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Calculus I · Full-sequence textbook draft

Calculus

From a changing quantity to a local rate, and from a rate back to accumulated change. Work through the mathematics, the proofs that justify it, and the assumptions each application needs.

Start the first chapter →Read or print the textbook →Fourteen-week study plan →

Editorial status: 21 original chapters form a first-pass Calculus I manuscript, with worked problems and solutions. Independent mathematical and classroom review remain pending. Calculus II and III remain planned courses.

Secants and tangents

Compare both signed approaches, including a continuous corner with no derivative.

Prerequisites

Algebra, elementary functions, graph interpretation, and trigonometry in radians. The first chapter includes diagnostic exercises.

Part 1

Limits and continuity

  1. 01Functions, Measurements, and ChangeDomains, units, average rates, and the algebra that an instantaneous rate will require. · 16 min read
  2. 02Limits and One-Sided BehaviorSeparate a function's value from its nearby behavior, and recognize when a limit fails. · 17 min read
  3. 03Computing Limits with ReasonsApply limit laws, factoring, conjugates, squeezing, and dominant terms without discarding their assumptions. · 20 min read
  4. 04The Epsilon–Delta DefinitionTurn arbitrarily close into a quantified claim, construct delta, and prove that a proposed limit fails. · 22 min read
  5. 05Continuity and ExistenceDiagnose discontinuities, apply the intermediate and extreme value theorems, and distinguish existence from uniqueness. · 18 min read
Part 2

Differentiation

  1. 06The Derivative at a PointDerive instantaneous rates and tangent lines, and prove why a corner can be continuous without being differentiable. · 19 min read
  2. 07The Derivative as a FunctionDerive power and reciprocal examples, track derivative domains, and distinguish a function from its rate function. · 16 min read
  3. 08Sum, Product, and Quotient RulesDerive the rules that make differentiation practical and learn to recognize the operation at the outside of an expression. · 18 min read
  4. 09Composition and Trigonometric DerivativesJustify the chain rule without dividing by a possibly zero increment, then derive sine and cosine derivatives. · 20 min read
  5. 10Exponentials, Logarithms, and InversesDifferentiate inverse functions and derive logarithmic, exponential, inverse-trigonometric, and real-power rules on their domains. · 20 min read
  6. 11Implicit Differentiation and Related RatesDifferentiate constraints, identify local branches, and build rate equations before substituting a particular instant. · 18 min read
Part 3

Consequences and applications of derivatives

  1. 12The Mean Value Theorem and Local ApproximationConnect interval change to instantaneous rates, prove monotonicity consequences, and bound linearization error. · 19 min read
  2. 13Extrema, Concavity, and Curve AnalysisUse sign charts and endpoint checks to classify behavior instead of treating every zero derivative as an extremum. · 17 min read
  3. 14Optimization as a Modeling ProblemTranslate a constraint into a feasible domain, then justify a global optimum with calculus and boundary analysis. · 18 min read
  4. 15Indeterminate Forms and Newton's MethodUse derivative-based limit and root tools while keeping their hypotheses, failure modes, and approximation status visible. · 18 min read
Part 4

Integration and accumulation

  1. 16Antiderivatives and Initial ValuesReverse differentiation on intervals, retain constants, and connect acceleration, velocity, and position. · 16 min read
  2. 17Accumulation and the Definite IntegralDefine accumulation through tagged sums, calculate a sum limit exactly, and state what integrability requires. · 21 min read
  3. 18The Fundamental Theorem of CalculusProve that continuous accumulation has the original rate, and derive evaluation by antiderivatives. · 21 min read
  4. 19Substitution and Integral StructureReverse the chain rule, transform bounds correctly, and recognize when a proposed substitution has not simplified the integral. · 17 min read
  5. 20Area, Volume, Work, and Average ValueBuild integral models from thin pieces with correct geometry, signs, and units. · 21 min read
  6. 21A Calculus Investigation from Rates to TotalsComplete a connected model, defend its assumptions, and review proof, computation, and interpretation across Calculus I. · 20 min read