Math Fundamentals
Formula-first lookup tables
1. Pre-Calculus
Functions & Transformations
| Concept | Formula |
|---|
| Function | |
| Composition | |
| Inverse | |
| Even | |
| Odd | |
| Vertical shift | |
| Horizontal shift | |
| Vertical scale | |
| Horizontal scale | |
Exponents & Radicals
| Rule | Formula |
|---|
| Product | |
| Quotient | |
| Power of power | |
| Negative exponent | |
| Zero exponent | |
| Rational exponent | |
| Square root | |
Logarithms
| Rule | Formula |
|---|
| Definition | |
| Product | |
| Quotient | |
| Power | |
| Change of base | |
| Inverse | |
Trigonometry
| Identity | Formula |
|---|
| Unit circle | |
| Pythagorean | |
| Tangent | |
| Angle sum (sin) | |
| Angle sum (cos) | |
| Double angle | |
| Euler | |
2. Algebra II
Polynomials
| Concept | Formula |
|---|
| Degree | highest power of |
| Quadratic | |
| Cubic | |
Factoring Identities
| Identity | Formula |
|---|
| Difference of squares | |
| Sum of cubes | |
| Difference of cubes | |
Quadratics
| Tool | Formula |
|---|
| Quadratic formula | |
| Discriminant | |
| Vertex | |
| Completing square | |
Polynomial Division
| Rule | Formula |
|---|
| Division algorithm | |
| Remainder theorem | remainder of |
| Factor theorem | |
| Partial fractions | |
Linear Systems (Intro Matrix View)
| Concept | Formula |
|---|
| Linear system | |
| 2×2 determinant | |
| 2×2 inverse | |
3. Calculus
Limits
| Concept | Formula |
|---|
| Limit | |
| One-sided | |
| ε–δ | |
Derivatives
| Function | Derivative |
|---|
| Constant | |
| |
| |
| |
| |
| |
Differentiation Rules
| Rule | Formula |
|---|
| Definition | |
| Product | |
| Quotient | |
| Chain | |
Integrals
| Concept | Formula |
|---|
| Indefinite | |
| Power rule | |
| Definite | |
| Substitution | |
Multivariable Calculus
| Concept | Formula |
|---|
| Partial | |
| Gradient | |
| Directional | |
| Jacobian | |
| Hessian | |
Taylor Approximation
4. Series & Convergence
Definitions
| Concept | Formula |
|---|
| Partial sum | |
| Convergence | |
| Necessary condition | |
Standard Series
| Series | Result |
|---|
| Geometric | converges iff |
| Harmonic | diverges |
| p-series | converges iff |
Comparison Tests
| Test | Criterion |
|---|
| Direct | , conv ⇒ conv |
| Limit | same behavior |
Ratio & Root Tests
| Test | Criterion |
|---|
| Ratio | |
| Root | |
| Result | conv, div |
Integral Test
| Condition | Result |
|---|
| , positive, decreasing | conv ⇔ conv |
Alternating Series
| Concept | Condition |
|---|
| Alternating test | decreasing , |
| Absolute convergence | conv |
| Conditional convergence | conv, div |
Power & Taylor Series
| Concept | Formula |
|---|
| Power series | |
| Radius | |
| Taylor | |
| Maclaurin | |