Calculus Study Guide and Chapter Summaries – David Guichard (Single Variable, Early Transcendentals) BS Math/BSCS

BS Math, BSCS and Engineering students can use this guide to study Calculus by David Guichard of Whitman College. This is the single-variable, early-transcendentals edition, so it covers exactly the two courses usually called Calculus I and Calculus II: limits and derivatives, curve sketching and applications, integration and its techniques, polar and parametric curves, and sequences and series up to Taylor’s Theorem. It does not include multivariable calculus.

The writing reads like a patient lecture. Ideas are discovered before they are stated: the Product Rule appears after the obvious guess fails, and the derivative of sine is built step by step around the one hard limit. Chapter 1 refreshes lines, circles and graph shifts, so first-year students who are shaky on precalculus can start there. The exercises come at the end of each section, and answers to the marked ones are collected in Appendix A, which suits students who want plenty of practice on their own.

Book Overview

Chapter List

Chapter 1: Analytic Geometry

Difficulty: Easy · Key topics: slope and lines, point-slope form, distance formula, circles, functions and domains, piecewise functions, shifts and dilations of graphs

The book opens with a short refresher of the coordinate geometry that every later chapter quietly relies on. It defines the change in x and y, slope as their ratio, and the equation of a line, with a tax-bracket example showing why slope can be read as a rate. Next come the distance formula and the equation of a circle, including how completing the square reveals a centre and radius. Section 1.3 treats a function as a rule, introduces domains (including domains restricted by a story problem) and piecewise formulas. The chapter ends with how replacing x or y in a formula shifts, stretches or flips its graph.

Key Points:

• The slope of the line through two points is m = (y2 - y1)/(x2 - x1), and a line can be written y = mx + b.
• The distance between two points comes from the Pythagorean theorem, and a circle with centre (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2.
• Completing the square turns an expanded circle equation into one where the centre and radius can be read off.
• The domain of a function is every input where the formula works, unless a story problem restricts it further (a box cannot have negative height).
• Replacing x by x - C moves a graph C units right; replacing y by y - D moves it D units up.
• Dilations replace x by x/A or y by y/B to stretch or squeeze, and a negative factor also flips the graph.

Memory Tip: Inside the bracket, the sign does the opposite of what it looks like: f(x - 2) slides the graph to the right, because every point now needs an input 2 larger to produce the old output.

Common Mistake: Treating f(x) + 2 and f(x + 2) as the same thing; the first lifts the graph two units, the second slides it two units to the left.

Important Questions:

• What is the slope of the line through (0, 1) and (2, 7)? The change in y is 6 and the change in x is 2, so the slope is 3.
• What does the graph of y = f(x – 3) + 1 look like compared with y = f(x)? It is the same shape moved 3 units to the right and 1 unit up.

Chapter 2: Instantaneous Rate of Change: The Derivative

Difficulty: Medium · Key topics: difference quotient, tangent line, instantaneous velocity, limits, one-sided limits, the derivative function, derivative notation, continuity, differentiability, Intermediate Value Theorem

Chapter 2 builds the derivative from a single question: how sensitive is y to a small change in x? Chords of a semicircle swing toward a tangent line as the gap shrinks, and a dropped ball shows why speed at one instant cannot be read from any average speed. That motivates limits, which are first explored informally and then defined precisely, with limit laws, one-sided limits and limits of composed functions. The derivative function f′ is then defined and given its alternative notations. The chapter closes with the vocabulary of bounded, continuous and differentiable functions and the Intermediate Value Theorem.

Key Points:

• The derivative is the limit of the difference quotient: f′(x) = lim (f(x + Δx) - f(x))/Δx as Δx → 0.
• Geometrically it is the slope of the tangent line; physically it is instantaneous velocity or any other instantaneous rate.
• A limit describes where the values are heading, not what happens at the point, which is why 0/0 can still have a perfectly good limit.
• The same derivative appears as f′(x), dy/dx, d/dx f(x) and y′; the book asks you to get comfortable with all of them.
• A function is continuous at a when lim f(x) = f(a), and differentiable at a when f′(a) exists.
• Intermediate Value Theorem: a continuous function on [a, b] takes every value between f(a) and f(b), which the book uses to trap a root to one decimal place.

Memory Tip: Picture the chord and the tangent: the difference quotient is the slope of a chord, and the derivative is what that slope settles on when the second point slides in to meet the first.

Common Mistake: Trying to find a limit by substituting the value when it gives 0/0 and then concluding the limit does not exist; the fraction usually simplifies first and then the limit is easy to see.

Important Questions:

• What does the derivative of a position function tell you? The instantaneous velocity: the limit of the average velocity over shorter and shorter time intervals.
• How can the Intermediate Value Theorem show that x^3 + x – 1 has a root between 0 and 1? The function is continuous, it equals -1 at 0 and 1 at 1, so it must cross 0 somewhere in between.

Chapter 3: Rules for Finding Derivatives

Difficulty: Medium · Key topics: Power Rule, linearity of the derivative, Product Rule, Quotient Rule, Chain Rule, composition of functions

This chapter turns the limit definition into a toolkit so that you rarely have to use it again. The Power Rule is proved for positive whole-number exponents by expanding (x + Δx)^n and then stated for any exponent. Linearity covers constant multiples and sums. The Product Rule is discovered rather than announced: the book tests the tempting guess of multiplying the two derivatives, shows it fails, and derives the correct rule. The Quotient Rule is built from the Product Rule plus the derivative of 1/g, and the Chain Rule handles compositions such as the square root of 625 minus x squared.

Key Points:

• Power Rule: d/dx x^n = n x^(n-1), valid for integer, fractional, negative and even irrational exponents.
• The derivative is linear: (cf)′ = c f′ and (f + g)′ = f′ + g′.
• Product Rule: (fg)′ = f′g + fg′; the derivative of a product is not the product of the derivatives.
• Quotient Rule: (f/g)′ = (f′g - fg′)/g^2, and it follows from the Product Rule.
• Chain Rule: d/dx f(g(x)) = f′(g(x)) g′(x), the derivative of the outside function evaluated at the inside, times the derivative of the inside.
• Composition order matters: f(g(x)) and g(f(x)) are different functions with different derivatives.

Practice Tip: Compute a product two ways, once by multiplying out and once with the Product Rule, and check that the answers match. The book does exactly this to show why the guess f′g′ fails.

Common Mistake: Differentiating the outside function of a composition and forgetting to multiply by the derivative of the inside.

Important Questions:

• What is the derivative of x^2 (x + 1)? By the Product Rule it is 2x(x + 1) + x^2 = 3x^2 + 2x.
• What is the derivative of the square root of (1 + x^2)? By the Chain Rule it is x divided by the square root of (1 + x^2).

Chapter 4: Transcendental Functions

Difficulty: Hard · Key topics: radians, derivatives of sine and cosine, the limit sin(x)/x, Squeeze Theorem, exponential and logarithmic functions, the number e, implicit differentiation, inverse trigonometric functions, limits at infinity, L’Hôpital’s Rule, hyperbolic functions

The longest differentiation chapter extends the toolkit to trigonometric, exponential, logarithmic and related functions. It starts with radians and the unit circle, then spends three short sections on the derivative of sine: the proof needs the hard limit of sin(x)/x, which is settled with the Squeeze Theorem. Exponentials come next, with e defined as the base whose derivative equals itself, then implicit differentiation, which gives the derivative of the natural logarithm and the inverse trigonometric functions. The chapter also widens the idea of a limit to infinity, introduces L’Hôpital’s Rule and ends with the hyperbolic functions.

Key Points:

• Calculus formulas for trigonometric functions assume radian measure; degrees would add an awkward constant everywhere.
• lim sin(x)/x = 1 as x → 0 is the one genuinely hard limit, and it leads to d/dx sin x = cos x and d/dx cos x = -sin x.
• The derivative of a^x is a constant times a^x; the number e is the base where that constant is exactly 1, so d/dx e^x = e^x.
• Implicit differentiation differentiates both sides of an equation and treats y as a function of x, and it yields d/dx ln x = 1/x.
• Inverse trigonometric functions are defined by restricting sine, cosine and tangent to intervals where they are one-to-one, which is why they only undo the truncated function.
• L’Hôpital’s Rule: if a quotient gives 0/0 or an infinity over an infinity, you may differentiate the top and bottom separately and take the limit again.
• cosh x = (e^x + e^(-x))/2 and sinh x = (e^x - e^(-x))/2 satisfy cosh^2 x - sinh^2 x = 1.

Memory Tip: Remember e as the base whose graph is its own slope: at the point (z, e^z) the tangent has slope e^z, no matter what z is.

Common Mistake: Using the Power Rule on 2^x as if it were x^2; the variable is in the exponent, so the derivative is a constant multiple of 2^x itself.

Important Questions:

• Why is the limit of sin(x)/x as x approaches 0 so important? It is the key step in proving that the derivative of sine is cosine, and its value is 1.
• When can L’Hôpital’s Rule be used? When the quotient has the form 0/0 or infinity/infinity, and the limit of the quotient of derivatives exists.

Chapter 5: Curve Sketching

Difficulty: Medium · Key topics: local maxima and minima, critical values, Fermat’s Theorem, first derivative test, second derivative test, concavity, inflection points, vertical and horizontal asymptotes, even and odd functions

Chapter 5 uses the first and second derivatives to read the shape of a graph. It defines local maximum and minimum points and shows that, at an interior extremum of a differentiable function, the derivative must be zero, so the candidates are the critical values. The first derivative test decides which candidates are maxima or minima by watching the sign change of f′. The second derivative test and concavity then use the sign of f′′, and inflection points mark where concavity changes. The last section adds a checklist: asymptotes, intercepts, and symmetry from even and odd functions.

Key Points:

• A local extremum of a differentiable function can only occur where f′(x) = 0 (Fermat’s Theorem), but a zero derivative does not guarantee an extremum.
• First derivative test: if f′ changes from positive to negative the point is a local maximum; from negative to positive, a local minimum.
• Second derivative test: with f′(a) = 0, a negative f′′(a) means a maximum and a positive one means a minimum; if f′′(a) = 0 the test says nothing.
• Where f′′ > 0 the graph is concave up; where f′′ < 0 it is concave down; an inflection point is where the concavity changes.
• A vertical asymptote often sits where a denominator is zero, but not always; (sin x)/x has a zero denominator at 0 and no asymptote.
• An even function satisfies f(-x) = f(x) (symmetric about the y-axis); an odd function satisfies f(-x) = -f(x) (symmetric about the origin).

Memory Tip: Think of f′ as the height of a second graph: wherever it crosses zero going down, the original function has just topped out.

Common Mistake: Declaring an extremum wherever f′(x) = 0; for x^3 the derivative is zero at the origin yet the function keeps increasing.

Important Questions:

• How do you find and classify the local extrema of a function? Solve f′(x) = 0 (and check where f′ is undefined), then use the sign change of f′ or the sign of f′′ at each critical value.
• What is an inflection point? A point on the graph where the concavity changes, so f′′ changes sign there.

Chapter 6: Applications of the Derivative

Difficulty: Medium · Key topics: optimization, Extreme Value Theorem, story problems, related rates, Newton’s method, linear approximation, differentials, Rolle’s Theorem, Mean Value Theorem

This chapter puts the derivative to work on applied problems. Optimization asks for the largest or smallest value of a function on an interval, using the Extreme Value Theorem and worked story problems such as the smallest perimeter for a given area, the cone inside a sphere, cylindrical containers and the quickest route to a point across the sand from a road. Related rates uses the Chain Rule when several quantities change with time. Newton’s method repeatedly follows tangent lines to approximate roots, and linear approximation and differentials extend that idea. The chapter closes with Rolle’s Theorem and the Mean Value Theorem.

Key Points:

• Extreme Value Theorem: a continuous function on a closed interval [a, b] has both a maximum and a minimum there, so check critical values and the endpoints.
• A good optimization solution writes one quantity as a function of one variable, finds the domain from the story, and then applies calculus.
• Related rates: write an equation linking the quantities, differentiate with respect to time, then substitute the known values; a dot over a letter is Newton’s shorthand for the rate dx/dt.
• Newton’s method improves a guess by moving to where the tangent line crosses the x-axis, and repeating.
• The linear approximation of f at a is L(x) = f′(a)(x - a) + f(a), and differentials define dy = f′(x) dx.
• Mean Value Theorem: some point c in (a, b) has f′(c) = (f(b) - f(a))/(b - a); two functions with the same derivative differ by a constant.

Practice Tip: Start any story problem by drawing a diagram and naming every quantity with a letter, as the book’s introduction advises, before writing a single equation.

Common Mistake: In related rates, substituting the numbers before differentiating, which turns a changing quantity into a constant and kills its derivative.

Important Questions:

• Among all rectangles with a fixed perimeter, which has the largest area? The square; writing area as a function of one side and setting its derivative to zero shows the sides must be equal.
• What does the Mean Value Theorem say about a car that averages 70 miles per hour on a trip? At some moment the speedometer must read exactly 70.

Chapter 7: Integration

Difficulty: Medium · Key topics: antiderivatives, area under a curve, sums of approximations, definite integral, Fundamental Theorem of Calculus, net area, properties of integrals

Chapter 7 runs the derivative backwards. Section 7.1 sets up two problems that look different but turn out to be the same: finding the distance traveled from a speed, and finding the area under a curve. Both are approximated by dividing an interval into small pieces and adding. Section 7.2 names the limit of those sums the definite integral and shows that it equals F(b) – F(a) for any antiderivative F, then states the second form of the Fundamental Theorem. Section 7.3 explains why integrals measure net distance and net area, and what happens when the limits are swapped.

Key Points:

• An antiderivative of f is any function F with F′ = f; two antiderivatives differ by a constant.
• The definite integral is the limit of sums f(t) Δt over smaller and smaller subintervals.
• Fundamental Theorem, evaluation form: ∫ f(x) dx from a to b equals F(b) - F(a).
• Fundamental Theorem, derivative form: if G(x) is the integral of f from a to x, then G′(x) = f(x).
• An integral of velocity gives net distance: motion backwards counts as negative, so total distance needs the pieces added with their signs removed.
• An integral measures net area, with regions below the x-axis counting as negative, and swapping the limits of integration multiplies the answer by -1.

Memory Tip: Differentiation and integration undo each other, and the constant of integration drops out of F(b) - F(a), so any antiderivative will do.

Common Mistake: Treating the integral of a velocity that changes sign as the total distance traveled; it only gives the net change in position.

Important Questions:

• How do you compute the definite integral of 3t^2 from 1 to 2? An antiderivative is t^3, so the answer is 8 – 1 = 7.
• Why can an integral of a positive-looking curve come out negative? Parts of the curve below the x-axis count as negative area, so the integral reports net area, not total area.

Chapter 8: Techniques of Integration

Difficulty: Hard · Key topics: substitution, powers of sine and cosine, trigonometric substitution, integration by parts, partial fractions, trapezoid rule, Simpson’s rule, error estimates

Because the Fundamental Theorem needs an antiderivative, this chapter collects the standard ways of finding one. Substitution recognises a Chain Rule pattern in reverse. Powers of sine and cosine are handled with identities, and trigonometric substitution replaces x by a sine or a tangent to remove square roots. Integration by parts reverses the Product Rule, and partial fractions break a rational function with a quadratic denominator into simple pieces. For functions with no simple antiderivative, the trapezoid and Simpson’s rules approximate the integral, each with an error theorem. Section 8.7 is a mixed set of review problems.

Key Points:

• Substitution undoes the Chain Rule: look for an inner function whose derivative also appears in the integrand.
• Odd powers of sine or cosine split off one factor and use sin^2 x + cos^2 x = 1; even powers use the double-angle identities.
• Trigonometric substitution replaces x with a trig function so that expressions like 1 - x^2 become perfect squares; the new variable’s range decides the sign of the square root.
• Integration by parts: ∫ u dv = uv - ∫ v du, a rewriting of the Product Rule for the case where the new integral is easier.
• Partial fractions in this book are worked out when the denominator is a quadratic, and a rational function whose degrees are too large is divided first.
• The trapezoid and Simpson’s rules approximate definite integrals, and their error bounds depend on the second and fourth derivatives respectively.

Memory Tip: Pick the technique by the shape of the integrand: a function and its inner derivative means substitution, a product of unlike functions suggests parts, a fraction of polynomials suggests partial fractions.

Common Mistake: Forgetting to change the limits (or convert back to x) after a substitution in a definite integral.

Important Questions:

• How do you integrate x e^x? Use integration by parts with u = x and dv = e^x dx, giving x e^x – e^x + C.
• Why do we need numerical integration? Some functions have no simple antiderivative, so the trapezoid or Simpson’s rule approximates the area, with a known error bound.

Chapter 9: Applications of Integration

Difficulty: Medium · Key topics: area between curves, velocity and position, volumes by cross-sections, disk and shell methods, average value, work, center of mass, improper integrals, probability density functions, arc length, surface area

Chapter 9 shows that the same recipe — cut into thin pieces, approximate each, add, take a limit — solves many different problems. It begins with the area between two curves and with recovering position from velocity. It then computes volumes from cross-sections, including solids of revolution by disks and shells, followed by the average value of a function, work done by a variable force, and the balance point of a beam or region. Section 9.7 pairs kinetic energy with improper integrals, and the probability section builds density functions, means and standard deviations. The chapter ends with arc length and surface area.

Key Points:

• The area between curves is the integral of upper minus lower, and it makes no difference whether you subtract the integrals or integrate the difference.
• Volumes by cross-sections integrate the area of a slice; a solid of revolution can be sliced into disks or washers or unrolled into cylindrical shells, and the answers agree.
• The average value of f on [a, b] is the integral of f divided by b - a.
• Work is force times distance, so with a force that varies along the path it becomes an integral of the force.
• Center of mass balances moments (mass times distance); the book builds the idea from weights on a beam before moving to regions.
• An improper integral is a limit of ordinary integrals; it converges if the limit is a finite number and diverges otherwise.
• A probability density function is non-negative with total integral 1, and the book computes the mean and standard deviation for uniform, exponential and normal distributions.
• Arc length and surface area come from approximating a curve by line segments and a surface by pieces of cones.

Memory Tip: Whenever a formula in this chapter looks new, it is the same four-step pattern: slice, approximate each slice, add, take the limit.

Common Mistake: Integrating the wrong way round for the area between curves (lower minus upper), or ignoring that the curves cross inside the interval, so the pieces must be split at the crossing.

Important Questions:

• How do you find the area between y = x and y = x^2? They meet at x = 0 and x = 1, with x above x^2 in between, so the area is the integral of (x – x^2), which is 1/6.
• What makes an integral improper? An infinite limit of integration (or an unbounded integrand); it is defined as a limit and may converge or diverge.

Chapter 10: Polar Coordinates, Parametric Equations

Difficulty: Medium · Key topics: polar coordinates, polar graphs, slopes in polar coordinates, areas in polar coordinates, parametric equations, cycloid, slopes and areas for parametric curves

Chapter 10 changes how a plane curve is described. Polar coordinates give a point by its distance r from the origin and its angle θ, and the section on them explains how they relate to rectangular x and y. Because x = r cos θ and y = r sin θ, a polar curve can be treated as a parametric curve, which gives its slope with the Chain Rule. Areas in polar coordinates use thin circular sectors instead of rectangles. The final two sections introduce parametric equations, where x and y both depend on a parameter t, and compute slopes and areas for them with a cycloid as the running example.

Key Points:

• Rectangular and polar coordinates are linked by x = r cos θ and y = r sin θ.
• For a polar curve r = f(θ), write x = f(θ) cos θ and y = f(θ) sin θ and use dy/dx = (dy/dθ)/(dx/dθ).
• Area in polar coordinates sums sectors of a circle, where a sector of angle α and radius r has area α r^2 / 2.
• Parametric equations describe a curve by x = f(t) and y = g(t), and t can be read as time.
• For parametric curves, dy/dx = (dy/dt)/(dx/dt); a horizontal tangent needs dy/dt = 0 with dx/dt nonzero.
• At points where the slope formula gives 0/0, such as the points where the cycloid touches the x-axis, the book notes there is no tangent line.

Memory Tip: Polar area uses pie slices, not rectangles, which is why the integrand is r^2/2 rather than r.

Common Mistake: Computing the slope of a polar curve as dr/dθ; that is only how fast the radius changes, not the slope of the curve in the xy-plane.

Important Questions:

• How do you convert the polar point (r, θ) to rectangular coordinates? Use x = r cos θ and y = r sin θ.
• How do you find the slope of a parametric curve? Divide dy/dt by dx/dt at the value of t you care about.

Chapter 11: Sequences and Series

Difficulty: Hard · Key topics: sequences, limits of sequences, series and partial sums, geometric series, integral test, p-series, alternating series, comparison tests, absolute convergence, ratio and root tests, power series, Taylor series, Taylor’s Theorem

The final chapter is the longest and treats sequences as functions on the natural numbers, whose limits behave like the limits already studied. A series is then defined through its sequence of partial sums, with the geometric series as the first complete example. Most of the chapter is about deciding whether a series converges, using the integral test, the alternating series test, comparison tests, absolute convergence, and the ratio and root tests. Power series turn the idea into functions with a radius of convergence, and can be differentiated and integrated term by term. Taylor series and Taylor’s Theorem then give approximations with an error bound.

Key Points:

• A series converges when its sequence of partial sums s_n converges; a geometric series with ratio x converges when |x| < 1.
• Integral test: for positive decreasing terms the series and the matching improper integral converge or diverge together; a p-series 1/n^p converges exactly when p > 1.
• Alternating series test: if the terms alternate in sign and shrink to zero, the series converges, and the error is at most the size of the next term.
• Comparison tests compare a new series with a known one; absolute convergence (the series of absolute values converges) guarantees convergence of the original.
• Ratio test: if lim |a(n+1)/a(n)| = L, the series converges for L < 1 and diverges for L > 1; the root test is the same idea with |a(n)|^(1/n).
• A power series has a radius of convergence R, and inside that radius it can be differentiated or integrated term by term.
• The Maclaurin series of sin x uses only odd powers and that of e^x has coefficients 1/n!; Taylor’s Theorem bounds the error of a partial sum.

Memory Tip: Keep a short list of known examples — geometric, p-series, alternating harmonic — and compare every new series with the closest one; most tests are just that comparison in disguise.

Common Mistake: Concluding that a series converges because its terms go to zero; the harmonic series 1/n has terms that go to zero and still diverges.

Important Questions:

• For which values of p does the series of 1/n^p converge? Exactly when p is greater than 1, by the integral test.
• How do you find the radius of convergence of a power series? Apply the ratio test to the terms and solve L < 1 for x; then check the endpoints separately if the interval is needed.

Appendices A and B: Selected Answers and Useful Formulas

Appendix · Difficulty: Easy · Key topics: answers to selected exercises, algebra rules, geometry formulas, analytic geometry, trigonometric identities

Appendix A collects answers to the exercises marked with an arrow in the text, organised by chapter and section, so you can check your result after you have tried a problem. The book says plainly that answers are meant as a final check, not a starting point. Appendix B is a compact reference of formulas you are expected to know or look up: algebra (including completing the square and the quadratic formula), areas and volumes of common shapes, the equations of lines, circles and ellipses, and a trigonometry sheet with identities and the law of cosines.

Key Points:

• Only exercises marked with an arrow have a published answer; the rest are left for you to check on your own.
• Answers appear in the same chapter and section numbering as the exercises, for example 11.10.1.
• Appendix B gathers the algebra, geometry and trigonometric identities that Chapters 4, 8 and 9 constantly use, but it does not contain a table of derivatives or integrals.
• Use the index when a term is unfamiliar; it points to the page where the definition first appears.

Practice Tip: Cover the printed answer, finish the problem, and only then look; if you do not match, redo the whole problem instead of tweaking your answer toward the key.

Common Mistake: Reading a worked answer in Appendix A before attempting the problem, which feels productive but skips the practice the book tells you to do every day.

Important Questions:

• How should you use the selected answers? As a final check after you have solved the problem yourself; the book says this directly in its introduction.
• Where can you look up a trig identity or the volume of a cone? In Appendix B, Useful Formulas, near the end of the book.

How to Study This Book

This PDF covers single-variable calculus only, which means the two courses usually called Calculus I and Calculus II. The book does not mark a boundary between the two, so check your own syllabus. A common split is Calculus I through the derivative chapters (1–6) and the opening of integration (7), with Calculus II taking Chapters 8 to 11, but your university may move the line by a chapter either way. Multivariable calculus is in a separate edition and is not in this file.

Read Chapter 1 quickly rather than skipping it. It is precalculus, but the shifting and dilating rules in Section 1.4 return when you sketch graphs in Chapter 5, and the habit of naming variables in a story problem pays off in Chapter 6. If you are confident with lines, circles and function notation, skim it and start properly at Chapter 2.

Chapters 2 to 4 are the foundation for everything after them. Spend time on the limit ideas in Section 2.3 and on the hard limit in Section 4.3, and learn the rules of Chapter 3 well enough that you can differentiate without thinking. Chapter 4 is long, so do not try to read it in one go; the three sections on the derivative of sine are really one argument and are best read together, and the hyperbolic functions in Section 4.11 can wait until a course actually asks for them.

Chapters 5 and 6 are where derivatives become useful, and they depend on each other: Chapter 5’s critical values and the first and second derivative tests are used in the optimization problems of Section 6.1. Work the story problems with a diagram. Related rates and Newton’s method in Chapter 6 are often examined, but the Mean Value Theorem in Section 6.5 matters more for later theory.

Chapter 7 is short but it is the hinge of the whole book: the Fundamental Theorem in Section 7.2 is what makes Chapters 8 and 9 possible. Chapter 8 is a toolbox, and the only way to learn it is to practise recognising which method fits; Section 8.7 is a good mixed review. In Chapter 9, you do not need to memorise every formula, because most come from the same slice-and-add idea, but the probability section (9.8) and the improper integrals of Section 9.7 are worth extra time if your course includes them.

Chapter 10 is often taught briefly. Polar coordinates and parametric equations connect to Chapter 11’s power series only loosely, so you can read it independently. Chapter 11 is the hardest material in the book, and it builds in order: sequences, then series, then the convergence tests, then power series, then Taylor series. Do not jump to the Ratio Test without the geometric series and the integral test behind you.

The book’s introduction says the best way to learn is to do many problems every day. It has no solutions manual beyond the selected answers in Appendix A, so if you need worked solutions, pair it with a book that includes more of them. It also does not cover differential equations or multivariable topics in this PDF, and it covers numerical integration only by the trapezoid and Simpson rules.

Used In These Programs

This book is used for the Calculus I and Calculus II courses in: BS Mathematics · BS Computer Science, and in BS Engineering programs that start with a two-semester single-variable calculus sequence. Browse all Calculus books or all Mathematics books.

Who Should Read This

This book suits a first-year BS Math, BSCS or Engineering student taking the two-semester single-variable calculus sequence who wants a clear, conversational text that reads like lecture notes and is free to download. It assumes algebra, trigonometry and function notation, and Chapter 1 reviews the geometry you need. It is probably not enough on its own for a student who wants hundreds of worked solutions, for a course that requires differential equations or multivariable calculus, or for a student who prefers a more guided, activity-based approach. For a more conventional textbook with a long exercise bank, compare it with Calculus Volume 1 from OpenStax, and when you are ready for multivariable calculus, move on to Calculus Volume 3. Students preparing for entry tests that include calculus will find the derivative and integration chapters especially useful.

Applicable Universities

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

Official Copy of Calculus (Guichard)

This book is free from its official source. Click below to open the official Whitman College page for the single-variable book, where you can download the full PDF or individual chapters.

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Guichard’s Calculus is a free, readable single-variable text for the full first-year calculus sequence, with plenty of exercises and answers to the marked ones. Browse more Mathematics books for the rest of your semester.

Single Variable Calculus, Early Transcendentals, by David Guichard, Whitman College. Free under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 licence (CC BY-NC-SA 4.0). Official source: https://www.whitman.edu/mathematics/calculus/