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This course is offered exclusively as a one-on-one class and is not available in a small-group format.
Syllabus
Syllabus
1 Basic Techniques for Solving Equations
1.1 Isolation
1.2 Substitution
1.3 Elimination
1.4 Larger Systems of Linear Equations
1.5 Summary
2 Functions Review
2.1 Function Basics
2.2 Graphing Functions
2.3 Composition
2.4 Inverse Functions
2.5 Summary
3 Complex Numbers
3.1 Arithmetic of Complex Numbers
3.2 The Complex Plane
3.3 Real and Imaginary Parts
3.4 Graphing in the Complex Plane
3.5 Summary
4 Quadratics
4.1 Factoring Quadratics
4.2 Relating Roots and Coe cients
4.3 Completing the Square
4.4 The Discriminant
4.5 Quadratic Inequalities
4.6 Summary
5 Conics
5.1 Parabolas
5.2 Problem Solving With Parabolas
5.3 Maxima and Minima of Quadratics
5.4 Circles
5.5 Ellipses
5.6 Hyperbolas
5.7 Summary
6 Polynomial Division
6.1 Polynomial Review
6.2 Introduction to Polynomial Division
6.3 Synthetic Division
6.4 The Remainder Theorem
6.5 Summary
7 Polynomial Roots Part I
7.1 The Factor Theorem
7.2 Integer Roots
7.3 Rational Roots
7.4 Bounds
7.5 Graphing and the Fundamental Theorem of Algebra
7.6 Algebraic Applications of the Fundamental Theorem
7.7 Summary
8 Polynomial Roots Part II
8.1 Irrational Roots
8.2 Nonreal Roots
8.3 Vieta’s Formulas
8.4 Using Roots to Make Equations
8.5 Summary
9 Factoring Multivariable Polynomials
9.1 Grouping
9.2 Sums and Di↵erences of Powers
9.3 The Factor Theorem for Multivariable Polynomials
9.4 Summary
10 Sequences and Series
10.1 Arithmetic Sequences
10.2 Arithmetic Series
10.3 Geometric Sequences
10.4 Geometric Series
10.5 Sequence, Summation, and Product Notation
10.6 Nested Sums and Products
10.7 Summary
11 Identities, Manipulations, and Induction
11.1 Brute Force
11.2 Ratios
11.3 Induction
11.4 Binomial Theorem
11.5 Summary
12 Inequalities
12.1 Manipulating Inequalities
12.2 The Trivial Inequality
12.3 AM-GM Inequality with Two Variables
12.4 AM-GM with More Variables
12.5 The Cauchy-Schwarz Inequality
12.6 Maxima and Minima
12.7 Summary
13 Exponents and Logarithms
13.1 Exponential Function Basics
13.2 Introduction to Logarithms
13.3 Logarithmic Identities
13.4 Using Logarithm Identities
13.5 Switching Between Logs and Exponents
13.6 Natural Logarithms and Exponential Decay
13.7 Summary
14 Radicals
14.1 Raising Radicals to Powers
14.2 Evaluating Expressions With Radicals
14.3 Radical Conjugates
14.4 Summary
15 Special Classes of Functions
15.1 Rational Functions and Their Graphs
15.2 Rational Function Equations and Inequalities
15.3 Even and Odd Functions
15.4 Monotonic Functions
15.5 Summary
16 Piecewise Defined Functions
16.1 Introduction to Piecewise Defined Functions
16.2 Absolute Value
16.3 Graphing Absolute Value
16.4 Floor and Ceiling
16.5 Problem Solving with the Floor Function
16.6 Summary
17 More Sequences and Series
17.1 Algebra of Recursive Sequences
17.2 Telescoping
17.3 Sums of Polynomial Series
17.4 Arithmetico-Geometric Series
17.5 Finite Di↵erences
17.6 Summary
18 More Inequalities
18.1 Mean Inequality Chain
18.2 The Rearrangement Inequality
18.3 When Formulas Fail
18.4 Summary
19 Functional Equations
19.1 Finding Values
19.2 Finding Functions with Substitution
19.3 Separation
19.4 Cyclic Functions
19.5 Summary
20 Some Advanced Strategies
20.1 Symmetry
20.2 Substitution for Simplification
20.3 Method of Undetermined Coe cients
20.4 Constructing Polynomials From Roots
20.5 Common Divisors of Polynomials
20.6 Symmetric Sums Revisited
20.7 Summary
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