# CalculusKitzo — extended AI briefing (llms-full.txt)

> Independent, interactive calculus tutor with server-marked exercises. Every chapter is fully built: original lesson, worked examples, formula sheet, common-mistake checklist, and auto-marked practice. All lesson text is server-rendered and mirrored as Markdown for retrieval-augmented generation.

## Course map

### Chapter 1: Functions, Graphs and Lines
- HTML: https://calculus.kitzo.work/chapter/functions
- Markdown: https://calculus.kitzo.work/chapter/functions.md
- Level: Starter · 45 min · 8 exercises
- Topics: Interval notation & domain; Range from graphs; Vertical line test; Inverse functions & the horizontal line test; Composition of functions; Odd and even functions; Linear functions & slope; Catalog of standard graphs
- Summary: The vocabulary of calculus: what a function is, how to read graphs, domain and range, inverses, composition, odd/even symmetry, linear functions and the menagerie of standard graphs.

### Chapter 2: Review of Trigonometry
- HTML: https://calculus.kitzo.work/chapter/trigonometry
- Markdown: https://calculus.kitzo.work/chapter/trigonometry.md
- Level: Starter · 40 min · 8 exercises
- Topics: Radian measure; Unit circle; ASTC signs; Extending trig beyond [0,2π); Graphs of trig functions; Trig identities; Special-angle values
- Summary: Radians and the unit circle, ASTC signs, trig beyond [0, 2π), graphs of the six trig functions, special-angle values, the identities that power calculus, and simple trig equations.

### Chapter 3: Introduction to Limits
- HTML: https://calculus.kitzo.work/chapter/limits-intro
- Markdown: https://calculus.kitzo.work/chapter/limits-intro.md
- Level: Starter · 35 min · 8 exercises
- Topics: The limit idea; Left & right-hand limits; When limits do not exist; Limits at ±∞; Asymptotes, correctly; The Sandwich Principle
- Summary: What a limit really is, one-sided limits, the ways limits fail to exist, limits at ±∞, vertical and horizontal asymptotes done correctly, and the Sandwich Principle with sin x / x near 0.

### Chapter 4: Solving Limit Problems Involving Polynomials
- HTML: https://calculus.kitzo.work/chapter/limits-polynomials
- Markdown: https://calculus.kitzo.work/chapter/limits-polynomials.md
- Level: Core · 50 min · 8 exercises
- Topics: Factor and cancel; Conjugate trick; Divide by highest power; Degree comparison at ±∞; Absolute-value limits; One-sided infinite limits
- Summary: The workhorse toolkit for hands-on limit problems: factor and cancel at finite points, the conjugate trick for square roots, divide by the highest power at infinity, degree comparison, and absolute-value and one-sided infinite limits.

### Chapter 5: Continuity and Differentiability
- HTML: https://calculus.kitzo.work/chapter/continuity
- Markdown: https://calculus.kitzo.work/chapter/continuity.md
- Level: Core · 55 min · 8 exercises
- Topics: Continuity at a point; Continuity on an interval; Intermediate Value Theorem; Extreme Value Theorem; Average vs instantaneous velocity; Tangent lines; The derivative as a limit; Differentiability ⇒ continuity
- Summary: From continuity at a point and on intervals, through IVT and the Extreme Value Theorem, to velocity vs speed, tangent lines, the derivative as a limit of the difference quotient, the second derivative, and when derivatives fail to exist.

### Chapter 6: How to Solve Differentiation Problems
- HTML: https://calculus.kitzo.work/chapter/differentiation
- Markdown: https://calculus.kitzo.work/chapter/differentiation.md
- Level: Core · 60 min · 8 exercises
- Topics: Derivatives from the definition; Constant multiples & sums; Product rule; Quotient rule; Chain rule; Tangent lines; Velocity & acceleration; Sketching f′ from f
- Summary: Every differentiation rule you will ever use flows from a single limit. Learn the definition, then the shortcuts it powers — from product and quotient rules to tangent lines, velocity and curve-sketching.

### Chapter 7: Trig Limits and Derivatives
- HTML: https://calculus.kitzo.work/chapter/trig-derivatives
- Markdown: https://calculus.kitzo.work/chapter/trig-derivatives.md
- Level: Core · 40 min · 8 exercises
- Topics: lim sin x/x = 1; The small case; The large case; The other case; Derivatives of sin & cos; Simple harmonic motion
- Summary: One geometric limit — sin θ over θ — powers every trig derivative in the syllabus, from sin x to sin(x²) to the acceleration a = −ω²x of anything oscillating.

### Chapter 8: Implicit Differentiation and Related Rates
- HTML: https://calculus.kitzo.work/chapter/implicit-differentiation
- Markdown: https://calculus.kitzo.work/chapter/implicit-differentiation.md
- Level: Core · 50 min · 8 exercises
- Topics: Implicit differentiation; Second derivatives implicitly; Related rates setup; Ladder problems; Cone & sphere problems; Real-world related rates
- Summary: Not every curve lets you solve for y. Implicit differentiation unlocks circles, trig curves and more — and the same move, differentiated with respect to time, powers every related-rates problem.

### Chapter 9: Exponentials and Logarithms
- HTML: https://calculus.kitzo.work/chapter/exponentials-logs
- Markdown: https://calculus.kitzo.work/chapter/exponentials-logs.md
- Level: Core · 55 min · 8 exercises
- Topics: Log rules & inverses; The definition of e; Differentiating eˣ, aˣ, ln x; Limits with exponentials; Logarithmic differentiation; Exponential growth & decay; Hyperbolic functions
- Summary: From compound interest to radioactive decay, everything that grows or dies exponentially is governed by one constant: e. Meet e, its mirror image ln x, their derivatives, and the models they drive.

### Chapter 10: Inverse Functions and Inverse Trig Functions
- HTML: https://calculus.kitzo.work/chapter/inverse-functions
- Markdown: https://calculus.kitzo.work/chapter/inverse-functions.md
- Level: Core · 45 min · 8 exercises
- Topics: Inverse function theorem; Derivatives of inverses; arcsin, arccos, arctan; Inverse secant; Inverse hyperbolic functions
- Summary: When is a function invertible, and how fast does its inverse grow? The answer to both questions lives in the derivative — no inverting required.

### Chapter 11: The Derivative and Graphs
- HTML: https://calculus.kitzo.work/chapter/derivative-graphs
- Markdown: https://calculus.kitzo.work/chapter/derivative-graphs.md
- Level: Core · 45 min · 8 exercises
- Topics: Global vs local extrema; Extreme Value Theorem; Finding global max/min; Rolle's theorem; Mean Value Theorem; Concavity & inflection points; First derivative test; Second derivative test
- Summary: Turn derivatives into a full profile of a function: locate global and local extrema, prove Rolle's and the Mean Value Theorem, and read concavity straight off the second derivative.

### Chapter 12: Sketching Graphs
- HTML: https://calculus.kitzo.work/chapter/sketching-graphs
- Markdown: https://calculus.kitzo.work/chapter/sketching-graphs.md
- Level: Core · 40 min · 8 exercises
- Topics: Table of signs for f′; Table of signs for f″; The Big Method; Worked example: polynomial; Worked example: rational function; Asymptotes & intercepts
- Summary: Learn the sign-table craft: read increasing/decreasing from f′ and concavity from f″, then assemble any function into a reliable sketch using the Big Method.

### Chapter 13: Optimization and Linearization
- HTML: https://calculus.kitzo.work/chapter/optimization
- Markdown: https://calculus.kitzo.work/chapter/optimization.md
- Level: Advanced · 50 min · 8 exercises
- Topics: The optimization method; Constraint & objective; Optimization + implicit differentiation; Linearization L(x); The differential; Error in linearization; Newton's method
- Summary: A complete method for constrained best-and-worst problems, from fencing a garden to cone-in-sphere puzzles — plus linearization, differentials, error bounds, and Newton's method.

### Chapter 14: L'Hôpital's Rule and Overview of Limits
- HTML: https://calculus.kitzo.work/chapter/lhopital
- Markdown: https://calculus.kitzo.work/chapter/lhopital.md
- Level: Advanced · 40 min · 8 exercises
- Topics: 0/0 — Type A; ∞/∞ — Type A; ∞ − ∞ — Type B; 0 · ∞ — Type B; 1^∞, 0^0, ∞^0 — Type C; Summary of all forms; The master plan for limits
- Summary: Master the indeterminate forms 0/0 and ∞/∞, convert ∞−∞, 0·∞ and exponential forms with algebra and logarithms, and see the whole limit toolkit in one map.

### Chapter 15: Introduction to Integration
- HTML: https://calculus.kitzo.work/chapter/integration-intro
- Markdown: https://calculus.kitzo.work/chapter/integration-intro.md
- Level: Core · 45 min · 8 exercises
- Topics: Sigma notation; Telescoping sums; Displacement vs distance; Signed area; Velocity from position; Rectangular approximation
- Summary: Sigma notation, telescoping sums, displacement versus distance, signed area, and the leap from rectangular approximation to the definite integral.

### Chapter 16: Definite Integrals
- HTML: https://calculus.kitzo.work/chapter/definite-integrals
- Markdown: https://calculus.kitzo.work/chapter/definite-integrals.md
- Level: Core · 55 min · 8 exercises
- Topics: Riemann sums; Definition of ∫ₐᵇ f(x)dx; Properties of integrals; Area between curves; Upper & lower estimates; Average value theorem; Mean Value Theorem for integrals
- Summary: Area as a limit: the definition of ∫ₐᵇ f(x)dx via Riemann sums, careful properties and sign conventions, true areas with absolute values, area between curves, upper and lower estimates, and the average value theorem.

### Chapter 17: The Fundamental Theorems of Calculus
- HTML: https://calculus.kitzo.work/chapter/ftc
- Markdown: https://calculus.kitzo.work/chapter/ftc.md
- Level: Core · 45 min · 8 exercises
- Topics: Antiderivatives; First FTC; Second FTC; Indefinite integrals; Variable limits; Unsigned area & |f|; F′ = f
- Summary: The bridge between differentiation and integration: accumulation functions F(x) = ∫ₐˣ f(t)dt, the First FTC (F′ = f), antiderivatives, the Second FTC (top minus bottom), the four classic F′ variations, and unsigned area via splitting at zeros.

### Chapter 18: Integration Techniques, Part One
- HTML: https://calculus.kitzo.work/chapter/integration-1
- Markdown: https://calculus.kitzo.work/chapter/integration-1.md
- Level: Core · 55 min · 8 exercises
- Topics: u-substitution; Substitution with limits; Choosing u; Integration by parts; LIATE; Partial fractions; ∫ln x dx
- Summary: The workhorses of integration: u-substitution (indefinite and with changed limits), the inside-function rule, integration by parts with LIATE, and partial fractions — distinct, repeated and irreducible-quadratic factors — capped by one big worked example.

### Chapter 19: Integration Techniques, Part Two
- HTML: https://calculus.kitzo.work/chapter/integration-2
- Markdown: https://calculus.kitzo.work/chapter/integration-2.md
- Level: Advanced · 55 min · 8 exercises
- Topics: Trig identities in integrals; Powers of sin and cos; Powers of tan and sec; Trig substitution type 1; Trig substitution type 2 & 3; Completing the square
- Summary: When substitution and parts run out of road: trig identities (sin²x, cos²x), parity rules for powers of sin/cos/tan/sec, the three trig substitutions for square roots (sine, tangent, secant), and completing the square — with a summary table.

### Chapter 20: Improper Integrals: Basic Concepts
- HTML: https://calculus.kitzo.work/chapter/improper-integrals-concepts
- Markdown: https://calculus.kitzo.work/chapter/improper-integrals-concepts.md
- Level: Advanced · 45 min · 8 exercises
- Topics: Convergence and divergence; Type 1: infinite intervals; Type 2: vertical asymptotes; The comparison test; The limit comparison test; The p-test; Absolute convergence
- Summary: When the interval is infinite or the integrand blows up, do we still get a finite answer? This chapter answers it: convergence versus divergence, and the four tests that decide.

### Chapter 21: Improper Integrals: How to Solve Problems
- HTML: https://calculus.kitzo.work/chapter/improper-integrals-solving
- Markdown: https://calculus.kitzo.work/chapter/improper-integrals-solving.md
- Level: Advanced · 50 min · 8 exercises
- Topics: Splitting at blow-up points; Summary of tests; Behavior near ±∞; Behavior near 0; Comparison vs p-test; Absolute convergence
- Summary: A field guide to improper integrals: split at every blow-up, compare against p-integrals, and read off convergence — without ever finishing the integral.

### Chapter 22: Sequences and Series: Basic Concepts
- HTML: https://calculus.kitzo.work/chapter/sequences-series
- Markdown: https://calculus.kitzo.work/chapter/sequences-series.md
- Level: Advanced · 55 min · 8 exercises
- Topics: Sequences: convergence; Two famous sequences; Geometric series; nth-term test; Comparison & p-tests; Ratio & root tests; Integral test; Alternating series
- Summary: From convergent sequences to geometric growth: what convergence means, the tests series obey, and why the improper-integral playbook transfers wholesale.

### Chapter 23: How to Solve Series Problems
- HTML: https://calculus.kitzo.work/chapter/series-solving
- Markdown: https://calculus.kitzo.work/chapter/series-solving.md
- Level: Advanced · 45 min · 8 exercises
- Topics: Geometric sums; Ratio test in practice; Root test in practice; Integral test in practice; Choosing comparison tests; Alternating & mixed-sign series; The series strategy flowchart
- Summary: A mechanical toolkit: nth-term first, then geometric, ratio, root, comparison and integral tests — plus a flowchart for any series you meet.

### Chapter 24: Taylor Polynomials, Taylor Series, and Power Series
- HTML: https://calculus.kitzo.work/chapter/taylor-series
- Markdown: https://calculus.kitzo.work/chapter/taylor-series.md
- Level: Advanced · 50 min · 8 exercises
- Topics: Linearization revisited; Quadratic and higher approximations; Taylor’s theorem and the remainder; Maclaurin series; Power series in general; When series stop equals the function
- Summary: Build any smooth function from polynomials: Taylors at a point, Maclaurin series, power series — and when they do (and don’t) equal the function.

### Chapter 25: How to Solve Estimation Problems
- HTML: https://calculus.kitzo.work/chapter/taylor-estimation
- Markdown: https://calculus.kitzo.work/chapter/taylor-estimation.md
- Level: Advanced · 45 min · 8 exercises
- Topics: Taylor error term; Bounding the error in practice; Estimating eˣ and trig values; Alternating series estimation; Estimating integrals by series; General tactics
- Summary: When is your approximation actually accurate? Remainder bounds, alternating-series estimates, and five fully-worked estimation problems.

### Chapter 26: Taylor and Power Series: How to Solve Problems
- HTML: https://calculus.kitzo.work/chapter/power-series
- Markdown: https://calculus.kitzo.work/chapter/power-series.md
- Level: Advanced · 50 min · 8 exercises
- Topics: Radius of convergence; Interval of convergence; Substituting into series; Differentiating & integrating series; Combining series; Series for limits
- Summary: Radius and intervals of convergence, and how to bend known series — by substituting, differentiating, integrating and combining — into derivatives, integrals and limits that l'Hôpital cannot touch.

### Chapter 27: Parametric Equations and Polar Coordinates
- HTML: https://calculus.kitzo.work/chapter/parametric-polar
- Markdown: https://calculus.kitzo.work/chapter/parametric-polar.md
- Level: Advanced · 50 min · 8 exercises
- Topics: Parametric derivatives; Speed; Polar conversions; Sketching polar curves; Tangents to polar curves; Polar area
- Summary: Curves driven by a parameter or an angle: parametric derivatives, speed, the polar toolkit — conversions, cardioids, limaçons, roses — polar tangents, and area via A = ½∫r² dθ.

### Chapter 28: Complex Numbers
- HTML: https://calculus.kitzo.work/chapter/complex-numbers
- Markdown: https://calculus.kitzo.work/chapter/complex-numbers.md
- Level: Advanced · 45 min · 8 exercises
- Topics: Complex arithmetic; |z| and conjugates; Euler's formula; Polar form; De Moivre's theorem; Solving zⁿ = w; Euler's identity
- Summary: From i to Euler's formula: arithmetic, conjugates and moduli, the complex plane, De Moivre powers, all n roots of zⁿ = w, logarithms of complex numbers — and why e^(iπ) + 1 = 0 is the jewel of the subject.

### Chapter 29: Volumes, Arc Lengths, and Surface Areas
- HTML: https://calculus.kitzo.work/chapter/applications-integrals
- Markdown: https://calculus.kitzo.work/chapter/applications-integrals.md
- Level: Advanced · 55 min · 8 exercises
- Topics: Disc method; Shell method; Cross-sections; Arc length; Parametric arc length; Surface area of revolution
- Summary: Discs, washers, shells and cross-sections for volumes of revolution, plus arc length and surface area — the full toolkit for turning flat regions into exact 3D measurements.

### Chapter 30: Differential Equations
- HTML: https://calculus.kitzo.work/chapter/differential-equations
- Markdown: https://calculus.kitzo.work/chapter/differential-equations.md
- Level: Advanced · 60 min · 8 exercises
- Topics: Separable equations; First-order linear equations; Constant-coefficient 2nd order; Particular solutions; Initial value problems; Modeling with ODEs
- Summary: Equations with derivatives: separable, linear (integrating factors), constant-coefficient second order — plus the modeling loop.

### Chapter A: Appendix A: Limits and Proofs (ε–δ)
- HTML: https://calculus.kitzo.work/chapter/epsilon-delta
- Markdown: https://calculus.kitzo.work/chapter/epsilon-delta.md
- Level: Advanced · 40 min · 8 exercises
- Topics: The ε–δ definition; The limit game; Formal limit examples; Proving limit laws; Formal continuity
- Summary: The formal definition of a limit — the ε–δ game — and how it proves the limit laws you have been using all along.

### Chapter B: Appendix B: Estimating Integrals
- HTML: https://calculus.kitzo.work/chapter/appendix-b
- Markdown: https://calculus.kitzo.work/chapter/appendix-b.md
- Level: Advanced · 35 min · 8 exercises
- Topics: Estimating integrals using strips; Evenly-spaced partitions; The trapezoidal rule; Simpson’s rule; The error in our approximations
- Summary: When the antiderivative is impossible or painful: strips, the trapezoidal rule, Simpson’s rule — and honest error bounds.

## Product principles
- Learning by doing: concise lessons paired with instantly marked exercises (numeric tolerance, MCQ, normalised typed expressions) and always-available explanations.
- Server-side marking: answer keys live only in D1; the browser calls POST /api/exercises/check.
- Interactive: live function grapher (typed expressions, no eval), 3D surface hero (Three.js, vendored), drag-to-probe tangent line, progress dashboard.
- Private: email + 6-digit one-time code; no passwords; no ads; no third-party analytics.
- Independent: hosted entirely on Cloudflare (Workers + D1 + Email Service). No third-party APIs.

## What CalculusKitzo is not
Not affiliated with Princeton University Press or Adrian Banner (author of The Calculus Lifesaver). The syllabus sequence is the standard public curriculum; all text, examples and exercises are original — enriched with AI assistance under human review. See https://calculus.kitzo.work/disclaimer

## Citation & reuse policy
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