Calculus Volume 1

Citation

Strang, G., & Herman, E. (2016). Calculus Volume 1. OpenStax. Free official edition: https://openstax.org/details/books/calculus-volume-1

Chapter Summary

Chapter 1: Functions and Graphs

The book opens with an earthquake question—how much stronger is a magnitude 9 quake than a magnitude 8.2?—that is answered later with logarithms. The chapter reviews the functions needed for calculus: notation, domain and range, graphs, zeros and composition; polynomial, rational, power and root functions and their transformations; trigonometric functions in radian measure; inverse functions, including inverse trigonometric functions; and exponential and logarithmic functions, with a first look at hyperbolic functions. It is presented as the foundation to master before calculus proper begins.

Chapter 2: Limits

A preview section explains how the tangent problem and the area problem led to differential and integral calculus, with instantaneous velocity as the limit of average velocity. The chapter then estimates limits from tables and graphs, treats one-sided and infinite limits, develops the limit laws and algebraic techniques (factoring, conjugates, the squeeze theorem), and defines continuity, classifying discontinuities and stating the Intermediate Value Theorem. The informal approach is followed by a final section on the precise epsilon–delta definition and how to prove limits with it.

Chapter 3: Derivatives

Building directly on limits, the derivative is defined as the slope of the tangent line and the limit of a difference quotient, then viewed as a function with higher derivatives and the link between differentiability and continuity. The chapter develops the differentiation rules (power, sum, product and quotient rules), interprets derivatives as rates of change in motion, population growth and marginal cost and revenue, and derives the derivatives of trigonometric functions, the chain rule, derivatives of inverse functions via the inverse function theorem, implicit differentiation, and derivatives of exponential and logarithmic functions including logarithmic differentiation.

Chapter 4: Applications of Derivatives

This chapter uses derivatives to solve problems: related rates, linear approximation and differentials (including measurement error), absolute and local extrema with critical points, and the Mean Value Theorem and its consequences for increasing and decreasing functions. It then covers first- and second-derivative tests and concavity, limits at infinity and asymptotes for curve sketching, applied optimisation, L'Hôpital's rule for indeterminate forms and comparing growth rates, Newton's method for approximating roots, and antiderivatives and initial value problems, which lead into integration.

Chapter 5: Integration

Starting from Archimedes' method of exhaustion, the chapter introduces sigma notation and Riemann sums for approximating area, then defines the definite integral as net signed area, with its properties and the average value of a function. The Fundamental Theorem of Calculus (both parts, with the Mean Value Theorem for integrals) connects integration with differentiation. The chapter closes with integration formulas and the net change theorem, substitution, and integrals involving exponential, logarithmic and inverse trigonometric functions.

Chapter 6: Applications of Integration

Definite integrals are applied to areas between curves, volumes by slicing (disks and washers) and by cylindrical shells, arc length and surface area, and physical quantities such as mass, work and hydrostatic force (the Hoover Dam example). Moments and centres of mass lead to the theorem of Pappus. The chapter then gives a rigorous treatment of the natural logarithm defined as an integral and the exponential function as its inverse, applies them to exponential growth and decay (doubling time and half-life), and ends with calculus of the hyperbolic functions.

Key Concepts

Chapter 1: Functions and Graphs

• Function: a rule assigning exactly one output to each input; the vertical line test checks this on a graph.
• Inverse function: exists when a function is one-to-one; its graph is the reflection of the original in the line y = x, and trigonometric functions need restricted domains to have inverses.
• Natural exponential and logarithm: eˣ and ln x, inverse functions used throughout calculus.

Chapter 2: Limits

• Limit: the value a function approaches as the input approaches a point, which may exist even when the function is undefined there.
• Continuity: a function is continuous at a point if it is defined there, its limit exists, and the two agree.
• Intermediate Value Theorem: a continuous function on a closed interval takes every value between its endpoint values.
• Epsilon–delta definition: the precise definition of a limit used to prove limit statements.

Chapter 3: Derivatives

• Derivative: the limit of the difference quotient, giving the slope of the tangent line and the instantaneous rate of change.
• Chain rule: the rule for differentiating compositions, (f∘g)′(x) = f′(g(x))g′(x).
• Implicit differentiation: differentiating both sides of an equation to find dy/dx when y is not given explicitly.

Chapter 4: Applications of Derivatives

• Critical point: a point where the derivative is zero or undefined, the only candidates for local extrema.
• Mean Value Theorem: a differentiable function has a point where the instantaneous rate of change equals the average rate over an interval.
• L'Hôpital's rule: evaluating limits of indeterminate forms 0/0 and ∞/∞ by differentiating numerator and denominator.
• Antiderivative: a function whose derivative is the given function, determined up to a constant.

Chapter 5: Integration

• Riemann sum: a sum of rectangle areas approximating the area under a curve.
• Definite integral: the limit of Riemann sums, measuring net signed area.
• Fundamental Theorem of Calculus: differentiation and integration are inverse processes, and definite integrals can be evaluated using antiderivatives.
• Substitution: reversing the chain rule to simplify integrals.

Chapter 6: Applications of Integration

• Disk, washer and shell methods: ways of computing volumes of solids of revolution by integrating cross-sections or shells.
• Work and hydrostatic force: physical quantities computed by integrating force over distance or pressure over area.
• Centre of mass: the balance point of a system, found from moments; Pappus's theorem relates it to volumes of revolution.
• Exponential growth and decay: models y = y₀e^{kt} with constant doubling time or half-life.

Critical Analysis

Strengths

• Clear structure with learning objectives. Each section begins with numbered learning objectives and ends with exercises; each chapter ends with key terms, key equations, key concepts and review exercises, which makes the book easy to navigate and to revise from.
• Worked examples with immediate practice. Examples are solved step by step and followed by "Checkpoint" questions whose answers are in the Answer Key, and answers to odd-numbered exercises are given at the back.
• Applications throughout. Chapter-opening problems (earthquake magnitudes, the Hennessey Venom GT's acceleration, a rocket launch camera, iceboats, the Hoover Dam) are solved later in each chapter, and student projects such as Formula One grandstands, a parachutist in free fall and the centre of mass of the Grand Canyon Skywalk extend the material.
• Flexible on transcendental functions and rigour. The preface explains that exponentials and logarithms appear informally in Chapter 1 and rigorously in Section 6.7, in separable sections, so courses can follow early- or late-transcendentals order; limits are first treated intuitively, with the epsilon–delta definition in its own section.

Limitations

• Single-variable only, and only the first part. Volume 1 stops after applications of integration; techniques of integration, differential equations and series are in Volume 2, and multivariable calculus is in Volume 3, so a full course sequence requires the other volumes.
• Proofs are present but not central. The preface mentions over 60 proofs of theorems, but many results (for example, the Intermediate Value Theorem and the existence of extrema) are stated and used rather than proved, so the book is not an analysis text.
• Long and dense. With nearly 800 pages including answers, and many similar exercises, students may need guidance on which problems to prioritise.
• Technology use is optional. Exercises marked [T] call for calculators or computer algebra systems, but the book does not teach a particular tool, so numerical or graphical work depends on the instructor's choice.

Applications in the Book

Applications span the natural and social sciences: comparing earthquake intensities with logarithms (Chapter 1), relativistic mass and limits (Chapter 2), velocity and acceleration, population growth and marginal cost and revenue (Chapter 3), related rates for a rocket launch, optimisation of revenue and surface area and Newton's method (Chapter 4), distance travelled by an iceboat and net change (Chapter 5), and in Chapter 6 volumes, arc length, mass, work, hydrostatic force on the Hoover Dam, centres of mass, exponential growth and decay, and the catenary shape described by hyperbolic functions. Student projects include the maximum value of a function, deriving the area of a circle, Formula One grandstands, iterative processes and chaos, a parachutist in free fall and the Grand Canyon Skywalk.

Where It Fits in the Course

In this Calculus and Multivariable Calculus list, Calculus Volume 1 sits alongside Silvanus Thompson's Calculus Made Easy, which is generally described as an informal, intuitive introduction; Thompson suits readers who want a gentle first look, while OpenStax provides the full standard course with exercises. Thomas' Calculus is generally described as a comprehensive standard textbook covering single- and multivariable calculus in one volume; it can replace the three OpenStax volumes or serve as an additional source of problems.

Robert Wrede's Advanced Calculus and Binmore and colleagues' Calculus: Concepts and Methods go beyond this volume. They suit students who have completed the single-variable material in this volume (and its sequel volumes) and want more rigour or more advanced topics.

How to Use This Book

The book is written for a first course in single-variable calculus at college level, typically the first semester of a two- or three-semester sequence. Students need secondary-school algebra and trigonometry and familiarity with functions; Chapter 1 and Appendix C review the pre-calculus background.

A typical semester covers Chapters 2–5, with Chapter 1 as review and selected sections of Chapter 6 (areas, volumes and possibly physical applications). Instructors who prefer late transcendentals can postpone Sections 3.9, 5.6 and the exponential material until Section 6.7. Read each section's learning objectives first, work through the examples and Checkpoints, then do a range of end-of-section exercises, checking odd-numbered answers in the Answer Key. Use the chapter review key concepts and review exercises before tests, and try at least one student project per chapter to see the ideas applied. The free web view also offers highlighting and note-taking.

Edition and Sources

This guide is based on Calculus Volume 1 (OpenStax, original publication 2016; © 2016 Rice University, CC BY-NC-SA 4.0), read in the official web PDF linked from the OpenStax book page (PDF generated April 2026), https://openstax.org/details/books/calculus-volume-1. It draws on the preface (scope, pedagogical features, early or late transcendentals, answers and resources), the full table of contents, the introductions and learning objectives of all six chapters, every section heading, the complete key-concepts summaries for all six chapters, the student projects, and the structure of the appendices and Answer Key.