Calculus

Theory part 1Practice part 2

average velocity, instantaneous velocity as a limiting idea, secant line, tangent line, difference quotient, slope, normal line and vertical tangents

Also uses Functions and coordinates and elementary algebra.

Theory part 3Practice part 4

approaching a value, nearby inputs, limit notation, function value versus limiting value, graphs and numerical tables, oscillation and existence and nonexistence

Also uses Functions, algebra, coordinates and basic sine notation.

Theory part 5Practice part 6

left-hand limit, right-hand limit, domain-side approach and one-sided behaviour of piecewise functions

Builds on Intuitive Definition of Limit.

Theory part 7Practice part 8

two-sided limit, agreement of finite one-sided limits, uniqueness, punctured neighbourhood and approach restricted to the domain

Builds on Intuitive Definition of Limit and One-Sided Limits.

Theory part 9Practice part 10

epsilon-delta definition, every epsilon before there exists delta, 0<|x-a|<delta, all nearby domain points, accumulation point, choosing delta and negating the definition

Builds on Intuitive Definition of Limit. Also uses quantifiers and inequalities.

Theory part 11Practice part 12

one-sided epsilon-delta definitions, 0<x-a<delta, 0<a-x<delta, domain endpoints and proofs and counterexamples

Builds on One-Sided Limits and The Formal Definition of a Limit.

Theory part 13Practice part 14

sum, difference, constant multiple, product, quotient with nonzero limiting denominator, powers, roots and real-domain conditions and proof of a limit law

Builds on Intuitive Definition of Limit. Also uses algebra.

Theory part 15Practice part 16

direct substitution for polynomials and rational functions, squeeze theorem, order and comparison, local boundedness, algebraic cancellation and rationalisation

Builds on Limit Laws. Also uses inequalities and factoring.

Theory part 17Practice part 18

sin(x)/x as x tends to zero in radians, (1-cos(x))/x, exponential and logarithmic limits near zero, geometric proof, algebraic transformations and noncircular derivations

Builds on Limit Laws and Limit Theorems. Also uses radians and trigonometry.

Theory part 19Practice part 20

x tending to positive or negative infinity, epsilon-M definition, finite limits at infinity, dominant terms, polynomial and rational growth and (1+1/x)^x and the number e

Builds on Intuitive Definition of Limit. Also uses rational functions.

Theory part 21Practice part 22

unbounded one-sided behaviour, positive and negative infinity, infinite limits at finite points and at infinity, sign analysis and infinity versus a real value

Builds on One-Sided Limits. Also uses inequalities.

Theory part 23Practice part 24

M-delta definition at a finite point, M-N definition at infinity, one-sided variants, positive and negative unboundedness and quantifier order

Builds on The Formal Definition of a Limit and Infinite Limits.

Theory part 25Practice part 26

vertical, horizontal, oblique asymptotes, rational functions, polynomial division, difference tending to zero, crossing an asymptote and one-sided conditions

Builds on Limits at Infinity and Infinite Limits. Also uses lines.

Theory part 27Practice part 28

continuity at a point, continuity relative to the domain, interior points, endpoints, left and right continuity and continuity on intervals

Builds on Intuitive Definition of Limit. Also uses functions and domains.

Theory part 29Practice part 30

removable, jump, infinite, oscillating discontinuities, piecewise functions, continuous extension and finite one-sided limits

Builds on One-Sided Limits and Continuous Functions.

Theory part 31Practice part 32

open and closed points, one-sided continuity, endpoints, graphical identification of discontinuities and limits of graph-based evidence

Builds on Continuous Functions.

Theory part 33Practice part 34

algebra of continuous functions, compositions, continuity of elementary functions on their domains, continuous images of intervals and max-min theorem on a closed bounded interval

Builds on Limit Laws and Continuous Functions.

Theory part 35Practice part 36

continuity on [a,b], values between endpoint outputs, existence of a root, no automatic uniqueness and counterexamples when continuity fails

Builds on Continuous Functions. Also uses intervals.

Theory part 37Practice part 38

pointwise versus uniform continuity, one delta for all points, continuous functions on closed bounded intervals, examples on unbounded and open intervals and connection to integrability

Builds on The Formal Definition of a Limit and Continuous Functions. Also uses quantifiers.

Theory part 39Practice part 40

derivative at a point, difference quotient limit, tangent slope, average and instantaneous rate, velocity and speed, units and normal line

Builds on Velocity and Tangent and Intuitive Definition of Limit. Also uses difference quotient.

Theory part 41Practice part 42

derivative function, Leibniz and prime notation, differentiation operator, differentiability on an interval, differentiability implies continuity, corners, cusps, vertical tangents and one-sided derivatives

Builds on Derivatives and Rates of Change. Also uses functions and domains.

Theory part 43Practice part 44

constant rule, power rule and real-domain restrictions, constant multiple, sum, difference, derivatives of polynomials and derivative of absolute value away from zero

Builds on Derivatives and Rates of Change. Also uses exponent laws.

Theory part 45Practice part 46

product rule, geometric explanation, proof from difference quotients and products of several functions

Builds on Derivatives and Rates of Change and Elementary Derivative Rules.

Theory part 47Practice part 48

quotient rule, reciprocal rule, nonzero denominator, rational functions and derivation from product rule

Builds on Derivatives and Rates of Change, Elementary Derivative Rules and The Product Rule.

Theory part 49Practice part 50

composition, inner and outer functions, derivative of a composite, nested compositions, rates with intermediate variables and differentiability assumptions

Builds on Derivatives and Rates of Change and Elementary Derivative Rules. Also uses composition of functions.

Theory part 51Practice part 52

sine, cosine, tangent, cotangent, secant, cosecant, radians, domains and derivation using trigonometric limits

Builds on Derivatives and Rates of Change and The Chain Rule. Also uses trigonometry and radians.

Theory part 53Practice part 54

e^x, a^x for a>0, natural logarithm, logarithm to a positive base other than one, ln|x| for x nonzero and derivative formulas and domains

Builds on Derivatives and Rates of Change and The Chain Rule. Also uses exponentials and logarithms.

Theory part 55Practice part 56

second and higher derivatives, notation, acceleration and jerk, recursive differentiation, higher derivatives of products and smoothness classes

Builds on The Derivative as a Function, Elementary Derivative Rules and The Chain Rule.

Theory part 57Practice part 58

implicit curves, differentiating an equation, local branches, dy/dx, higher implicit derivatives and points where the solved derivative formula fails

Builds on The Product Rule, The Quotient Rule and The Chain Rule.

Theory part 59Practice part 60

inverse-function derivative, nonzero derivative condition, local invertibility, inverse trigonometric derivatives, branch ranges and endpoint restrictions

Builds on The Chain Rule. Also uses inverse functions and trigonometry.

Theory part 61Practice part 62

products, quotients, variable exponents, logarithmic differentiation, positive-base assumptions and absolute values for nonzero real factors

Builds on The Chain Rule, Derivatives of Exponential and Logarithmic Functions and Implicit Differentiation.

Theory part 63Practice part 64

sinh, cosh, tanh, reciprocal hyperbolic functions, exponential definitions, identities, catenary, inverse hyperbolic functions and derivative formulas and real domains

Builds on Derivatives of Exponential and Logarithmic Functions and Derivatives of Inverse Functions. Also uses hyperbolic functions.

Theory part 65Practice part 66

global and local maxima and minima, extreme values, critical points, stationary and singular points, domain points, endpoints and extrema on open intervals

Builds on Derivatives and Rates of Change. Also uses intervals and extrema.

Theory part 67Practice part 68

extreme value theorem, continuity on a compact interval, Fermat’s theorem at differentiable interior extrema, closed interval method and endpoints and nondifferentiable candidates

Builds on Continuous Functions and Extreme Values.

Theory part 69Practice part 70

Rolle’s theorem, mean value theorem, continuity on [a,b] and differentiability on (a,b), Cauchy’s generalised mean value theorem, corollaries and constant functions

Builds on Continuous Functions and Derivatives and Rates of Change. Also uses Rolle theorem.

Theory part 71Practice part 72

strict and nonstrict monotonicity, derivative sign tests, first derivative test for extrema, isolated zero derivatives and interval assumptions

Builds on Derivatives and Rates of Change. Also uses derivative sign.

Theory part 73Practice part 74

concave up and down, second derivative sign, inflection points require a change of concavity, second derivative test and inconclusive cases

Builds on Higher-Order Derivatives. Also uses derivative sign.

Theory part 75Practice part 76

domain, intercepts, symmetry, asymptotes, monotonic intervals, critical points, concavity, inflection points and coherent curve sketch

Builds on Asymptotes, Extreme Values, Increasing and Decreasing Functions and Concavity and Inflection Points.

Theory part 77Practice part 78

time-dependent quantities, geometric constraints, differentiating before substituting, unit consistency, sign conventions and interpreting a rate

Builds on The Chain Rule and Implicit Differentiation. Also uses geometry and rates.

Theory part 79Practice part 80

objective function, feasible domain, constraint reduction, local versus global optimum, endpoints, unbounded domains and model validation

Builds on Elementary Derivative Rules and The Chain Rule. Also uses extrema and closed intervals.

Theory part 81Practice part 82

rates in physics, chemistry, biology, economics, marginal cost and revenue, elasticity and interpretation of units and modelling assumptions

Builds on Derivatives and Rates of Change. Also uses units and modeling.

Theory part 83Practice part 84

constant relative growth rate, law of natural growth and decay, population, radioactive decay and half-life, Newton’s law of cooling, continuously compounded interest and checking a model by differentiation

Builds on Derivatives of Exponential and Logarithmic Functions. Also uses elementary separable equations.

Theory part 85Practice part 86

0/0 and infinity/infinity, local differentiability and nonzero denominator derivative, one-sided and infinite limits, indeterminate products, differences, powers, repeated use and inconclusive cases

Builds on Intuitive Definition of Limit, Elementary Derivative Rules and The Chain Rule. Also uses theorem hypotheses.

Theory part 87Practice part 88

linearisation, differential versus actual change, tangent approximation, error tending to zero faster than the increment, Big-O and little-o notation and mean value error bounds

Builds on Derivatives and Rates of Change and Elementary Derivative Rules. Also uses linear functions.

Theory part 89Practice part 90

antiderivative, family of antiderivatives, indefinite integral, constant of integration on an interval, antiderivative theorems, initial conditions and verification by differentiation

Builds on Derivatives and Rates of Change and Elementary Derivative Rules. Also uses reverse differentiation.

Theory part 91Practice part 92

constant, constant multiple, sum, difference, power rule except exponent -1, logarithmic rule, exponential, trigonometric and inverse trigonometric rules and domains

Builds on Introduction to Indefinite Integrals. Also uses elementary derivative formulas.

Theory part 93Practice part 94

sigma notation, index, summand, limits of summation, reindexing, sum rules, sums of powers and finite geometric sums

Also uses Algebra and sequences and finite sums.

Theory part 95Practice part 96

area by polygons and rectangles, lower and upper sums, area approximation, refinement, limiting area and nonnegative regions

Builds on Intuitive Definition of Limit and Sums and Sigma Notation. Also uses area.

Theory part 97Practice part 98

partition, mesh or norm, tagged partition, Riemann sum, lower and upper Darboux sums, refinement and arbitrary sample points

Builds on Intuitive Definition of Limit, Sums and Sigma Notation and The Area Problem.

Theory part 99Practice part 100

Riemann integral as a common limit of tagged sums as mesh tends to zero, integrability, integrand, bounds, dummy variable, differential notation, signed net area, continuity implies integrability and midpoint sums

Builds on Partitions and Riemann Sums. Also uses signed area.

Theory part 101Practice part 102

linearity, interval additivity, reversed bounds, comparison, absolute-value inequality, symmetry, piecewise continuous integrands and mean value theorem for integrals under continuity

Builds on Introduction to Definite Integrals.

Theory part 103Practice part 104

accumulation function, derivative of an integral with variable upper bound, continuity at the evaluation point, signed area and variable bounds and chain rule

Builds on Derivatives and Rates of Change and Introduction to Definite Integrals. Also uses accumulation.

Theory part 105Practice part 106

evaluation using an antiderivative, F(b)-F(a), continuity and antiderivative hypotheses, evaluation notation, net change theorem and total distance versus displacement

Builds on Introduction to Indefinite Integrals, Introduction to Definite Integrals and Accumulation Functions and the Fundamental Theorem, Part 1.

Theory part 107Practice part 108

reverse chain rule, substitution for indefinite integrals, definite-integral bounds, change of variable, back-substitution and interval and regularity conditions

Builds on The Chain Rule, Introduction to Indefinite Integrals and Integration Rules for Indefinite Integrals.

Theory part 109Practice part 110

product-rule derivation, definite and indefinite formulas, boundary terms, repeated integration by parts and reduction formulas

Builds on The Product Rule, Introduction to Indefinite Integrals and Integration Rules for Indefinite Integrals.

Theory part 111Practice part 112

proper and improper rational functions, polynomial division, partial fraction decomposition, repeated linear factors, irreducible quadratic factors, coefficient matching and real-domain intervals

Builds on Integration Rules for Indefinite Integrals. Also uses partial fractions and logarithms.

Theory part 113Practice part 114

powers of sine and cosine, powers of tangent and secant, product-to-sum identities, parity strategies and trigonometric identities

Builds on Derivatives of Trigonometric Functions, Integration Rules for Indefinite Integrals and The Substitution Method. Also uses trig identities.

Theory part 115Practice part 116

sine substitution, tangent substitution, secant substitution, inverse trigonometric forms, triangles, branch restrictions and absolute values in square roots

Builds on The Substitution Method. Also uses inverse trigonometry and radicals.

Theory part 117Practice part 118

hyperbolic substitutions, inverse hyperbolic forms, tangent-half-angle or Weierstrass substitution, rationalising trigonometric expressions and substitution intervals

Builds on Derivatives of Hyperbolic Functions and The Substitution Method. Also uses trig identities.

Theory part 119Practice part 120

undetermined coefficients in antiderivatives, differential binomials and elementary-integrability criteria for rational exponents, reduction strategies, tables of integrals, computer algebra, non-elementary antiderivatives and differentiation check

Builds on The Substitution Method, Integration by Parts, Integration of Rational Functions, Trigonometric Integrals and Trigonometric and Inverse Substitutions.

Theory part 121Practice part 122

limit definition at infinity, separate tails on the real line, convergence and divergence, p-integrals and principal value distinguished from an improper integral

Builds on Limits at Infinity, Introduction to Definite Integrals and The Fundamental Theorem, Part 2, and Net Change.

Theory part 123Practice part 124

unbounded integrands, endpoints and interior singularities, separate limits, direct and limit comparison for nonnegative integrands and absolute and conditional convergence

Builds on Infinite Limits and Introduction to Definite Integrals. Also uses comparisons.

Theory part 125Practice part 126

geometric area versus signed integral, intersections, vertical and horizontal slices, splitting at sign changes and area between a curve and an axis

Builds on Introduction to Definite Integrals and The Fundamental Theorem, Part 2, and Net Change. Also uses intersections.

Theory part 127Practice part 128

integral average over an interval, average versus midpoint value, continuous-function mean value theorem and weighted averages

Builds on Introduction to Definite Integrals and The Fundamental Theorem, Part 2, and Net Change.

Theory part 129Practice part 130

volume as a limit of slices, prisms, oblique solids, cylinders, cross-sectional area, solids of revolution, disks and washers, ball, cone, pyramid and bounds and radii

Builds on Introduction to Definite Integrals and The Fundamental Theorem, Part 2, and Net Change. Also uses areas and volumes.