routine calculations of ordinary limits.
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Derivatives: Definition and Notation
- Calculating Derivatives
- A short summary of the key rules for calculating derivatives.
- Calculating Derivatives
- Also a short summary of the key rules for calculating derivatives, but without including trigonometric, exponential and logarithmic functions.
- Derivation of the Chain Rule
- This short note gives a derivation of the Chain Rule.
- An Example and a Proof
- An example of a derivative which is not continuous and a proof that derivatives have the Intermediate Value Property.
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Implicit Differentiation
- Inequalities
- Analyzing the sign of a function, a skill that is needed time and time again, particularly in analyzing monotonicity and concavity.
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Monotonicity, Extrema, Concavity and Sketching Graphs
- Differentials and Tangent Approximations
- Integration Using the Substitution Method
- Numerical Integration
- Differentials and Tangent Approximations
- Derivations of the Left Hand Rule, Right Hand Rule, Midpoint Rule, the Trapezoid Rule and Simpson's Rule. Examples of how to use error estimates along with a glimpse at how formulas for error estimates may be derived.
- Improper Integrals
- Definitions. Comparison Test and P-Test, along with variations.
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Sequences and Series
- Velocity, Acceleration and Curvature
- Vectors, Lines and Planes
- Velocity, Acceleration and Curvature
- A brief introduction using vectors.
- Dot Products
- The rationale for the dot product of two vectors, basic properties and the Cauchy-Schwarz Inequality.
- Cross Products
- Definition and rationale for the cross product of two vectors in R3.
- Line Integrals
Notes for Linear Algebra
- Change of Basis
- Notes relating to how the matrix for a linear transformation changes under a change of basis.
Notes for Differential Equations
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