Book: Calculus (Apex) - Mathematics

Book: Calculus (Apex) - Mathematics

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This text comprises a three–volume series on Calculus. The first part covers material taught in many “Calculus 1” courses: limits, derivatives, and the basics of integration, found in Chapters 1 through 6. The second text covers material often taught in “Calculus 2”: integration and its applications, along with an introduction to sequences, series and Taylor Polynomials, found in Chapters 7 through 10. The third text covers topics common in “Calculus 3” or “Multivariable Calculus”: parametric equations, polar coordinates, vector–valued functions, and functions of more than one variable, found in Chapters 11 through 15. All three are available separately for free.

Printing the entire text as one volume makes for a large, heavy, cumbersome book. One can certainly only print the pages they currently need, but some prefer to have a nice, bound copy of the text. Therefore this text has been split into these three manageable parts, each of which can be downloaded separately.

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Corral, Michael Dawkins, Paul Christian Heinold, Brian Siemers, Troy Chalishajar, Dimplekumar Bowen, Jennifer(Jennifer Lynne)



Starting with the source at, faculty at the University of North Dakota made several substantial changes to create APEX Late Transcendentals. The most obvious change was to rearrange the text to delay proving the derivative of transcendental functions until Calculus 2. UND added Sections 7.1 and 7.3, adapted several sections from other resources, created the prerequisite sections, included links to videos and Geogebra, and added several examples and exercises. In the end, every section had some changes (some more substantial than others), resulting in a document that is about 20% longer. The source files can now be found at

Extra thanks are due to Michael Corral for allowing us to use portions of his Vector Calculus, available at (specifically, Sections 11.7, 13.9, and 14.7, and Chapter 15) and to Paul Dawkins for allowing us to use portions of his online math notes from (specifically, Sections 8.5 and 9.7, as well as “Area with Parametric Equations” in Section 10.3). The work on Calculus III was partially supported by the NDUS OER Initiative.

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Inside the AP Calculus AB Exam

Before you plunge into studying for the AP Calculus AB exam, let's take a step back and look at the big picture, What's the AP Calculus AB exam all about? How can I prepare for it? How's it scored? This chapter and the next will answer these questions and more.

What is the Advanced Placement (AP) Program?

Through the Advanced Placement Program, you can take college-level courses while you are in high school. Based on your grade on an AP Exam, colleges and universities can grant you placement or college credit or both.

In addition to getting a head start on your college coursework, you can improve your chances of acceptance to competitive schools since colleges know that AP students are better prepared for the demands of college courses. There's also the money you can save on tuition if you receive credit!

About the AP Calculus AB Exam

The AP Calculus AB exam is administered in May by the College Board's AP Services. The exam is 3 hours and 15 minutes long, so developing your stamina is very important. The exam consists of two equally weighted sections: Section I, Multiple-Choice, and Section II, Free-Response.

Section I -- Multiple-Choice

There are 45 multiple choice questions, This section is split into two parts: part A consists of 28 questions with no calculator allowed (55 minute time limit) and part B consists of 17 questions for which a calculator is required (50 minute time limit). Each multiple choice question is worth 1 point.

The second section of the examination consists of 6 free-response questions worth 9 points each. Part A consists of 3 questions for which a calculator is required (45 minutes). Part B consists of 3 questions for which calculators may be not be used (45 minutes). During Part B, you may go back to the first three questions in Part A if you have time, but you will not be permitted to use your calculator.

There are two AP Calculus exams and the AB distinguishes this exam from the AP Calculus BC exam. The AB exam covers one semester of college-level calculus, whereas the BC exam covers a full year of college calculus, although individual colleges may differ in their course offerings and placement/credit policies. You may take only one of the two exams in any given year.

The AP Calculus exam is scored on a scale of 1 to 5, with 5 being the highest grade. The scores are defined as follows:

5 Extremely well qualified
4 Well qualified
3 Qualified
2 Possibly qualified
1 No recommendation

In 2000, the average grade of the 137,276 candidates who took the AB test was 3.03. Typically, to obtain a score of 3 or higher, you need to answer about 50 percent of the multiple choice section correctly, and do acceptable work on the free-response section. Remember that you can miss questions and still get a perfect 5 score.

Raw scores are calculated in points. The multiple-choice section of the AP Calculus exam is worth the same number of points as the free-response section. There is a 1.2 weighting factor used so that the multiple-choice and free-response sections of the exam have equal weight. Each section is worth 54 points, for a total of 108 points.

It would not be correct to say that the AP examination is graded on a curve. How other students do on the same exam will not affect your score. However, some multiple-choice questions are used from one year to the next to allow for a calibration of the scores so that a 4 one year reflects the same statistical strength of performance. For this reason, the cut-offs for each score level do not stay constant from year to year.

For multiple choice questions, there is a penalty for incorrect answers as opposed to simply leaving answers blank. This is sometimes called a guessing penalty, but it is really a wrong answer penalty. If you guess fight, you're in great shape!

Here's how the scoring works: You receive 1 point for a correct answer, 0 points for no answer, and -1/4 point for a wrong answer. For example, getting 30 correct and 15 wrong would give you a score of 30-(15x1/4) = 26.25. We'll talk more about guessing on the AP Calculus AB multiple-choice section in the next chapter, Taking AP Calculus AB -- Strategies for Success.

* Two calculators (see below) and batteries
* Photo I.D. (driver's license, school I.D., or a valid passport are acceptable)
* Your secondary school code number (see your Guidance Counselor or AP Coordinator)
* Your Social Security number
* Several sharpened No. 2 pencils
* Pencil sharpener
* Eraser
* A watch (in case your exam room doesn't have a clock you can see easily. You need to be able to pace yourself during this long test!)

* DON'T bring scratch paper. You'll make your notes in the test booklet in the spaces provided.
* Don't bring books, compasses, correction fluid, dictionaries, highlighters, notes, or rulers.
* Don't bring beepers or cellular phones, or watches that have beepers or alarms.
* Don't bring food or drinks.
* Don't even wear a T-shirt with math of any kind on it!

You can only bring two calculators, both of which must be approved types. To be an approved calculator, it should be able to:

* Produce the graph of a function within an arbitrary viewing window.
* Find the zeros of a function.
* Compute the derivative of a function numerically.
* Compute definite integrals numerically.

You can't use a calculator with a QWERTY keypad, because it makes it too easy to type text (such as test questions) into the calculator.

The multiple-choice section of the exam is graded by computer. The free-response booklets are graded by faculty consultants. These are college professors and AP teachers who are specially trained to assess the questions. Usually about six faculty consultants review each free-response booklet.

AP Grade Reports are sent in July to each student's home, high school, and any colleges designated by the student. Students may designate the colleges they would like to receive their grade on the answer sheet at the time of the test. Students may also contact AP Services to forward their grade to other colleges after the exam, or to cancel or withhold a grade.

AP Grades by phone are available for $13 a call beginning in early July. A touch-tone phone is needed. The toll-free number is (888) 308-0013.

To register for the AP Calculus AB exam, contact your school guidance counselor or AP Coordinator. If your school does not administer the exam, contact AP Services for a listing of schools in your area that do.

The fee for each AP Exam is $77. The College Board offers a $22 credit to qualified students with acute financial need. A portion of the exam fee may be refunded if a student does not take the test. There is a $20 late fee for late exam orders. Check with AP Services for applicable deadlines.

The College Board offers a number of publications about the Advanced Placement Program, including: Advanced Placement Program Course Description -- Calculus, A Guide to the Advanced Placement Program, and the AP Bulletin for Students and Parents.

For more information about the AP Program and/or the AP Calculus AB exam, contact your school's AP Coordinator or guidance counselor, or contact AP Services at:

Table of Contents

  • Chapter 1: Limits
  • Chapter 2: Derivatives
  • Chapter 3: The Graphical Behavior of Functions
  • Chapter 4: Applications of the Derivative
  • Chapter 5: Integration
  • Chapter 6: Techniques of Antidifferentiation
  • Chapter 7: Applications of Integration
  • Chapter 8: Sequences and Series
  • Chapter 9: Curves in the Plane
  • Chapter 10: Vectors
  • Chapter 11: Vector Valued Functions
  • Chapter 12: Functions of Several Variables
  • Chapter 13: Multiple Integrations
  • Chapter 14: Vector Analysis

Other Texts

APEX PreCalculus: A precalc book written by my colleague Amy Chapman, with Meagan Herald & Jessica Libertini. It matches the look of APEX Calculus and covers introductory material well. Available at for under $8. Source files are available on Github.

Active Calculus: A full calculus text written by Matt Boelkins with David Austin and Steve Shlicker, written in an "active learning" style. This is intentionally not a traditional calculus text. It is very well written with Geogebra and WeBWorK support. It is very popular the .pdf's have 70,000+ downloads and a number of schools use it as their calc text. Available at for under $20 find out more at

Fundamentals of Matrix Algebra: A text I've written for students needing a background in matrix operations without most of the theory included with "Linear Algebra" courses. The text covers Gaussian Elimination, matrix algebra, determinants, eigenvalues/vectors and a few other topics. Available at for $10 download a pdf version here.

An Introduction to MATLAB and Mathcad: A text written by my colleague Troy Siemers that covers the basics of using MATLAB and Mathcad, along with basic programming principles/structures such as if/then, loops, etc. Available at for $8 download a pdf version here.

Essential Precalculus: Another precalculus book produced in the APEX Calculus style, by Sean Fitzpatrick, based on the precalculus text by Stitz and Zeager. He has also adapted the APEX Calculus texts to better suit the needs of his classes/students at the University of Lethbridge see more of his work here.


The PDF's are free. If you want to support ongoing improvements to the text, or just want to say "Thank you", click the "Support APEX Calclulus" button below. Any and all support is appreciated.

What's different about the b&w versions?

Printers do a fine job of printing b&w versions of color pages, though we are all familiar with the way certain detail is lost. In particular, pure blue, red and green portions of a picture are rendered in the same shade of gray. As the default pen colors of graphs in APEX Calculus are pure blue and red, each will be printed as the same shade of gray, making it difficult to distinguish which curve in a graph is which.

So when I created the graphics for the text, I produced two copies of each picture. One had a "color" color scheme, and the other had a "b&w" color scheme. The result is that the "b&w" figures look better printed on a grayscale printer than the color figures do.

Best Calculus Books for Beginners

Book Name & Author Image Rating Price
The Calculus Lifesaver: All the Tools you Need to Excel at Calculus (Princeton Lifesaver Study Guides) by Adrian Banner 9.2 View on Amazon
Essential Calculus Skills Practice Workbook with Full Solutions by Chris McMullen 9.2 View on Amazon
The Humongous Book of Calculus Problems by W. Michael Kelley 9.2 View on Amazon
Calculus: An Intuitive and Physical Approach (Dover Books on Mathematics) by Morris Kline 9.2 View on Amazon
Calculus: Early Transcendentals by James Stewart 8.8 View on Amazon

1. The Calculus Lifesaver: All the Tools you Need to Excel at Calculus (Princeton Lifesaver Study Guides)

Calculus brought about motion to the field of mathematics. If you want to accelerate your learning motion to master the subject, you may go for this book. The innovative book comes with some extraordinary materials to help you excel at calculus. As a beginner, you will have a firm grasp of all the major topics of calculus to enhance your thinking competence. The book will teach you the complex chapters in a humorous and appropriate manner. You could ace calculus within a few days if you consistently follow the book.

It contains more than 475 examples with step-by-step reasoning to be highly effective for your academic examinations. As it illustrates the topics with diagrams, real-life examples, and in-depth explanations, you could master the subject without taking help from a teacher. This supplementary guidebook will aid you to obtain a clear understanding of derivative, differentiation, and anti-differentiation, the definition of the chain rule, techniques of integration, and many more topics. Besides, you will have rock-solid skills in polar coordinates, logarithmic and exponential functions, trigonometric functions, and other core concepts.

The easy-to-follow format, engaging tune, and expressive writing style will make your learning route smooth. The simplicity and adaptability features of this self-teaching book will help you to easily absorb it. It incorporates a complete course and can be used as a formal textbook. All the theorems and methods are justified in this comprehensive book. As this single-variable calculus book is developed to provide the utmost benefit to the newbies, therefore, it’s rewarded to be one of the best calculus books for beginners.

Inside the Book

  • The book will present you with all the concepts of calculus from basic to advance to elevate your expertise.
  • It will improve your problem-solving and creative thinking competence through a broad range of exercises.
  • This book will aid you to solve some real-life problems by applying the principles and formulas of calculus.
  • You could excel at your academic calculus syllabus with the support of this self-directed book.
  • The book will enhance your brain functioning, imagination, and mathematical skills.
  • It covers the concepts of integration, differentiation, polar coordinates, and many more complex topics.
  • The book breaks down the principles of trigonometric functions, limit, logarithms, and exponential functions.

2. Essential Calculus Skills Practice Workbook with Full Solutions

Chris McMullen, the author of this book followed an application-oriented approach to write this book. He talked about all the major concepts of calculus in an easy and concise approach. You will find step-by-step guidelines with enriched examples to grasp the complete ideas of calculus. The book will present you with all the essential formulas, principles, and terms to bring your skill to the upper level. It’s the best calculus textbook for high school students to ace the academic syllabus. The book covers the complex topics in an interesting and digestible manner to engage the student in such a challenging subject.

You will get a good collection of examples with appropriate answers to solidify your understanding. Besides, it will show you how to apply calculus to solve real-life problems. You will also acquire some techniques for mastering the subject overnight. Most importantly, the book will guide you towards the right learning curve with some great resources. It features diagrams and a variety of exercises to visualize all the concepts to you. With this self-study book, you could spontaneously experience the explicit beauty of calculus. One of the great features of this book is its simplicity.

The book will teach you the concepts of integration, differentiation, limit, trig function, exponential, and logarithm. In addition, it will aid you to excel in second derivatives, anti-derivatives of polynomial, the chain rule, product rule, and quotient rule. You could easily determine the extreme value of a function and grab the techniques of multiple integrals. The easy-to-understand writing format and engaging tone will stimulate you to stay with calculus for long hours. It’s a productive book for mastering all the essential calculus skills.

Inside the Book

  • The book covers the techniques of integration and differentiation in various forms to level up your expertise.
  • It explains the concepts of limit, second derivatives, and derivatives of polynomial, exponentials, and trig function.
  • This book illustrates all the required topics of high school students in an in-depth explanation.
  • It features a wide range of exercises, examples, and other inevitable resources to improve your calculus skill.
  • You will find some secret techniques of acing calculus at home through this self-directed textbook.
  • The book meets all the common standards and comes in an easy-to-understand format.
  • It will advance your thinking capability, brain functionality, problem-solving skill, and imagination.

3. The Humongous Book of Calculus Problems

The Humongous Book of Calculus Problems is one of the best calculus textbook for self-study. It will present you with all the required stuff to immerse you in the subject of calculus. You will find calculus no longer a frustrating subject if you can stick to this great book. It will allow you to become an expert in solving mathematical problems. Your academic stress over calculus will be relieved through the provided guidelines and strategies of this masterpiece. Most interestingly, it will enable you to grab the complete course within a short time. You even no need to reach out for a tutor if you carry such a self-teaching book.

No matter how strong you’re in mathematics, it will explain the complex concepts in a digestible way to level up your understanding. The book will show you the application and relation of all the formulas of calculus in the real world. Surprisingly, you will become efficient in solving real-life problems by taking advocacy from such an excellent book. The author of this best-selling book focused on the annotated notes, diagrams, keywords, and other visualizations to make you competent in mastering calculus. He also integrated some pointers to show you the skills you need to command.

The simple, engaging, and humongous writing style will help you to clear all the core concepts of calculus. You could learn limits, continuity, integrals, differentiation, infinite series, and other topics at ease through the book. Moreover, it will aid you to acquire the best expertise in derivatives, velocity, tangent line, volume, area, and so on. One of the great features of this book is the proof of each topic. To push your mathematical skill to the next level, it incorporates up to 1000 problems with comprehensive solutions. You will find very few books as helpful for beginners as it.

Inside the Book

  • The book covers all the core concepts of calculus in an easy, concise, and digestible manner to aid the students.
  • It explains every chapter with proofs and gives some mysterious techniques to master the subject.
  • You could discover the real-world application of calculus through this self-directory book.
  • The book will elevate your problem-solving and critical thinking skills with 1000 plus exercises.
  • Every lesson of this book contains some useful diagrams, keynotes, quotes, and extra explanations.
  • This book will aid you to obtain full marks in your academic examinations by overcoming all fears.
  • You could easily grasp all the complex including integration, limit, differentiation, polar coordination, and so on.

4. Calculus: An Intuitive and Physical Approach (Dover Books on Mathematics)

It’s obvious that calculus is related to science. You could explore the contextual connection between science and mathematics through this extraordinary guidebook. It will teach you to concepts of the derivative, theorems on differentiation, techniques of integration, polar coordinates, and so on. Moreover, you could grab the ideas of trigonometric, logarithmic, and exponential functions along with the definition of the chain rule. A broad range of illustrative examples, drills, and clear-cut explanations will play a crucial role to elevate your expertise in calculus.

Each chapter includes an extensive exercise section to allow you to grow your skills. The relevant and real-world instance of each complex topic will aid you to grasp everything completely. An engaging and expressive writing approach will aid you to obtain full marks in your academic examinations. As it’s developed for beginners, you could easily comprehend all the principles. Most importantly, the proofs of every topic will boost your understanding at a significant margin. You will hardly need to take help from others if you carry such a well-written book.

Thanks to its integrated diagrams, symbols, keynotes, and other visualizations as you could digest everything more quickly through them. It will build your mathematical confidence and empower your understanding through structured data. In addition, it will reinforce your problem-solving skill and critical thinking competence. You could also observe the coordination point of all the mathematical sectors through this persuasive book. The pathway in reaching your calculus goal will be no longer rugged if you keep practicing with this great book.

Inside the Book

  • The book will reinforce your mathematical skills and help you to build a bright career in the future.
  • This book will guide you to the right route to conquer calculus fear and reach your goal.
  • It’s relatively easy to follow and covers all the prime topics of calculus to enhance your expertise.
  • You could master the subject through its step-by-step guidelines and provided strategies.
  • It will teach you the concepts of derivatives, techniques of integration, and theorems of differentiation.
  • You could acquire rock-solid skills in polar coordination, limit, trigonometric, logarithmic, and exponential functions, and chain rule.
  • This book is composed for absolute beginners to grasp all the lesions at ease.

5. Calculus: Early Transcendentals

It’s composed of a solid teaching method to fit into your brain. The author of this book knows well how to break down such an overwhelming subject in a simple and manageable manner. That’s why he covered every topic in a specialized technique to encourage you. Achieving success in calculus seems cozy with the guidance of this masterpiece. It has been one of the most-sold books worldwide over the years because of the firm explanation and relevant examples. This book will benefit you to take challenges on several mathematical competitions through effective study guides.

Exploring a higher degree in science or math-related subjects will become approachable for you when you carry it. To develop your technical competence, the book holds all the necessary utilities. It covers the concepts of integration, differentiation, limits, Leibnitz Theorem, derivatives, and so on. You can gain a solid understanding of exponential, logarithmic, and trigonometric functions, and polar coordinates through it. Do you want the assumption of instinctive beauty of calculus? If so, then this book is for you as it explains every topic with proofs.

You could gain optimal confidence in reaching your ultimate goal through this built-in learning aid. It will promote your problem-solving skill, brain functionality, critical thinking capability, and imagination to a great extent. Inside the book, the author added some extra chapters to boost your job skills and guide you to pursue the best career options. Some frequently asked questions regarding calculus are described in the book in very clear terms. The text is rich, vivid, deep, clear, and easy-to-grasp. You will face no interruptions to master the subject by dint of this best calculus textbook.

Inside the Book

  • The book explains calculus from basic to advance in an appropriate, effective, and concise manner.
  • This book contains all the essential utilities, formulas, principles, and ideas to have a good command of calculus.
  • It will present you the complex topics in an innovative approach and provide you immense motivation.
  • You will find a massive amount of exercises, examples, keynotes, and many more helpful theories to master the subject.
  • The author included some real-life examples to show you the importance of calculus in human life.
  • It will allow applying the principles of calculus to solve real-life problems as well as to secure marks in academic examinations.
  • You could experience the concepts of derivatives, Leibniz Theorem, limits, integration, differentiation, and more complex topics.

The Final Words

Calculus is a principal branch of mathematics. It’s somewhat difficult than other mathematical fields. Students hardly feel comfortable with it because of not having accurate guidance. It’s not all right to rely on merely the teachers to have the best solution a well-composed book can also help a student to release the calculus fear. If you’re going to start your journey with calculus before long, you may keep reliance on one of the best calculus books for beginners here.

However, you obviously know the characteristics of a good book. You need to stay determined to pick such a book as will become capable to meet your requirements. Since you’ve seen some of the great books from here, you can opt for one of them confidently. All the mentioned books are from genius scholars. It will be easy to reach your ultimate goal while you stick to an effective book by an acknowledged author.

An inexpensive, open-source alternative to the traditional Calculus textbook.

APEX Calculus is a calculus textbook written for traditional college/university/high school calculus courses. It has the look and feel of the calculus book you likely use right now (Stewart, Thomas & Finney, etc.). The explanations of new concepts is clear, written for someone who does not yet know calculus. Each section ends with an exercise set with ample problems to practice & test skills (odd answers are in the back).

APEX Calculus looks like a "regular" calculus book in most all respects except the price. Anyone can download the .pdf version of the text for free. Want a print copy? Feel free to print it yourself, or buy a b&w copy from the APEX Store for $12 or on for $15.

While APEX Calculus looks like an inexpensive version of the traditional calculus text, a closer look will show it's a better calculus book for an even better price. Check out the Compare page for more details.

Got questions? Explore the site, including the Frequently Asked Questions section. If you still need more information, just ask!

7. Calculus: Early Transcendentals

Calculus: Early Transcendentals is another book by James Stewart that presents calculus in clear language, empowering the reader to understand the subject using easy to understand examples. These examples present calculus as used in real-life situations in the most accurate manner.

The reader is expected to have developed technical competence as a result of using this book and an appreciation of the application of calculus to real-life.

The book comes with learning aids aimed at building mathematical confidence as well as achieving set calculus goals. Readers have commended the author’s approach and labeled the book as one of the best on calculus.

Calculus: Early Transcendentals presents you with more in-depth information compared to that which is taught in class. To enjoy the added benefits, you should open a Cengage account online.

  • Author: James Stewart
  • Publisher: Brooks Cole 8 Edition (February 4, 2015)
  • Pages: 1368 pages
  • Edition: Eighth Edition

Book: Calculus (Apex) - Mathematics

Calculus 1 (APEX Calculus v3.0)

A Calculus text covering limits, derivatives and the basics of integration. This book contains numerous examples and illustrations to help make concepts clear.

Publication date: 04 Jun 2015

ISBN-10: 1514225158

ISBN-13: 9781514225158

Paperback: 304 pages

Views: 8,305

Post time: 20 May 2016 11:00:00

Calculus 1 (APEX Calculus v3.0)

A Calculus text covering limits, derivatives and the basics of integration. This book contains numerous examples and illustrations to help make concepts clear.

Share — copy and redistribute the material in any medium or format
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The licensor cannot revoke these freedoms as long as you follow the license terms.

Click here to read the full license.

A Calculus text covering limits, derivatives and the basics of integration. This book contains numerous examples and illustrations to help make concepts clear. The follow-up to this text is Calculus 2, which reviews the basic concepts of integration, then covers techniques and applications of integration, followed by sequences and series. Calculus 3 finishes this series by covering parametric equations, polar coordinates, vector valued functions, and multivariable functions.

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