Calculus Volume 1 PDF Download – Strang & Herman (OpenStax)

Students taking Calculus I can download the complete textbook “Calculus Volume 1” by Gilbert Strang and Edwin “Jed” Herman, free as a PDF from OpenStax. It is a traditional, example-and-exercise-driven calculus text — every section works through several fully-solved examples before a large practice-problem set, and every chapter closes with a Key Terms, Key Equations, and Key Concepts review before its Chapter Review Exercises.

This is Volume 1 of OpenStax’s three-volume calculus series, covering the full first semester: a pre-calculus function review, limits, derivatives and their applications, and integration through its applications. Calculus II and Multivariable Calculus continue in Volumes 2 and 3, published separately.

Book Overview

Course	Calculus I (single-variable)
Degree Programs	BS Mathematics, BSCS, Engineering — general first-year requirement
Level	University — first semester (first-year calculus)
Edition	OpenStax edition — published March 30, 2016
Author	Gilbert Strang, Edwin “Jed” Herman
Structure	6 chapters, from a pre-calculus function review through applications of integration
Exercises	Every section ends with a substantial practice-problem set, and every chapter closes with Key Terms, Key Equations, Key Concepts, and a full Chapter Review Exercises set
Language	English
License	Creative Commons Attribution-NonCommercial-ShareAlike 4.0 (CC BY-NC-SA 4.0) — Model: Link-only
Format	Free PDF and web/HTML reader; also available as a low-cost print edition through third-party printers

Chapter List

Chapter 1: Functions and Graphs

Difficulty: Easy · Calculus I · Key topics: reviewing functions, polynomial/rational/algebraic function classes, trigonometric functions, inverse functions, exponential and logarithmic functions

Before any calculus begins, Chapter 1 reviews the function toolkit the rest of the book depends on. It opens by revisiting what makes a relation a function, then classifies the building-block functions used throughout calculus — polynomial, rational, and algebraic — before turning to the trigonometric functions and the identities calculus relies on most. The chapter closes with two function families that behave unlike anything in ordinary algebra: exponential functions, which grow by repeated multiplication rather than repeated addition, and their inverses, the logarithmic functions, alongside the general idea of what it means for a function to have an inverse at all.

Key Points:

• A relation is a function only if every input maps to exactly one output — the vertical line test checks this graphically.
• Polynomial, rational, and algebraic (root-based) functions form the core toolkit; combining or composing them builds most functions used in calculus.
• sin, cos, tan and their reciprocals satisfy identities (Pythagorean, angle-sum) that calculus uses constantly, not just in a trigonometry course.
• A function has an inverse only where it is one-to-one; restricting a function’s domain — e.g. sine to [−π/2, π/2] — is what makes an inverse exist.
• Exponential functions a^x grow (or decay) by a constant multiplicative factor per unit of x, unlike a linear function’s constant additive rate.
• log_a(x) is defined as the inverse of a^x, so log_a(a^x) = x and a^{log_a x} = x for x > 0.

Practice Tip: Before Chapter 2 begins, make sure you can sketch — without a calculator — the graphs of y=xⁿ, y=sin x, y=e^x, and y=ln x from memory. Nearly every limit and derivative example later in the book assumes you already know these shapes.

Common Mistake: Assuming every function has an inverse. A function needs to be one-to-one (pass the horizontal line test) before an inverse function can even be defined — this is exactly why arcsin, arccos, and arctan are defined only on restricted domains of sine, cosine, and tangent.

Important Questions:

• Is f(x) = x² one-to-one on the domain of all real numbers? If not, what restricted domain makes it one-to-one? No — f(−2)=f(2)=4, so it fails the horizontal line test on all reals; restricting the domain to x ≥ 0 (or x ≤ 0) makes it one-to-one, which is exactly how √x is defined as its inverse.
• Simplify log₂(8) + log₂(4) using properties of logarithms, and explain your steps. log₂(8)=3 and log₂(4)=2, so the sum is 5; equivalently log₂(8)+log₂(4) = log₂(8×4) = log₂(32) = 5, using the product rule for logarithms.

Chapter 2: Limits

Difficulty: Easy · Calculus I · Key topics: a preview of calculus, the limit of a function, the limit laws, continuity, the precise (epsilon-delta) definition of a limit

Chapter 2 opens calculus proper by asking what it means for a function to approach a value without necessarily reaching it. After a preview showing how limits let you answer questions ordinary algebra cannot — such as the slope of a curve at a single point — the chapter builds an informal, table-and-graph sense of a limit, then formalizes it with limit laws that let you compute limits algebraically instead of estimating them. It uses limits to define continuity precisely, and closes with the formal epsilon-delta definition that makes every earlier informal statement rigorous.

Key Points:

• lim_x→a f(x) = L means f(x) gets arbitrarily close to L as x gets arbitrarily close to a — it says nothing about f(a) itself.
• The Limit Laws let you compute limits of sums, products, quotients, and compositions from the limits of their pieces, avoiding a table of values every time.
• A limit can fail to exist by going to different values from the left and right (a jump), by growing without bound (a vertical asymptote), or by oscillating.
• f is continuous at x=a exactly when three things hold: f(a) is defined, lim_x→a f(x) exists, and the two are equal.
• The Intermediate Value Theorem — a consequence of continuity — guarantees a root exists between any two points where a continuous function changes sign.
• The formal ε-δ definition says: for every ε>0 there exists δ>0 such that 0<|x−a|<δ forces |f(x)−L|<ε — this is what “arbitrarily close” means, made precise.

Practice Tip: When a limit produces 0/0 by direct substitution, don’t stop — that’s not “no limit,” it’s a signal to factor, rationalize, or simplify first. Almost every worked limit example in this chapter follows exactly that two-step pattern.

Common Mistake: Treating lim_x→a f(x) and f(a) as automatically the same number. They agree at every point where f is continuous, but a function can have a well-defined limit at a point where it’s undefined, or even defined to a different value — that gap is exactly what a removable discontinuity is.

Important Questions:

• Evaluate lim_x→3 (x²−9)/(x−3). Factor the numerator: (x−3)(x+3)/(x−3) = x+3 for x≠3, so the limit as x→3 is 6, even though the original expression is undefined at x=3 itself.
• Is f(x) = 1/x continuous at x=0? Explain using the three-part definition of continuity. No — f(0) is not defined at all (the function has a vertical asymptote there), which alone fails the first of the three conditions, so continuity cannot hold regardless of what the limit does.

Chapter 3: Derivatives

Difficulty: Medium · Calculus I · Key topics: defining the derivative, the derivative as a function, differentiation rules, derivatives as rates of change, derivatives of trigonometric functions, the chain rule, derivatives of inverse functions, implicit differentiation, derivatives of exponential and logarithmic functions

Chapter 3 turns the limit machinery of Chapter 2 into the derivative — the single idea calculus is built around. It defines the derivative at a point as a limit of difference quotients, generalizes that to the derivative function itself, and then develops the differentiation rules (power, product, quotient) that make computing derivatives fast, rather than requiring every derivative to be found from the limit definition directly. The chapter works through the derivatives of trigonometric, inverse, exponential, and logarithmic functions in turn, and introduces the chain rule and implicit differentiation as the two techniques needed once functions start being combined or defined implicitly.

Key Points:

• f′(a) = lim_h→0 [f(a+h)−f(a)]/h is the instantaneous rate of change of f at x=a, and geometrically the slope of the tangent line there.
• Power rule: d/dx[xⁿ] = n·xⁿ⁻¹; this rule, plus linearity, differentiates every polynomial without any limit computation.
• Product rule (fg)′ = f′g+fg′ and quotient rule (f/g)′ = (f′g−fg′)/g² are both necessary — you cannot differentiate a product or quotient by differentiating each piece separately.
• d/dx[sin x] = cos x and d/dx[cos x] = −sin x are the two facts every other trig derivative in the chapter is built from.
• The chain rule, d/dx[f(g(x))] = f′(g(x))·g′(x), lets you differentiate a function applied to another function, and it underlies implicit differentiation.
• d/dx[e^x] = e^x (the defining property of e) and d/dx[ln x] = 1/x round out the toolkit needed for the rest of the book.

Practice Tip: Practice differentiating the same function two different ways when a problem allows it — e.g. expand (x+1)² and differentiate term-by-term, then redo it with the chain rule. Getting the same answer both ways is the fastest way to trust the chain rule instead of just memorizing it.

Common Mistake: Forgetting the inner-function derivative when applying the chain rule — writing d/dx[sin(x²)] as cos(x²) instead of cos(x²)·2x. The chain rule always has two factors: the derivative of the outside, evaluated at the inside, times the derivative of the inside.

Important Questions:

• Differentiate f(x) = 3x⁴ − 2x² + 5 using the power rule. f′(x) = 12x³ − 4x.
• Find dy/dx if x² + y² = 25, using implicit differentiation. Differentiating both sides gives 2x + 2y·y′ = 0, so y′ = −x/y.

Chapter 4: Applications of Derivatives

Difficulty: Hard · Calculus I · Key topics: related rates, linear approximations and differentials, maxima and minima, the Mean Value Theorem, derivatives and the shape of a graph, limits at infinity and asymptotes, applied optimization, L’Hôpital’s Rule, Newton’s Method, antiderivatives

Chapter 4 is the longest chapter in the book, and it’s where derivatives start doing real work. Related rates problems connect the rates of several changing quantities through a single equation; linear approximation uses the tangent line to estimate a function’s value nearby; and the Mean Value Theorem guarantees a point where the instantaneous rate of change matches the average rate over an interval. From there the chapter uses the sign of the first and second derivative to determine a graph’s shape, before tackling limits at infinity, applied optimization, L’Hôpital’s Rule for indeterminate limits, Newton’s Method for approximating roots, and antiderivatives — which quietly reverse the entire chapter’s direction and set up the rest of the book.

Key Points:

• Related rates problems differentiate a single equation connecting several quantities with respect to time, then substitute known rates and values.
• Linear approximation: f(x) ≈ f(a) + f′(a)(x−a) for x near a — the tangent line as a local stand-in for the function.
• Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) has f′(c) equal to the average rate of change on [a,b].
• f increasing/decreasing is read from the sign of f′; concave up/down (and inflection points) is read from the sign of f″.
• L’Hôpital’s Rule resolves 0/0 or ∞/∞ limits by differentiating numerator and denominator separately, not by the quotient rule.
• An antiderivative F of f satisfies F′(x)=f(x); this single idea, introduced almost as an afterthought at the end of Chapter 4, is what the rest of the book is built on.

Practice Tip: For any optimization word problem, write the quantity to optimize as a function of ONE variable before differentiating anything — if you have two variables, find the constraint equation the problem gives you and use it to eliminate one of them first.

Common Mistake: Using the First Derivative Test and Second Derivative Test as if they always agree, and giving up when f″(c)=0. The Second Derivative Test is inconclusive exactly when f″(c)=0 — the First Derivative Test (checking the sign of f′ on either side of c) always works and should be the fallback.

Important Questions:

• A rectangular box with a square base has a fixed volume of 32 cubic units. Set up the function for its surface area in terms of one variable, ready to minimize. With side x and height h, V=x²h=32 gives h=32/x², so S(x) = 2x² + 4xh = 2x² + 128/x, a single-variable function ready for differentiation.
• Use L’Hôpital’s Rule to evaluate lim_x→0 (sin x)/x. Direct substitution gives 0/0; differentiating top and bottom gives lim_x→0 cos(x)/1 = cos(0) = 1.

Chapter 5: Integration

Difficulty: Medium · Calculus I · Key topics: approximating areas, the definite integral, the Fundamental Theorem of Calculus, integration formulas and the net change theorem, substitution, integrals involving exponential and logarithmic functions, integrals resulting in inverse trigonometric functions

Chapter 5 reverses the direction Chapter 4 pointed toward at its very end. It begins by approximating the area under a curve with a sum of rectangles, then takes the limit of that sum as the number of rectangles grows without bound to define the definite integral precisely. The Fundamental Theorem of Calculus is the chapter’s hinge point: it proves that this area-based definition and the antiderivative from Chapter 4 are secretly the same idea, which is what makes definite integrals computable without ever summing an infinite number of rectangles by hand. The chapter closes by extending substitution — already familiar from differentiation’s chain rule — to integrals, and works out the antiderivatives of exponential, logarithmic, and inverse trigonometric functions.

Key Points:

• A Riemann sum approximates area under a curve using rectangles: ∑f(x_i)·Δx, with left, right, or midpoint sample points.
• The definite integral ∫_a^b f(x)dx is defined as the limit of Riemann sums as the number of rectangles → ∞.
• FTC Part 1: if F is any antiderivative of f, then ∫_a^b f(x)dx = F(b)−F(a) — no limit of sums is needed once an antiderivative is known.
• FTC Part 2: the accumulation function A(x)=∫_a^x f(t)dt satisfies A′(x)=f(x), so integration and differentiation genuinely undo each other.
• u-substitution reverses the chain rule for integrals: choosing u=g(x) and rewriting the whole integral in terms of u.
• ∫(1/x)dx = ln|x|+C and ∫e^xdx = e^x+C are the two integral facts the rest of the chapter’s exponential/log work depends on.

Practice Tip: When approximating a definite integral with a Riemann sum, sketch the curve and shade the actual rectangles for a small n (like n=4) by hand before trusting a formula — seeing whether the rectangles overshoot or undershoot the true area builds the intuition that the Fundamental Theorem later makes exact.

Common Mistake: Forgetting that a definite integral’s value can be negative when the curve dips below the x-axis. ∫_a^bf(x)dx measures signed area — a curve entirely below the x-axis on [a,b] gives a negative integral, not zero or an error.

Important Questions:

• Use the Fundamental Theorem of Calculus to evaluate ∫₁⁴ 2x dx. An antiderivative of 2x is x², so the integral equals 4²−1² = 16−1 = 15.
• Evaluate ∫ 2x·cos(x²) dx using substitution. Let u=x², du=2x dx, giving ∫cos(u)du = sin(u)+C = sin(x²)+C.

Chapter 6: Applications of Integration

Difficulty: Hard · Calculus I · Key topics: areas between curves, determining volumes by slicing, volumes of revolution (cylindrical shells), arc length and surface area, physical applications, moments and centers of mass, integrals of exponential functions and logarithms, exponential growth and decay, calculus of the hyperbolic functions

Chapter 6 closes Volume 1 by showing how far the definite integral’s core idea — summing infinitely many infinitesimal pieces — extends beyond area. Area between two curves and arc length come first, followed by two different methods (slicing into disks/washers, and cylindrical shells) for finding the volume of a solid of revolution. The chapter then applies the same accumulation idea to physical quantities: work done by a variable force, and moments and centers of mass for an object with varying density. It closes by returning to the exponential and logarithmic functions from Chapter 1, defining them properly in terms of integrals, and introducing the hyperbolic functions as a genuinely new function family built from e^x.

Key Points:

• Area between curves: ∫_a^b[f(x)−g(x)]dx, where f(x)≥g(x) on [a,b].
• The disk/washer method and the shell method both compute volumes of revolution by integrating cross-sectional area or shell surface area — pick whichever gives a simpler integral for a given axis of rotation.
• Arc length of a curve y=f(x) on [a,b] is ∫_a^b√(1+[f′(x)]²)dx — a direct consequence of summing infinitesimal straight-line segments.
• Work done by a variable force F(x) moving an object from a to b is W=∫_a^bF(x)dx.
• The center of mass of a region balances the region’s weighted position, computed by dividing a moment integral by a mass (or area) integral.
• The hyperbolic functions sinh x and cosh x are built from e^x and e^−x and satisfy identities that mirror ordinary trig functions, but with different signs in places (e.g. cosh²x − sinh²x = 1).

Practice Tip: For any volume-of-revolution problem, sketch the 2D region AND the axis of rotation before choosing disks/washers or shells — the picture alone usually tells you which method gives an integral in terms of a single, easy variable.

Common Mistake: Using the disk method when the region being revolved doesn’t touch the axis of rotation. A gap between the region and the axis means you need the washer method — subtracting an inner radius — not a single disk radius.

Important Questions:

• Set up the integral for the area between y=x² and y=x on [0,1]. Since x≥x² on [0,1], the area is ∫₀¹(x−x²)dx = [x²/2 − x³/3]₀¹ = 1/6.
• A variable force F(x)=3x newtons moves an object from x=0 to x=4 meters. Find the work done. W=∫₀⁴3x dx = [3x²/2]₀⁴ = 24 joules.

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Download Calculus Volume 1 PDF (Free)

This book is free from its official source, OpenStax. Click below to download the complete PDF — a free web-based reader edition (with per-section, linkable pages) is also available on the OpenStax site if you’d rather read online.

↓ Download PDF

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How to Study This Book

Read the six chapters in order — each one leans directly on the last. Chapter 1 is a pre-calculus review of the function toolkit (polynomial, trig, exponential, logarithmic); skim it quickly if you’re already comfortable with these, but don’t skip it if any of them feel shaky, since every later chapter assumes fluency with all of them.

Chapters 2 and 3 (Limits, Derivatives) are the conceptual core of the course and should be worked through carefully and in order — Chapter 3’s differentiation rules are proved using the limit definition from Chapter 2.

Chapter 4 (Applications of Derivatives) is the longest chapter in the book, covering ten distinct techniques. It does not need to be read start-to-finish in one sitting — related rates, optimization, and curve-sketching (4.1–4.7) are commonly covered before L’Hôpital’s Rule and Newton’s Method (4.8–4.9), which some courses treat as optional or move later in the term. Check your syllabus.

Chapters 5 and 6 (Integration, Applications of Integration) form the second half of a standard Calculus I course. Chapter 5’s Fundamental Theorem of Calculus (5.3) is the hinge the entire second half of the book depends on — do not move on to Chapter 6 until it is comfortable.

This is Volume 1 of a three-volume OpenStax series. If your course continues into Calculus II (techniques of integration, series, differential equations), that content is in Calculus Volume 2, not this book.

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Used In These Programs

This book is used for the Calculus I course in: BS Mathematics, BSCS, and Engineering programs. Browse all Calculus books or all Mathematics books.

Who Should Read This

Calculus Volume 1 is written for a first-year student taking their required Calculus I course in a BS Mathematics, BSCS, or Engineering program — it assumes a solid pre-calculus/algebra background (reviewed in Chapter 1) but no prior calculus. Its example-and-exercise-driven style, with large practice sets at the end of every section, suits a student who learns by working through many worked examples and problems rather than one who wants an activity-first, discovery-based approach. It is also a strong fit for self-study, since the free web reader links every term to its definition and every chapter closes with a Key Terms/Key Equations/Key Concepts review before the practice problems.

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Applicable Universities

This book is useful for students at Pakistani universities offering BS Mathematics, BSCS, or Engineering programs, including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

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Calculus Volume 1 is OpenStax’s own entry point into its three-volume calculus series, and its large, traditional practice-problem sets make it a strong match for students who learn best by working through many worked examples. Browse more Mathematics books for the rest of your semester.

Calculus Volume 1, by Gilbert Strang and Edwin “Jed” Herman. OpenStax, Rice University. Free under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 licence. Access for free at https://openstax.org/details/books/calculus-volume-1