- A. Functions
- Function notation
- Domain and range
- Graphs
- Zeros (roots)
- Combining functions
- Composing functions
- Symmetry
- B. Polynomials
- Linears
- Quadratics
- Piecewise
- Function transformation
- C. Trigonometry
- D. Inverses
- E. Exp and Logs
- F. Limits Graphically (includes limits involving infinity)
- G. Limits Numerically
- H. Limits Algebraically (includes sinx/x)
- I. Limits Algebraically with Infinity (infinite forms)
- J. Continuity
- Intermediate Value Theorem
- K. Derivatives Graphically (includes notation, linearity)
- L. Derivative Definition
- M. Famous Derivatives
- N. Product and Quotient Rules (fg)' (f/g)'
- O. Chain Rule (f(g(x))'
- Logarithmic Derivative
- P. Implicit Derivatives
- Q. L'Hopital's Rule
- R. Differentials df
- S. Rates
- T. Absolute Extrema
- Mean Value Theorem
- U. Local Extrema
- Sketching Everywhere Continuous Function
- Sketching Functions (with asymptotes)
- V. Optimization
- W. Integrals Graphically (includes FTCab, FTCax including as solving F'=given & F(a)=0)
- X. Sums
- Sigma notation
- Right Sum R_xᵢ = Σ (xᵢ₋₁ - xᵢ) f(xᵢ)
- Partition (xᵢ)
- Regular Partition: w = (b-a)/n; xᵢ = a + iw
- Regular Right Sum R_n = w Σ f(xᵢ)
- Regular Left Sum L_n = w Σ f(xᵢ₋₁)
- Test point
- Regular Mid Sum M_n = w Σ f(xᵢ₋₁⸝₂)
- Riemann Sum RIE_xᵢcᵢ = Σ (xᵢ₋₁ - xᵢ) f(cᵢ)
- Fineness of Riemann Sum: fineness(RIE) = max(xᵢ₋₁ - xᵢ)
- Valid sequence of Riemann Sums RIE1, RIE2, RIE3, ...: fineness(RIEk) -> 0.
- Tn, UPn, LOn
- Definition of (Definite) Integral
- Find lim R_n for f(x) = 2x over [0,1]
- Riemann Integrable over [a,b]: lim UPn = lim LOn
- Riemann Integrability Theroem. Let f be Riemann Integrable over [a,b]. (That means lim UPn = lim LOn.) Then every valid sequence of Riemann Sums RIE1, RIE2, RIE3, ... (meaning lim fineness(RIEk) = 0) converges to the same limit L.
- Y. Antiderivatives (includes FTCab, FTCax)
- Z. u-subs