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Calculus

Limits to Taylor series: the full mechanical toolkit.

Prerequisites: None (entry point)

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  1. 1LimitsWhat it means to approach a value without reaching it, and why the number e shows up in compound interest.
  2. 2DerivativesThe derivative as an instantaneous rate of change, built from the limit definition.
  3. 3Differentiation RulesProduct, quotient, and chain rules, plus log differentiation and the classic x^x problem.
  4. 4Integration FundamentalsAntiderivatives, the Fundamental Theorem of Calculus, and area under a curve.
  5. 5U-SubstitutionThe chain rule in reverse: spotting composite structure inside integrals.
  6. 6Integration by PartsThe product rule in reverse, the LIATE heuristic, and the tabular shortcut.
  7. 7Advanced Integration TechniquesPartial fractions, trig substitution, improper integrals, and a strategy decision tree.
  8. 8Partial DerivativesDifferentiating functions of several variables: partials, the gradient, and directional derivatives.
  9. 9Multiple IntegralsDouble and triple integrals, Fubini's theorem, and polar coordinates.
  10. 10Taylor Series & SequencesApproximating functions with polynomials, key expansions, and convergence tests.
  11. 11Optimization: Gradients, the Hessian and Gradient DescentFinding the best point of a function: set the gradient to zero, read the Hessian to tell a minimum from a maximum or a saddle, walk downhill with gradient descent when you can't solve by hand, and size a Kelly bet with the same tools.
  12. 12Constrained Optimization: Lagrange MultipliersOptimizing a function when you have to stay on a curve or inside a region: Lagrange multipliers, the multiplier as a shadow price, and the KKT conditions for inequality constraints.
  13. 13Differential EquationsEquations whose answer is a function: separable equations, the integrating factor, damping in brief, and how adding noise to an ODE gives an SDE.
  14. 14Numerical Methods: Newton, Bisection and Finite DifferencesRoot finding by bisection and Newton's method, solving for implied volatility, and finite-difference Greeks with the right step size.

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