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• Pre-Calculus
  1. Algebra
     1. Introduction   1
        1. Introduction
     2. Polynomials   1   2   3   4
        1. Definitions
        2. Multiplying Polynomials
        3. Factoring Polynomials
        4. Roots of Polynomials
     3. Sequences & Series   1   2   3
        1. Definitions
        2. Arithmetic Sequences & Series
        3. Geometric Sequences & Series
     4. Mathematical Induction   1   2
        1. Induction Procedure
        2. Summation Formulas
     5. Combinatorics   1   2   3   4   5
        1. Permutations
        2. Combinations
        3. Pascal's Triangle
        4. Binomial Expansion
        5. Binomial Coefficients Count Combinations
     6. Determinants   1   2   3   4   5
        1. Method of Diagonals
        2. Expansion on Minors
        3. More Permutations
        4. Precise Determinants
        5. Properties of Determinants
     7. Typical Misconceptions   1
        1. Fractions
     8. Exercises   (HW)   1
        1. Exercises
  2. Functions
     1. Definition and Properties   1   2   3   4   5
        1. Formulas
        2. Graphs
        3. Vertical Line Test
        4. Domain and Image
        5. One-to-One and the Horizontal Line Test
     2. Polynomials, Etc.   1   2   3   4   5   6
        1. Linear Functions
        2. Positive Powers
        3. Polynomials
        4. Negative Powers
        5. Rational Powers
        6. Generalized Polynomials
     3. New Functions from Old   1   2   3   4   5   6
        1. Algebraic Operations
        2. Shifting Functions
        3. Scaling Functions
        4. Reflecting Functions
        5. Composition
        6. Inverse Functions
     4. Equations and Implicitly Defined Functions   1   2   3
        1. Equations
        2. Graphs of Equations
        3. Implicitly Defined Functions
     5. Exercises   (HW)   1
        1. Exercises
  3. Trigonometry Available as Sample
     1. Angular Measure   1
        1. Angular Measure
     2. Triangle Definitions   1
        1. Triangle Definitions
     3. Circle Definitions   1   2   3   4   5   6
        1. Formulas
        2. Angles Bigger than \(90^\circ\)
        3. Unit Circle
        4. Plots of Trig Functions
        5. Properties of Sine and Cosine Curves
        6. Tutorial on the Properties of Sine and Cosine Curves
     4. Trig Identities   1   2   3   4   5   6   7
        1. Pythagorean Identities
        2. Parity, Complementarity and Supplementarity Identities
        3. Law of Sines and Law of Cosines
        4. Sum and Difference Identities
        5. Sum and Difference Consequences
        6. The Triangle Inequality
        7. Summary of Trig Identities
     5. Inverse Trig Functions   1   2   3   4
        1. Inverse Functions (Review)
        2. Inverse Sine & Cosine
        3. Inverse Tangent and Cotangent
        4. Inverse Secant & Cosecant
     6. Applications   1   2   3   4
        1. Similar & Congruent Triangles
        2. Solving Triangles
        3. Solving Triangles- Almost (Optional)
        4. Slope
     7. Exercises   (HW)   1
        1. Exercises
  4. Exponentials and Logarithms
     1. Exponentials   1   2
        1. Definitions
        2. Identities
     2. Logarithms   1   2
        1. Definitions
        2. Identities
     3. Plots   1
        1. Plots
     4. The Natural Exponential and Logarithm   1   2
        1. The Number \(\large\text{e}\)
        2. Identities
     5. Applications of Exponentials and Logs   1   2   3   4
        1. Exponential Growth
        2. Exponential Decay
        3. Compound Interest
        4. Continuously Compounded Interest
     6. Typical Misconceptions   1   2
        1. Exponentials
        2. Logarithms
     7. Hyperbolic Trig Functions (Optional)   1   2   3   4   5
        1. Definitions
        2. Plots
        3. Identities
        4. Inverse Function Plots
        5. Inverse Function Formulas
     8. Exercises   (HW)   1
        1. Exercises
  5. Vectors Available as Sample
     1. Definition, Magnitude and Direction   1   2   3   4
        1. Vectors as Arrows
        2. Displacements
        3. Magnitude
        4. Direction
     2. Vector Addition   1   2   3   4
        1. Algebraic Definition
        2. Geometric Construction
        3. Properties
           1. Triangle Inequality - Proof
        4. Application - Balancing Forces
     3. Scalar Multiplication   1   2   3   4   5
        1. Algebraic Definition
        2. Geometric Construction
        3. Properties
        4. Linear Combinations
        5. \(\hat\imath\hat\jmath\) Notation
     4. Dot Product   1   2   3   4   5   6   7
        1. Algebraic Definition
        2. Properties
        3. Geometric Formula
        4. Angles
        5. Vector Projections
        6. Scalar Projections
        7. Application - Work
     5. Parametric Curves   1   2   3
        1. Parametric Lines
        2. Intersections of Lines
        3. Parametric Curves
     6. Exercises   (HW)   1
        1. Exercises
• Limits and Continuity
  6. Intuitive Limits and Continuity Available as Sample
     1. What my Algebra Teacher Never Told me!   1
        1. What my Algebra Teacher Never Told me!
     2. Intuitive Definitions   1   2
        1. Left and Right Limits
        2. 2-Sided Limits and Continuity
     3. Tutorial on Limits and Continuity   1
        1. Tutorial on Limits and Continuity
     4. Continuity on an Interval   1
        1. Continuity on an Interval
     5. Limits at Infinity   1
        1. Limits at Infinity
     6. Exercises   (HW)   1
        1. Exercises
  7. Computing Limits Available as Sample
     1. Limits at Finite Numbers   1   2
        1. Special Limits
        2. Limit Laws
           1. Examples
     2. When Limit Laws Don't Apply   1
        1. Limit Tricks
           1. Deriving Indeterminate Forms
           2. Factor and Cancel
           3. Expand and Cancel
           4. Divide by the Largest Term in the Denominator
           5. Put Terms over a Common Denominator
           6. Multiply by the Conjugate
     3. Sandwich Theorem   1   2
        1. The Theorem
        2. Sine and Cosine Limits
     4. Limits at Infinity   1   2   3
        1. Special Limits
        2. Limit Laws
           1. Examples
        3. When Limit Laws Don't Apply
           1. Divide by the Largest Term in the Denominator
           2. Put Terms over a Common Denominator
           3. Multiply by the Conjugate
           4. Replace Variable by Reciprocal
     5. Exercises   (HW)   1
        1. Exercises
  8. Applications of Limits and Continuity
     1. Intermediate & Extreme Value Theorems   1   2   3
        1. Intermediate Value Theorem
        2. Numerical Solution of Equations
        3. Extreme Value Theorem
     2. Asymptotes   1   2
        1. Horizontal Asymptotes
        2. Vertical Asymptotes
     3. Interest   1   2   3
        1. Simple Interest
        2. Compound Interest
        3. Continuous Interest
     4. Rates of Change   1   2   3   4   5   6
        1. Introduction
        2. Average Velocity
        3. Instantaneous Velocity
        4. Slope of a Secant Line
        5. Slope of a Tangent Line
        6. Other Rates of Change
     5. Accumulations of Change (Optional)   1   2   3   4
        1. Introduction (Optional)
        2. Distance (Optional)
        3. Area (Optional)
        4. Other Accumulations of Change (Optional)
     6. Exercises   (HW)   1
        1. Exercises
  9. Precise Limits and Continuity
     1. Precise Definitions   1   2   3
        1. Precise Left and Right Limits
        2. Precise \(2\)-Sided Limits
        3. Precise Continuity
     2. Proofs   1   2
        1. For Linear Functions
        2. For Non-Linear Functions
     3. Limit Laws   1   2
        1. Limit Laws
        2. Sandwich Theorem
     4. Infinite Limits & Limits at Infinity   1   2
        1. Precise Infinite Limits
        2. Precise Limits at Infinity
     5. Intermediate and Extreme Value Theorems   1   2
        1. Intermediate Value Theorem
        2. Extreme Value Theorem
     6. Exercises   (HW)   1
        1. Exercises
• Derivatives
  10. Derivatives and Tangent Lines Available as Sample
      1. Derivative at a Point   1   2
         1. Definition
         2. Applications
      2. Tangent Lines   1
         1. Tangent Lines
      3. Derivative as a Function   1   2   3
         1. Definition
         2. Slope, Increasing and Decreasing
         3. Velocity, Acceleration and Other Rates
      4. The First Derivative Test for Maxima and Minima   1
         1. The First Derivative Test for Maxima and Minima
      5. Exercises   (HW)   1
         1. Exercises
  11. Derivative Rules
      1. Monomial Rules   1
         1. Monomial Rules
      2. Sum & Difference Rules   1
         1. Sum & Difference Rules
      3. Product & Quotient Rules   1   2   3
         1. Product Rule
         2. Quotient Rule
         3. Negative Power Rules
      4. Trigonometric Rules   1   2
         1. Sine & Cosine Rules
         2. Tangent, Cotangent, Secant & Cosecant Rules
      5. Exponential Rules   1   2
         1. Derivation
         2. Hyperbolic Trig Rules (Optional)
      6. Derivative Rules Summary   1
         1. Derivative Rules Summary
      7. Exercises   (HW)   1
         1. Exercises
  12. Differentials & Linear Approximation Available as Sample
      1. Differences & Differentials   1
         1. Differences & Differentials
      2. Linear Approximation   1
         1. Linear Approximation
      3. Mean Value Theorem   1
         1. Mean Value Theorem
            1. Mean Value Theorem - Proof
      4. Exercises   (HW)   1
         1. Exercises
  13. Chain Rule & Implicit Differentiation Available as Sample
      1. Compositions   1
         1. Compositions
      2. Chain Rule   1   2
         1. Chain Rule
         2. Nested Chain Rules
      3. Tutorial on Chain Rule   1
         1. Tutorial on Chain Rule
      4. Implicit Differentiation   1
         1. Implicit Differentiation
      5. Exercises   (HW)   1
         1. Exercises
  14. Inverse Functions
      1. Inverse Function Theorem   1
         1. Inverse Function Theorem
      2. Rational Power Law   1   2
         1. Root Functions
         2. Rational Power Functions
      3. Inverse Trigonometric Functions   1   2
         1. Inverse Sine & Cosine Functions
         2. Inverse Tangent, Cotangent, Secant & Cosecant Functions
      4. Logarithm Functions   1   2   3
         1. Logarithm Functions
         2. Logarithmic Differentiation
         3. Complete Power Rule
      5. Inverse Hyperbolic Trig Functions (Optional)   1
         1. Inverse Hyperbolic Trig Functions (Optional)
      6. Other Inverse Functions   1
         1. Other Inverse Functions
      7. Exercises   (HW)   1
         1. Exercises
  15. Higher Derivatives
      1. Higher Derivatives   1
         1. Higher Derivatives
      2. Concavity   1
         1. Concavity
      3. The Second Derivative Test for Maxima and Minima   1
         1. The Second Derivative Test for Maxima and Minima
      4. Acceleration and Jerk   1
         1. Acceleration and Jerk
      5. Higher Order Approximations (Optional)   1
         1. Higher Order Approximation (Optional)
      6. Extended Mean Value Theorem (Optional)   1
         1. Extended Mean Value Theorem (Optional)
            1. Proof
      7. Exercises   (HW)   1
         1. Exercises
• Applications of Derivatives
  16. Parametric Curves
      1. Position & Velocity   1   2
         1. Position Vector
         2. Velocity Vector
      2. Tangent Vectors & Tangent Lines   1   2
         1. Tangent Vectors
         2. Tangent Lines
      3. Slopes   1   2
         1. Rise and Run
         2. Horizontal and Vertical Tangent Lines
      4. Ballistic Motion   1
         1. Ballistic Motion
      5. Exercises   (HW)   1
         1. Exercises
  17. Limits and Solving Equations
      1. l'Hopital's Rule   1   2
         1. For the Indeterminate Form \(\dfrac{0}{0}\)
         2. For the Indeterminate Form \(\dfrac{\infty}{\infty}\)
      2. Indeterminate Forms   1
         1. Indeterminate Forms
            1. Deriving Indeterminate Forms
            2. Indeterminate Forms \(\dfrac{0}{0}\) and \(\dfrac{\infty}{\infty}\)
            3. Indeterminate Form \(0\cdot\infty\)
            4. Indeterminate Form \(\infty-\infty\)
            5. Indeterminate Forms \(0^0\) and \(1^\infty\) and \(\infty^0\)
      3. Newton's Method   1
         1. Newton's Method
      4. Exercises   (HW)   1
         1. Exercises
  18. Related Rates Available as Sample
      1. For Variables Related by an Equation   1
         1. For Variables Related by an Equation
      2. Word Problems   1
         1. Word Problems
      3. Exercises   (HW)   1
         1. Exercises
  19. Properties of Graphs
      1. Overview   1
         1. Overview
      2. Value & Limit Information   1   2   3   4
         1. Intercepts & Positive and Negative Values
         2. Vertical Asymptotes
         3. Horizontal Asymptotes
         4. Slant Asymptotes (Optional)
      3. First Derivative Information   1   2
         1. Increasing & Decreasing
         2. Maxima & Minima by the First Derivative Test
      4. Second Derivative Information   1   2
         1. Concave Up & Down
         2. Maxima & Minima by the Second Derivative Test
      5. Curve Sketching   1
         1. Curve Sketching
      6. Interpreting Graphs   1
         1. Interpreting Graphs
      7. Exercises   (HW)   1
         1. Exercises
  20. Max/Min Problems
      1. Local Maxima and Minima   1   2   3
         1. Qualitative Behavior
         2. \(1^{\text{st}}\) Derivative Test
         3. \(2^{\text{nd}}\) Derivative Test
      2. Global Maxima and Minima   1   2
         1. On a Finite Interval
         2. On an Infinite Interval
      3. Constrained Max/Min Problems   1   2
         1. Eliminating a Variable
         2. Word Problems
      4. Exercises   (HW)   1
         1. Exercises
• Integrals
  21. Antiderivatives, Areas and the FTC Available as Sample
      1. Definition of Antiderivatives   1   2
         1. Definition of Antiderivatives
         2. Non-Uniqueness of Antiderivatives
      2. Antiderivative Rules   1   2   3   4
         1. Algebraic Operations
         2. Exp and Log Functions
         3. Trig Functions
         4. Hyperbolic Trig Functions (Optional)
      3. From Acceleration to Velocity and Position   1   2   3
         1. In General
         2. Motion with Constant Acceleration
         3. Ballistic Motion
      4. Differential Equations and Initial Value Problems   1
         1. Differential Equations and Initial Value Problems
      5. Area as an Antiderivative   1
         1. Area as an Antiderivative
      6. Fundamental Theorem of Calculus for Area   1
         1. Fundamental Theorem of Calculus for Area
      7. Exercises   (HW)   1
         1. Exercises
  22. Riemann Sums, Integrals and the FTC Available as Sample
      1. Definition of Riemann Sums   1   2   3
         1. Right Riemann Sums and Areas
         2. General Riemann Sums and Integrals (Optional)
         3. Limits of Riemann Sums (Optional)
      2. Fundamental Theorem of Calculus for Integrals   1   2
         1. The \(1^\text{st}\) and \(2^\text{nd}\) Fundamental Theorems
         2. Leibniz's Method
      3. Indefinite Integrals   1
         1. Indefinite Integrals
      4. Integration Rules and Properties   1   2   3   4   5
         1. Algebraic Operations
         2. Exp and Log Functions
         3. Trig Functions
         4. Hyperbolic Trig Functions (Optional)
         5. Definite Integral Properties
      5. Exercises   (HW)   1
         1. Exercises
  23. Integration by Substitution
      1. Indefinite Integrals   1   2
         1. Indefinite Integrals
         2. Adjusting the Differential
      2. Definite Integrals   1   2
         1. Substituting Back
         2. Changing Limits
      3. Exercises   (HW)   1
         1. Exercises
  24. Applications of Integration
      1. Area   1   2   3   4
         1. Area Below a Curve
         2. Area Between Curves
         3. Area Between Curves in Pieces
         4. \(y\)-Integrals
      2. Average Value and Mean Value Theorem   1   2   3
         1. Average Value of a Function
         2. Average Value as Height of Rectangle
         3. Mean Value Theorem
      3. Tutorial on Average Value   1
         1. Tutorial on Average Value
      4. Mass and Center of Mass   1   2   3
         1. Mass: Bar with Non-Uniform Density
         2. Center of Mass: Discrete Masses
         3. Center of Mass: Bar with Non-Uniform Density
      5. Exercises   (HW)   1
         1. Exercises
  25. Sample Exams
      1. Math 171 Fall 2021   1
         1. Math 171 Fall 2021
      2. MATH 151 Fall 2011   1
         1. MATH 151 Fall 2011