Calculus that
finally clicks.
30 interactive chapters from limits to differential equations — with live 3D visuals, a real function grapher, and exercises that mark themselves and explain why.
30 chapters, one beautiful sequence
Follows the classic path — limits → derivatives → integrals → series → ODEs. Every chapter is fully built with original lessons and 8 auto-marked exercises.
Functions, Graphs and Lines
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.
Review of Trigonometry
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.
Introduction to Limits
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.
Solving Limit Problems Involving Polynomials
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.
Continuity and Differentiability
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.
How to Solve Differentiation Problems
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.
Trig Limits and Derivatives
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.
Implicit Differentiation and Related Rates
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.
Exponentials and Logarithms
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.
Inverse Functions and Inverse Trig Functions
When is a function invertible, and how fast does its inverse grow? The answer to both questions lives in the derivative — no inverting required.
The Derivative and Graphs
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.
Sketching Graphs
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.
Optimization and Linearization
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.
L'Hôpital's Rule and Overview of Limits
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.
Introduction to Integration
Sigma notation, telescoping sums, displacement versus distance, signed area, and the leap from rectangular approximation to the definite integral.
Definite Integrals
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.
The Fundamental Theorems of Calculus
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.
Integration Techniques, Part One
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.
Integration Techniques, Part Two
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.
Improper Integrals: Basic Concepts
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.
Improper Integrals: How to Solve Problems
A field guide to improper integrals: split at every blow-up, compare against p-integrals, and read off convergence — without ever finishing the integral.
Sequences and Series: Basic Concepts
From convergent sequences to geometric growth: what convergence means, the tests series obey, and why the improper-integral playbook transfers wholesale.
How to Solve Series Problems
A mechanical toolkit: nth-term first, then geometric, ratio, root, comparison and integral tests — plus a flowchart for any series you meet.
Taylor Polynomials, Taylor Series, and Power Series
Build any smooth function from polynomials: Taylors at a point, Maclaurin series, power series — and when they do (and don’t) equal the function.
How to Solve Estimation Problems
When is your approximation actually accurate? Remainder bounds, alternating-series estimates, and five fully-worked estimation problems.
Taylor and Power Series: How to Solve Problems
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.
Parametric Equations and Polar Coordinates
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.
Complex Numbers
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.
Volumes, Arc Lengths, and Surface Areas
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.
Differential Equations
Equations with derivatives: separable, linear (integrating factors), constant-coefficient second order — plus the modeling loop.
Appendix A: Limits and Proofs (ε–δ)
The formal definition of a limit — the ε–δ game — and how it proves the limit laws you have been using all along.
Appendix B: Estimating Integrals
When the antiderivative is impossible or painful: strips, the trapezoidal rule, Simpson’s rule — and honest error bounds.
Built like the best tutor — multiplied
Auto-marked practice
Numeric answers with tolerance, MCQs, and typed expressions — checked instantly in your browser. Every miss comes with the why.
3D & interactive visuals
Drag a point along a curve, watch the tangent slope update live; see Riemann sums converge. Math you can touch.
Real progress tracking
Email + 6-digit code sign-in (no passwords). XP, per-chapter completion and best scores sync across devices.
The classic sequence
Follows the track made famous by The Calculus Lifesaver — 30 chapters in the exact order students actually learn them.
Private by design
100% on Cloudflare's edge. No ads, no analytics beacons, no data brokers. Your email is only ever used to send your code.
Actually free
Every chapter, widget and exercise is free. Supported by the site's modest hosting costs — and by people buying the book.
Frequently asked
Is CalculusKitzo free?
Yes — lessons, exercises and progress tracking are all free. You only need an email address to save your place.
Do I need to install anything or create a password?
No. Sign in with your email and a 6-digit code. Nothing to remember, nothing to install.
How are exercises marked?
In your browser, instantly: tolerance for numeric answers, exact match for multiple choice, and smart normalisation for typed expressions. Each question includes a hint and a full explanation.
Who is this for?
AP Calculus AB/BC, A-level/IB Maths, first-year university, and self-learners. If you've heard of the "calculus lifesaver" sequence of topics — limits, derivatives, integrals, series, ODEs — this is that, interactive.
Is this affiliated with The Calculus Lifesaver by Adrian Banner?
No. We follow the same classic chapter sequence (which is public knowledge), but every word and exercise here is original, and we are not endorsed by the author or Princeton University Press. The book is excellent — buy it.