15 core topics + 96 prerequisite topics taught as needed · approximately 29 hours of instruction including spaced review
An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.
| Differential Equations: Verifying Solutions [M] | A solution is a function that satisfies the equation. |
| Slope Fields [M] | A picture of dy/dx at every point. |
| Euler's Method (BC) [H] | Stepping along tangent lines. |
| Separation of Variables [H] | Move all y's left, all x's right, integrate both sides. |
| Exponential Growth & Decay Models [M] | dy/dt = ky means y = y₀e^{kt}. |
| Logistic Growth (BC) [H] | Growth limited by a carrying capacity L. |
| First-Order Linear ODEs [M] | y′ + ay = b: solved with an integrating factor. |
| Second-Order Linear ODEs [H] | Constant coefficients: guess e^{rt}, factor the characteristic equation. |
| Complex & Repeated Roots [H] | Complex roots −a ± bi mean decaying oscillations. |
| Undetermined Coefficients [H] | Guess a particular solution shaped like the forcing term. |
| Initial Value Problems [H] | Initial conditions pin down the constants. |
| Systems of ODEs [H] | x′ = Ax grows and decays along eigenvector directions. |
| Laplace Transforms [H] | Turn calculus into algebra: differential equations become polynomial ones. |
| Equilibria & Stability [H] | Where y′ = 0, and whether solutions are attracted or repelled. |
| Oscillators & Mixing Models [H] | The same equations describe springs, circuits, and tanks. |
| Adding & Subtracting Whole Numbers | Multi-digit addition and subtraction. |
| Multiplication | Multiplying whole numbers. |
| Division | Dividing whole numbers. |
| Order of Operations | Parentheses first, then multiplication/division, then addition/subtraction. |
| Negative Numbers: Adding & Subtracting | Working with numbers below zero on the number line. |
| Negative Numbers: Multiplying & Dividing | Sign rules for products and quotients. |
| Exponents | Repeated multiplication in shorthand. |
| Square Roots | Undoing a square. |
| Equivalent Fractions | Different fractions can name the same amount. |
| Simplifying Fractions | Reducing a fraction to lowest terms. |
| Adding Fractions (Like Denominators) | Same-denominator addition. |
| Adding Fractions (Unlike Denominators) | Rewrite over a common denominator first. |
| Multiplying Fractions | Multiply straight across. |
| Dividing Fractions | Multiply by the reciprocal. |
| Fractions ↔ Decimals | Converting between the two notations. |
| Percent of a Number | Percent means per hundred. |
| Percent Increase & Decrease | Applying a percent change to a quantity. |
| Ratios & Proportions | Two quantities that scale together. |
| Evaluating Expressions | Substituting a value for a variable. |
| Combining Like Terms | Adding the coefficients of matching variable parts. |
| The Distributive Property | Multiplying across a sum. |
| One-Step Equations | Undoing a single operation. |
| Two-Step Equations | Undo addition/subtraction first, then multiplication. |
| Linear Inequalities | Solving with <, >, ≤, ≥. |
| The Coordinate Plane | Locating points with (x, y) pairs. |
| Slope of a Line | Rise over run between two points. |
| Slope-Intercept Form | y = mx + b describes a whole line. |
| Adding & Subtracting Polynomials | Combining polynomials by collecting like terms. |
| Multiplying Binomials (FOIL) | Expanding products of binomials. |
| Factoring Out the GCF | Undoing the distributive property. |
| Factoring Trinomials | Reversing FOIL: finding two numbers that multiply to c and add to b. |
| Special Factoring Patterns | Difference of squares and perfect-square trinomials. |
| Solving Quadratics by Factoring | Zero-product property: if a·b = 0 then a = 0 or b = 0. |
| Solving x² = k | Taking square roots of both sides — remembering ±. |
| Completing the Square | Turning any quadratic into a perfect square plus a constant. |
| The Quadratic Formula | x = (−b ± √(b² − 4ac)) / 2a solves any quadratic. |
| Product Rule for Exponents | Multiplying powers of the same base adds the exponents. |
| Quotient & Power Rules | Dividing powers subtracts exponents; a power of a power multiplies them. |
| Zero & Negative Exponents | Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal. |
| Simplifying Radicals | Pulling perfect-square factors out of a square root. |
| Rational Exponents | Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p. |
| Exponential Growth & Decay | Quantities that multiply by the same factor each time step: y = a·bᵗ. |
| Angle Relationships | Vertical, complementary, and supplementary angle pairs. |
| Triangle Angle Sum | The three angles of a triangle always add to 180°. |
| The Pythagorean Theorem | In a right triangle, a² + b² = c². |
| Distance & Midpoint | Measuring segments in the coordinate plane. |
| Similar Triangles | Same shape, different size: corresponding sides are proportional. |
| Function Notation & Evaluation | Reading f(x) notation and plugging in inputs. |
| Function Composition | Feeding one function's output into another: f(g(x)). |
| Piecewise Functions | Functions defined by different rules on different intervals. |
| Complex Numbers | The imaginary unit i = √(−1) and numbers of the form a + bi. |
| Operations with Complex Numbers | Multiplying complex numbers with FOIL and i² = −1. |
| Quadratics with Complex Roots | When the discriminant is negative, the roots come in a conjugate pair a ± bi. |
| Polynomial Division | Dividing a polynomial by (x − a) with long or synthetic division. |
| Remainder & Factor Theorems | The remainder when p(x) is divided by (x − a) is simply p(a). |
| Zeros of Polynomials | Finding all the roots of a cubic by factoring it down. |
| End Behavior of Polynomials | Far from the origin, only the leading term matters. |
| Simplifying Rational Expressions | Factor top and bottom, then cancel the common factor. |
| Operations on Rational Expressions | Multiplying and dividing algebraic fractions. |
| Logarithms | log_b(x) asks: to what power must b be raised to get x? |
| Properties of Logarithms | Logs turn products into sums, quotients into differences, powers into multiples. |
| Arithmetic Sequences | Sequences that grow by a constant difference each step. |
| Geometric Sequences | Sequences that grow by a constant ratio each step. |
| Right-Triangle Trigonometry | SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle. |
| Asymptotes of Rational Functions | Where rational functions blow up and where they level off. |
| Vectors: Components & Magnitude | A vector is a displacement: components ⟨Δx, Δy⟩ and a length. |
| Vector Operations | Scaling, adding, and dotting vectors — all component by component. |
| Sigma Notation & Series | Σ compresses a sum: read the limits, add up the terms. |
| Average Rate of Change | The slope of the secant line: (f(b) − f(a)) / (b − a). |
| Limits: Graphical & Numerical | What value a function approaches — which need not be the value it takes. |
| Evaluating Limits Algebraically | Direct substitution — and the factor-and-cancel fix for 0/0. |
| One-Sided Limits | Approaching from the left or right — and when the two disagree. |
| Infinite Limits & Vertical Asymptotes | Where a function blows up: reading the sign of an infinite limit. |
| Limits at Infinity | End behavior of rational functions: compare the degrees. |
| The Limit Definition of the Derivative | The derivative is the limit of average rates of change. |
| Derivatives Graphically | Reading slopes off a graph. |
| The Power Rule | d/dx xⁿ = n·xⁿ⁻¹ for any real n. |
| Sum & Constant-Multiple Rules | Differentiate term by term. |
| The Product Rule | (fg)′ = f′g + fg′. |
| Derivatives of Exponentials & Logs | eˣ is its own derivative; (ln x)′ = 1/x. |
| The Chain Rule | d/dx f(g(x)) = f′(g(x)) · g′(x). |
| Implicit Differentiation | Differentiating equations that mix x and y. |
| Tangent Lines & Linear Approximation | The tangent line is the best local linear stand-in for f. |
| Antiderivatives | Reversing differentiation: the power rule backwards. |
| Riemann Sums | Approximating area with rectangles. |
| Properties of Definite Integrals | Linearity, additivity, and orientation. |
| The Fundamental Theorem: Evaluating Integrals | ∫ₐᵇ f = F(b) − F(a). |
| u-Substitution | The chain rule in reverse. |
| Partial Fractions (BC) | Splitting rational functions to integrate them. |
| Improper Integrals (BC) | Integrals to infinity, defined by limits. |
| Vectors in Rⁿ | Ordered lists of numbers, added and scaled componentwise. |
| Dot Product & Norm | Multiply matching components and add; lengths and angles follow. |
| Matrix Addition & Scalar Multiples | Matrices add entry by entry; scalars multiply every entry. |
| Matrix Multiplication | Row times column: each entry of AB is a dot product. |
| Determinants | A single number that measures how a matrix scales area or volume. |
| Eigenvalues of a 2×2 Matrix | The scaling factors along a matrix's special directions. |