Active Calculus PDF Download – Matthew Boelkins (2nd Edition)

Students taking Calculus I or Calculus II can download the complete two-semester textbook “Active Calculus” by Matthew Boelkins, free as a PDF in its current second edition. It is built around an “active learning” approach — every one of its roughly 49 sections opens with a Preview Activity that has students explore an idea graphically or numerically before any formal definition appears, and the book pairs that with 147 guided in-class Activities and a mix of WeBWorK and longer written exercises.

Unlike a book written for one semester, Active Calculus covers the full first-year sequence: Chapters 1–4 build differential calculus from limits through the definite integral (a complete Calculus I course), and Chapters 5–8 build integral calculus onward through differential equations and Taylor series (a complete Calculus II course). It is endorsed by the AIM Open Textbook Initiative Editorial Board and is the primary calculus text at Grand Valley State University.

Book Overview

Course	Calculus I & Calculus II (single-variable)
Degree Programs	BSCS, BS Mathematics, Engineering — general first/second-year requirement
Level	University — first and second semester (first-year calculus sequence)
Edition	2nd Edition — released August 7, 2025
Author	Matthew Boelkins (Grand Valley State University), with contributions from David Austin and Christina Safranski
Structure	8 chapters, covering the first two semesters of single-variable calculus
Exercises & Activities	58 non-WeBWorK exercises across the book, plus anonymous WeBWorK problem sets per section and 147 guided Preview/in-class Activities
Language	English
License	Creative Commons BY-SA — no specific version stated by the publisher — Model: Link-only
Format	Free HTML and PDF; a print edition is available for purchase separately, starting January 2026

Chapter List

Chapter 1: Understanding the Derivative

Difficulty: Easy · Calculus I · Key topics: average vs. instantaneous velocity, limits, the derivative at a point, the derivative function, the second derivative, continuity vs. differentiability, the tangent line approximation

Calculus begins with a simple question: given a changing quantity, how fast is it changing right now? Chapter 1 builds the derivative from the ground up, starting with a tossed ball’s average velocity over shrinking time intervals and using limits to pin down its instantaneous velocity exactly. The chapter then generalizes the pattern to any function, defining the derivative at a point as the slope of the tangent line and extending it to the derivative function itself. It closes by connecting differentiability to continuity and introducing the tangent line as the simplest local approximation of a function.

Key Points:

• Average velocity = change in position ÷ change in time = slope of the secant line between two points.
• Instantaneous velocity is the limit of average velocity as the time interval shrinks to 0.
• f′(a) = lim_h→0 [f(a+h) − f(a)] / h defines the derivative of f at a point a.
• Differentiable ⇒ continuous, but continuous does not imply differentiable — e.g. f(x) = |x| is continuous but not differentiable at x = 0.
• The second derivative f″ measures concavity: how the rate of change is itself changing.
• The tangent line L(x) = f(a) + f′(a)(x−a) is the simplest linear approximation of f(x) near x = a.

Practice Tip: Before computing any derivative symbolically, sketch what you expect the derivative graph to look like from the original function’s graph. Active Calculus builds every section around a Preview Activity for exactly this reason — graphical intuition first, formulas second.

Common Mistake: Treating “differentiable” and “continuous” as interchangeable. A function can be continuous everywhere — like f(x) = |x| — yet fail to be differentiable at specific points where the graph has a sharp corner.

Important Questions:

• If f(x) = x² − 3x, use the limit definition to find f′(2) and interpret what the number means. f′(2) = lim_h→0 [(2+h)²−3(2+h) − (4−6)] / h = 1, meaning that at x = 2 the function’s value is instantaneously increasing at a rate of 1 unit of f per unit of x.
• Explain why a function that is differentiable at x = a must also be continuous at x = a, but not conversely. Differentiability requires the graph to have a well-defined tangent line at a, which rules out both jumps and sharp corners; continuity only rules out jumps, so it is a weaker condition that differentiability implies but does not require.

Chapter 2: Computing Derivatives

Difficulty: Easy · Calculus I · Key topics: elementary derivative rules, derivatives of sine and cosine, the product and quotient rules, other trigonometric derivatives, the chain rule, derivatives of inverse functions, implicit differentiation

Having defined the derivative from first principles, Chapter 2 develops the shortcut rules that make differentiation practical. Starting from the elementary rules for sums, constants, and power functions, the chapter builds outward to the derivatives of sine and cosine, the product and quotient rules, and the derivatives of the remaining trigonometric functions. The chain rule follows, giving a systematic way to differentiate composite functions, and the chapter closes by extending these tools to inverse functions and to curves defined implicitly, such as circles and other equations that cannot easily be solved for y.

Key Points:

• Power rule: d/dx[xⁿ] = n·xⁿ⁻¹.
• Product rule: (fg)′ = f′g + fg′. Quotient rule: (f/g)′ = (f′g − fg′)/g².
• Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x) — differentiate the outside, then multiply by the derivative of the inside.
• d/dx[sin x] = cos x and d/dx[cos x] = −sin x, the base case every other trig derivative in the chapter builds on.
• Implicit differentiation finds dy/dx directly from an equation in x and y, without first solving for y.
• Inverse function derivative: if g = f⁻¹, then g′(x) = 1 / f′(g(x)).

Practice Tip: Before reaching for a rule, identify the function’s outermost operation first — a sum, a product, a quotient, or a function applied to another function. That single identification decides which rule you actually need.

Common Mistake: Applying the power rule to an exponential function like 2^x, treating x as the base when it is really the exponent. Functions of the form a^x need the exponential differentiation rule, not the power rule.

Important Questions:

• Differentiate h(x) = x² sin(x) using the product rule and simplify. h′(x) = 2x·sin(x) + x²·cos(x), obtained by treating x² and sin(x) as the two factors in the product rule.
• If y² + xy = 4, use implicit differentiation to find dy/dx in terms of x and y. Differentiating both sides gives 2y·y′ + y + x·y′ = 0, so y′ = −y / (2y + x).

Chapter 3: Using Derivatives

Difficulty: Medium · Calculus I · Key topics: related rates, using derivatives to evaluate limits, identifying extreme values, families of functions, global optimization, applied optimization

With the mechanics of differentiation in hand, Chapter 3 turns to what derivatives are for. Related rates problems — such as finding how fast a balloon’s radius grows as air is pumped in at a constant rate — show how implicit differentiation connects the rates of change of several quantities linked by a single equation. The chapter then uses derivatives to evaluate otherwise-indeterminate limits, locate a function’s extreme values, characterize families of functions that share a parameter, and solve global and applied optimization problems — the kind of real-world questions, like minimizing cost or maximizing area, that motivated calculus in the first place.

Key Points:

• Related rates problems link several time-dependent quantities through one equation, then differentiate implicitly with respect to t.
• Critical points occur where f′(x) = 0 or f′(x) is undefined.
• First Derivative Test: f′ changing from + to − signals a local max; − to + signals a local min.
• Second Derivative Test: f″(a) > 0 suggests a local min at a critical point a; f″(a) < 0 suggests a local max.
• On a closed interval, a global (absolute) extreme value occurs either at a critical point or at an endpoint.
• Applied optimization requires translating a word problem into a single-variable function before any calculus is applied.

Practice Tip: In a related rates problem, write down every rate you’re given and the rate you want before differentiating anything. A related rates equation only works once every quantity involved is expressed as a function of time.

Common Mistake: Substituting a fixed numeric value into a related rates equation before differentiating. Differentiate first, treating the varying quantities symbolically, and only substitute the specific numbers afterward.

Important Questions:

• A 10-foot ladder slides down a wall; if the base moves away from the wall at 2 ft/s, how fast is the top sliding down when the base is 6 ft from the wall? Using x²+y²=100 and differentiating, the top is sliding down at 1.5 ft/s at that instant (dy/dt = −x/y · dx/dt = −6/8 · 2).
• Find the absolute maximum and minimum of f(x) = x³ − 3x on the interval [−2, 2]. Critical points at x = ±1 give f(−1)=2 and f(1)=−2; checking endpoints f(−2)=−2 and f(2)=2 shows the absolute maximum is 2 (at x=−1 and x=2) and the absolute minimum is −2 (at x=1 and x=−2).

Chapter 4: The Definite Integral

Difficulty: Medium · Calculus I · Key topics: determining distance from velocity, Riemann sums, the definite integral, the Fundamental Theorem of Calculus

Chapter 4 reverses the question that opened the book: instead of finding velocity from position, it asks how to recover position — or any accumulated quantity — from a known rate of change. Starting from the same tossed-ball scenario used to introduce the derivative, the chapter builds Riemann sums that approximate the area under a velocity curve, then takes a limit to define the definite integral precisely. It culminates in the Fundamental Theorem of Calculus, which shows that differentiation and integration are inverse processes — the result that connects the two central ideas of the entire course.

Key Points:

• A Riemann sum approximates the area under a curve with rectangles: ∑ f(x_i)·Δx.
• The definite integral ∫_a^b f(x) dx is the limit of Riemann sums as the number of rectangles → ∞.
• For a nonnegative velocity function, ∫_a^b v(t) dt gives the total distance traveled on [a,b].
• Fundamental Theorem of Calculus (Part 1): if F′(x) = f(x), then ∫_a^b f(x) dx = F(b) − F(a).
• Left, right, and midpoint Riemann sums can each over- or under-estimate the true area, depending on the function’s shape.
• A definite integral can be negative when f(x) is negative on part of the interval — it measures signed area, not raw area.

Practice Tip: Before evaluating any definite integral, sketch the region. Knowing whether the curve dips below the x-axis tells you whether you’re computing a signed area or a true distance/area before you touch any algebra.

Common Mistake: Confusing “total distance traveled” with “net displacement.” When velocity is negative on part of an interval, the definite integral gives net displacement, not total distance — total distance requires integrating |v(t)|.

Important Questions:

• Use a right Riemann sum with 4 subintervals to approximate ∫₀⁴ x² dx, then compare it to the exact value. The right sum gives 1²+2²+3²+4² = 30, versus the exact value 64/3 ≈ 21.33 — the right sum overestimates because x² is increasing on [0,4].
• Explain, using the Fundamental Theorem of Calculus, why d/dx[∫_a^x f(t) dt] = f(x). The accumulation function A(x) = ∫_a^x f(t) dt is itself an antiderivative of f by the FTC’s construction, so differentiating it simply undoes the integration and returns f(x).

Chapter 5: Evaluating Integrals

Difficulty: Medium · Calculus II · Key topics: constructing graphs of antiderivatives, the Second Fundamental Theorem of Calculus, integration by substitution, integration by parts, other algebraic antiderivative techniques, numerical integration

Chapter 5 asks how much can be determined about a function f from information about its derivative f′ alone — a question the book first raised graphically, sketching a plausible graph of f from only a picture of f′. Building on that idea, the chapter introduces the Second Fundamental Theorem of Calculus and develops the two workhorse techniques for finding antiderivatives algebraically: substitution, which reverses the chain rule, and integration by parts, which reverses the product rule. It closes by acknowledging that not every function has an elementary antiderivative, which is what motivates numerical integration methods.

Key Points:

• An antiderivative F of f satisfies F′(x) = f(x); any two antiderivatives of the same function differ only by a constant.
• Substitution reverses the chain rule: let u = g(x), then rewrite the whole integral in terms of u.
• Integration by parts reverses the product rule: ∫ u dv = uv − ∫ v du.
• The Total Change Theorem: ∫_a^b f′(x) dx = f(b) − f(a).
• Some functions, such as e^−x², have no elementary antiderivative at all, which is exactly when numerical methods are needed.
• Numerical integration methods approximate a definite integral directly when an exact antiderivative can’t be found in closed form.

Practice Tip: When choosing u for integration by parts, pick the factor that gets simpler when differentiated (often a polynomial) and the factor that’s easy to integrate for dv — getting this choice backward usually makes the new integral harder, not easier.

Common Mistake: Forgetting the “+C” when reporting an indefinite integral, or dropping it partway through a multi-step substitution. Every family of antiderivatives needs the constant, except when you’re evaluating a definite integral, where it cancels automatically.

Important Questions:

• Evaluate ∫ x·cos(x²) dx using substitution. Let u = x², du = 2x dx, so the integral becomes ½∫cos(u) du = ½sin(x²) + C.
• Evaluate ∫ x·e^x dx using integration by parts. With u = x and dv = e^xdx, the result is x·e^x − e^x + C.

Chapter 6: Using Definite Integrals

Difficulty: Hard · Calculus II · Key topics: area and length, volume, density, mass, and center of mass, physics applications (work, force, pressure), improper integrals

Chapter 6 shows just how much the definite integral can measure once its role in computing area is generalized. Beyond area between curves and arc length, the same accumulation idea computes the volume of a solid formed by revolving or slicing a region, the mass and center of mass of an object with varying density, and physical quantities such as work done against a variable force or force exerted by fluid pressure. The chapter closes with improper integrals, extending the definite integral to unbounded intervals or unbounded integrands, and asking when that infinite accumulation still settles on a finite value.

Key Points:

• Area between two curves: ∫_a^b [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a,b].
• Volumes of revolution can be computed with the disk/washer method or the shell method, both built by summing thin slices.
• Density × volume (or length) gives mass; center of mass balances the weighted position of that mass.
• Work = ∫ F(x) dx when force varies with position — e.g. pumping water out of a tank or stretching a spring.
• Improper integrals handle infinite intervals or unbounded integrands by rewriting the integral as a limit.
• An improper integral “converges” if that limit exists and is finite, and “diverges” otherwise.

Practice Tip: For any volume, work, or mass problem, start by drawing and labeling a single representative slice — a disk, a shell, or a thin layer — before writing any integral. The entire setup follows directly from that one picture.

Common Mistake: Using the disk method when the region doesn’t touch the axis of revolution. A gap between the region and the axis means you need the washer method, which subtracts an inner radius, not a plain disk.

Important Questions:

• Set up (but don’t necessarily evaluate) the integral for the volume obtained by rotating the region bounded by y = x² and y = 4 about the x-axis. Using washers from x = −2 to x = 2: V = π∫₋₂² [4² − (x²)²] dx.
• Determine whether ∫₁^∞ 1/x² dx converges or diverges, and find its value if it converges. Rewriting as lim_b→∞ ∫₁^b x⁻² dx = lim_b→∞ [1 − 1/b] gives a finite limit of 1, so the integral converges to 1.

Chapter 7: Differential Equations

Difficulty: Hard · Calculus II · Key topics: an introduction to differential equations, qualitative behavior of solutions, Euler’s method, separable differential equations, modeling with differential equations, the logistic equation

Chapter 7 introduces differential equations — equations that describe a function through its rate of change rather than through an explicit formula. Building on the idea that integration recovers a function from its derivative, the chapter first studies what can be learned about solutions qualitatively, directly from the equation itself, before introducing Euler’s method to approximate solutions numerically. It then develops separable differential equations, one of the few types solvable algebraically, and applies the whole toolkit to real modeling problems, culminating in the logistic equation, which models population growth that levels off as it approaches a carrying capacity.

Key Points:

• A differential equation relates a function to its own derivative(s), e.g. dy/dt = ky.
• Qualitative analysis (slope fields, equilibrium solutions) reveals a solution’s long-term behavior without solving the equation.
• Euler’s method approximates a solution numerically by taking small steps along the slope field.
• A separable equation is solved by rewriting it as g(y) dy = h(t) dt and integrating both sides.
• Exponential growth/decay solves dy/dt = ky; every solution has the form y = Ce^kt.
• The logistic equation dy/dt = ky(1 − y/L) models growth that slows as the population y approaches a carrying capacity L.

Practice Tip: Before solving any differential equation algebraically, sketch its slope field or reason about its equilibrium solutions first. Knowing what the answer should look like catches sign errors in the algebra that follows.

Common Mistake: Losing the constant of integration when solving a separable equation, or forgetting to solve for it using the given initial condition. An unresolved constant leaves you with a whole family of curves instead of the one solution the problem actually asked for.

Important Questions:

• Solve the separable differential equation dy/dt = 0.5y with initial condition y(0) = 20. Separating and integrating gives y = Ce^0.5t; applying y(0)=20 gives C=20, so y(t) = 20e^0.5t.
• Explain why solutions to the logistic equation dy/dt = ky(1 − y/L) approach L as t → ∞, regardless of the (positive) starting population. As y approaches L, the factor (1 − y/L) approaches 0, driving dy/dt toward 0, so L is a stable equilibrium that any positive starting population is pulled toward over time.

Chapter 8: Taylor Polynomials and Taylor Series

Difficulty: Hard · Calculus II · Key topics: extending local linearization, Taylor polynomials, geometric sums, Taylor series, finding and using Taylor series, the accuracy of approximations, using series to solve differential equations

Chapter 8 extends the tangent line approximation from Chapter 1 into a far more powerful tool. Using e^x near x = 0 as a running example, the chapter shows how adding higher-degree polynomial terms produces increasingly accurate local approximations, leading to the general Taylor polynomial. After a detour through geometric sums, it builds the full Taylor series — an infinite polynomial that, for well-behaved functions, converges exactly to the original function — and develops techniques for finding and manipulating these series. The chapter closes by quantifying how accurate a finite-degree approximation actually is, and by using series to solve differential equations that resist algebraic methods.

Key Points:

• The degree-n Taylor polynomial matches a function’s value and its first n derivatives at the center point.
• Taylor series: f(x) = ∑ [f⁽ⁿ⁾(a)/n!]·(x−a)ⁿ, built from every derivative of f at a.
• A geometric series ∑ arⁿ converges to a/(1−r) when |r| < 1.
• Known series for e^x, sin x, cos x, and 1/(1−x) can be combined or manipulated to find new series, rather than deriving each one from scratch.
• The remainder (error) of a Taylor polynomial approximation can be bounded, telling you how many terms are needed for a target accuracy.
• Every Taylor series has a radius of convergence — an interval around the center where the infinite sum actually equals the function.

Practice Tip: Before computing a new Taylor series from scratch, check whether it can be built from a known series (e^x, 1/(1−x), sin x) using substitution, differentiation, or integration — it’s almost always faster than starting over.

Common Mistake: Assuming a Taylor polynomial is accurate everywhere just because it matches the function closely near the center. The approximation typically gets worse — sometimes drastically — the farther x moves from the center point a.

Important Questions:

• Find the degree-3 Taylor polynomial for f(x) = e^x centered at a = 0. P₃(x) = 1 + x + x²/2 + x³/6, using f⁽ⁿ⁾(0) = 1 for every derivative of e^x.
• Use the geometric series formula to find the Taylor series for 1/(1−x) centered at 0, and state its radius of convergence. 1/(1−x) = ∑_n=0^∞ xⁿ = 1 + x + x² + x³ + …, with radius of convergence 1 (i.e. |x| < 1).

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Download Active Calculus PDF (Free)

This book is free from its official source. Click below to open the author’s own site and download the complete second-edition PDF — the HTML edition (better on phones and tablets, with live WeBWorK exercises built in) is also available there.

↓ Download PDF

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How to Study This Book

Read Chapters 1 through 4 in order first — together they form a complete, self-contained Calculus I course (limits, the derivative, its applications, and the definite integral), and each section’s Preview Activity is meant to be attempted before you read the section itself, not after.

Chapters 5 through 8 form the Calculus II half of the book and assume Chapters 1–4 are already comfortable, especially the definite integral and the Fundamental Theorem from Chapter 4. At Grand Valley State University, where the book is developed, Chapters 1–4 correspond to the Math 201 course and Chapters 5–8 to Math 202.

Within Chapter 5, work through substitution before integration by parts — substitution is used again inside several integration-by-parts examples later in the same chapter.

Chapter 7 (Differential Equations) leans on the integration techniques from Chapter 5, and Chapter 8 (Taylor Series) leans on the geometric-series and general-series ideas it introduces in its own early sections plus the tangent line idea first introduced in Chapter 1 — read those chapters in order rather than skipping ahead.

One honesty note: the WeBWorK exercises and the print Activities Workbook are optional classroom supports built by the author, not requirements — every section, exercise, and activity in this book is fully readable and workable for free in the HTML and PDF editions without either one.

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Used In These Programs

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

Who Should Read This

Active Calculus is written for a first- or second-year student taking their required Calculus I or Calculus II course in a BSCS, BS Mathematics, or Engineering program — it assumes only a solid pre-calculus/algebra background, not any prior calculus. It suits a complete beginner well, since every section opens with a Preview Activity that builds intuition graphically before any formal definition or rule is introduced. It also suits a student who wants extra worked practice, since each section pairs a small set of anonymous WeBWorK problems (unlimited attempts, immediate feedback) with several longer, written-response exercises. Students who prefer a fully worked-example-driven text with less activity-based self-discovery may want to pair this book with supplementary lecture notes.

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Applicable Universities

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

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Active Calculus (Single Variable), 2nd Edition, by Matthew Boelkins, with contributions from David Austin and Christina Safranski. Free under a Creative Commons Attribution-ShareAlike licence (no specific version stated by the publisher). Official source: https://activecalculus.org/