## Calculus: Volume 1 (OpenStax)

April 29, 2016 | Updated: February 10, 2022
Author: Gilbert Strang, Edwin “Jed” Herman

Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 1 covers functions, limits, derivatives, and integration.

Subject Areas
Math/Stats, Calculus

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##### Reviews (3) Avg: 4.27 / 5

Natasha Mandryk

Institution:Vancouver Community CollegeTitle/Position: Instructor

##### Q: The text covers all areas and ideas of the subject appropriately and provides an effective index and/or glossary

The topics in this text are sufficient for a first-term course involving limits, derivatives, and applications of the derivative. The book opens with a review chapter on functions and graphs, which may be helpful for some students. There is an introduction to integral calculus, but not much in terms of integration techniques or differential equations. Polar coordinates and parametric equations are not included in this volume.

One helpful inclusion in this text is a list of key terms and brief definitions at the end of each chapter. This serves as a good review tool, as well as a point of reference.

The index could be more robust. It appears to have been generated automatically. As noted below, there are two separate entries for Rolle's theorem and rolle's [sic] theorem. There are a number of similar duplications, e.g. "differential calculus" and "Differential calculus", "average rate of a function" and "average rate of the function".

There are also a few oddities: Euler's number appears in the index under both "number e" (where e is in plaintext) and "number e" (where e is italicised), but not under "e" or "Euler's number".

Comprehensiveness Rating: 4 out of 5

##### Q: Content is accurate, error-free and unbiased

I didn't note any problems.

Content Accuracy Rating: 5 out of 5

##### Q: Content is up-to-date, but not in a way that will quickly make the text obsolete within a short period of time. The text is written and/or arranged in such a way that necessary updates will be relatively easy and straightforward to implement

The principles behind calculus haven't changed in years, so there is no danger of concepts becoming irrelevant. However, the applications of calculus, and the way we learn and teach it, change.

One potentially-dated feature of this text is the wording used for certain exercises to be done using a graphing calculator. (These are marked with a T, so that they can be avoided for classes where this technology is not used.)

Students may be using computer algebra systems or web apps to solve problems, so the particular feature recommended by the problem may not be appropriate. (e.g. p. 34 "Use the INTERSECT feature on a graphing calculator"; later, INTERCEPT and INTERSECTION are referred to.) This is minor.

A much larger potential problem is the question of how to integrate the role of internet-capable technology in calculus. This text makes a preliminary attempt at such integration by including a few links to third-party web apps that illustrate concepts. It's a nice touch. One feature of an open textbook is the ability to link to other work rather than have to make everything "in-house." As technology and websites change, links can be updated. To that end, I'd like to see more from these links: perhaps a brief description of what the applet does, in case the link is removed or the reader can't use the applet (because of Java or accessibility issues).

While texts will probably continue to include methods of hand calculation/analysis for many years, it seems odd for an electronic textbook not to feature exercises that take advantage of technology. Why not take advantage of web tools for mathematical analysis for student practice and exercises, not just for illustration of one idea?

Relevance Rating: 3 out of 5

##### Q: The text is written in lucid, accessible prose, and provides adequate context for any jargon/technical terminology used

Language is fairly conversational and accessible. Terms are defined when used, and reviewed at the end of the chapter.

Clarity Rating: 4 out of 5

##### Q: The text is internally consistent in terms of terminology and framework

This is largely the case. As noted above, there are some problems with the index; identical concepts are sometimes given slightly different names (capital letters changed to lowercase, etc) and then are indexed separately.

Consistency Rating: 4 out of 5

##### Q: The text is easily and readily divisible into smaller reading sections that can be assigned at different points within the course (i.e., enormous blocks of text without subheadings should be avoided). The text should not be overly self-referential, and should be easily reorganized and realigned with various subunits of a course without presenting much disruption to the reader.

Chapters and sections in the text are long, but clearly divided into readable subsections.
Examples, theorems, and exposition are all clearly distinguished from one another.

It is not too difficult to rearrange ordering of topics (within reason, obviously); for instance, the section on limits at infinity and asymptotes could be easily placed in the chapter on limits, rather than with applications of the derivative.

Modularity Rating: 5 out of 5

##### Q: The topics in the text are presented in a logical, clear fashion

The placing of limits at infinity and asymptotes with curve-sketching topics rather than in the chapter on limits seems odd, but will please those taking a visual approach to calculus. Some readers or instructors may prefer a slightly different organisation, but the topics as presented are fine.

Organization Rating: 4 out of 5

##### Q: The text is free of significant interface issues, including navigation problems, distortion of images/charts, and any other display features that may distract or confuse the reader

This is an area where the text could use some improvement.

Line spacing is inconsistent throughout. The reason is undoubtedly because of the demands of mathematical notation; however, by making overall line spacing greater, and/or using display-style rather than in-line expressions, the visual flow of the text would be improved.

There is a related problem of alignment of question numbers and the questions themselves; in places, the numbers are about half a line above the questions, which is visually distracting.

Readability of exercises would be improved if exercise instructions were bolded/highlighted and appeared on the same page as the exercises they apply to. For instance, the instruction for exercises 245-252 (section 3.6) appears on p. 300 even though the exercises are on the reverse, on p. 301. There is also a table preceding said exercises (and necessary for them), but it has no name/number, and its purpose is clear only from the directions on the preceding page.

Interface Rating: 3 out of 5

##### Q: The text contains no grammatical errors

There are minor syntax errors. For instance, on p. 505 (chapter 4 review), "rolle's theorem [sic]" is listed as a "Key Term", with no capitalisation. As a result, in the index, there are separate entries for "Rolle's theorem" and "rolle's theorem".

There are missing or inconsistent punctuation marks at times (e.g. no terminal period on question 239 p. 300, which is inconsistent with questions 238 and 240).

I did not come across serious syntactical errors in my review.

Grammar Rating: 4 out of 5

##### Q: The text is not culturally insensitive or offensive in any way. It should make use of examples that are inclusive of a variety of races, ethnicities, and backgrounds

This is not a relevant concern to most of the text. Many examples involve applications of physics, as is common. Others I noticed involve health (modelling blood pressure), environmental science (sea level increase, atmospheric CO2 levels), or economics (cost and revenue models). The cultural knowledge required for applications-type problems is not excessive.

Cultural Relevance Rating: 5 out of 5

For Canadian students, a greater proportion of problems using the metric system rather than Imperial units would be helpful. There are far more problems using feet and miles rather than metres and kilometres.

I would very much like to see an update to this text that takes advantage of existing web-based technology (e.g. Desmos, Geogebra) in student exercises. I have yet to see a textbook that takes full advantage of current graphing technology.

Jerry Chik

Institution:Grande Prairie Regional CollegeTitle/Position: Power Engineering Instructor

##### Q: The text covers all areas and ideas of the subject appropriately and provides an effective index and/or glossary

The text provides comprehensive overview on first-year university/college calculus, including limits, derivatives, and integrals. An index has been provided at the end of the book, and a glossary in the form of review key terms at the end of every chapter is provided. These terms can be brought to the back of the book under glossary is better so that the student doesn’t have to figure out which chapter to look up a particular term and instead, he/she can look up any term in the book.

Comprehensiveness Rating: 5 out of 5

##### Q: Content is accurate, error-free and unbiased

Explanations and examples/solutions appear to be accurate and error-free. There appears to be no bias either. Answers for the practice questions in each chapter are there, but full solutions to the problems would be preferred.

Content Accuracy Rating: 5 out of 5

##### Q: Content is up-to-date, but not in a way that will quickly make the text obsolete within a short period of time. The text is written and/or arranged in such a way that necessary updates will be relatively easy and straightforward to implement

Calculus principles are the same regardless of when it is presented, so unless proven false, the content will remain up-to-date. It can be easily updated, as each section for explanations, examples, and questions are distinct and can be separately updated.

Relevance Rating: 5 out of 5

##### Q: The text is written in lucid, accessible prose, and provides adequate context for any jargon/technical terminology used

Concise, to-the-point, with technical glossaries at the end of each chapter. Explanations are sometimes written as though an instructor is speaking to the reader. (lots of uses of “we, you” etc.

Clarity Rating: 5 out of 5

##### Q: The text is internally consistent in terms of terminology and framework

The book is coherent when it comes to the consistency of terminology throughout each chapter. There appears to be no double-meaning of any terminology.

Consistency Rating: 4 out of 5

##### Q: The text is easily and readily divisible into smaller reading sections that can be assigned at different points within the course (i.e., enormous blocks of text without subheadings should be avoided). The text should not be overly self-referential, and should be easily reorganized and realigned with various subunits of a course without presenting much disruption to the reader.

For most calculus concepts, there is a section on introducing the concept of the calculus principle, followed by a section with examples or steps on how to apply the principle, then another section showing problems with solutions.

Modularity Rating: 5 out of 5

##### Q: The topics in the text are presented in a logical, clear fashion

yes, the book starts with the basics like functions, graphs, and limits then progresses up to derivatives and their applications and finally integrals and their applications, as derivatives are learned before integrals can be understood. Also, each math principle starts off with an introductory explanation of the concept, followed by steps on how to do it, examples with detailed step-by-step solutions, followed by problems to try out.

Organization Rating: 4 out of 5

##### Q: The text is free of significant interface issues, including navigation problems, distortion of images/charts, and any other display features that may distract or confuse the reader

the chapters appear to be free of such any interface issues.

Interface Rating: 5 out of 5

##### Q: The text contains no grammatical errors

no grammatical errors were found.

Grammar Rating: 5 out of 5

##### Q: The text is not culturally insensitive or offensive in any way. It should make use of examples that are inclusive of a variety of races, ethnicities, and backgrounds

the text does not appear to contain any culturally offensive references.

Cultural Relevance Rating: 5 out of 5

some of the cultural and social references appear to pertain to American geography, e.g. references to places like the San Andreas fault and the Grand Canyon skywalk, so perhaps content that reflects Canadian context may be more relevant for the Canadian learner.

Jung-Lynn Jonathan Yang

Institution:University of Alberta (Edmonton)Title/Position: Postdoctoral scholar, Open Learning consultant

##### Q: The text covers all areas and ideas of the subject appropriately and provides an effective index and/or glossary

The text is comprehensive for typical Calculus I courses in British Columbia. The examples in the text include a variety of subjects, e.g. physics, sports, and finance, to illustrate the versatility of calculus and appeal to students in multiple disciplines.

Several features of the text set the framework for an effective learning experience. The learning objectives are stated, and there is a glossary and a search bar. The chapter summary of key terms, equations, and concepts reinforce the learning objectives. There are many images and colour-coded graphs (functions) to assist in understanding mathematical concepts. The learning experience is supplemented with links to web pages, e.g. interactive applets, GeoGebra, and Wolfram. The range of easy and tough example questions have fully worked out solutions to guide the reader. There are hundreds of exercise questions for each section, including section-specific and chapter review exercises, to allow students to put mathematical concepts into practice.

The answer key does not have fully worked solutions. As well, “Answers may vary” as the only response is not illuminating. If students are unable to obtain the correct answer, they may need more guidance toward the correct solution.

There are concepts that could be included for better comprehensiveness. In the section about determining the area bounded between two functions of y, the concept can be extended to include functions that intersect. Parallel to the case for functions of x, where the integrand is the top function minus the bottom function, the case for functions of y will have right function minus the left function in the integrand. It would be nice to have an example for solids of revolution involving functions of y in Section 6.8.

Comprehensiveness Rating: 4 out of 5

##### Q: Content is accurate, error-free and unbiased

There are slight typographical errors. For example:
- There are mismatched parentheses in the solution to Example 1.10c and in Section 3.6, in the expression k′ (x) = h′ (f (g (x)) f ′(g (x)) g′(x)).
- Example 4.39a has an extra factor of x in the denominator that does not appear in the original question.
- In Figure 6.3, the figure caption should have a space between (a) and “We”.
- The text preceding Figure 3.30 has the incorrect domain for x as “0 < x < 25”. The correct domain is 0 < x < 5.

There are issues with some solutions. For example:
- The solution to Example 1.32c is awkwardly written: cos(cos−1 (5π/4)) = cos(cos−1 (−√2 / 2)) = 3π/4. The function cos−1 cannot have 5π/4 > 1 as an argument, and cos(cos−1 (−√2 / 2)) = −√2 / 2 not 3π/4.
- The answer to Exercise 97 is
“Since the absolute maximum is the function (output) value rather than the x value, the answer is no”
and appears incorrect because Figure 4.13(c) presents an example of a function f (x) = cos(x) that has multiple absolute maxima.
- The answer to Exercise 313 states that “y = −x2 has a minimum only.” This answer does not appear correct because the graph of y = −x2 actually has a maximum value and no minimum value in the interval (−∞, ∞).

Other than these, the content is relatively accurate and error-free.

Content Accuracy Rating: 4 out of 5

##### Q: Content is up-to-date, but not in a way that will quickly make the text obsolete within a short period of time. The text is written and/or arranged in such a way that necessary updates will be relatively easy and straightforward to implement

The content is up-to-date and can be easily updated.

Relevance Rating: 5 out of 5

##### Q: The text is written in lucid, accessible prose, and provides adequate context for any jargon/technical terminology used

At times, the text presents concepts in an overly complicated manner. Calculus, at either the high school or undergraduate level, is often the first encounter to advanced math. These students may not have the background to interpret text that is written in formal language or uses mathematical symbols. Sometimes the wording is difficult for students to understand. For example, in Section 4.6, the phrase “Similarly, for x < 0, as the values |x| get larger, the values of f (x) approaches 2” can be simplified as “Similarly, for negative x-values, as the magnitude of the x-value gets larger, the y-value of the function approaches 2.”

In Section 2.3, the examples of limit laws are a bit overly complicated by breaking down the function into smaller terms and evaluating the limit of each term individually. A simpler method is to mentally substitute in the limit and observe that there is no indeterminate form. Alternatively, conceptual questions about limit laws can be used.

Given that limx → 3 f (x) = L, limx → 3 g(x) = M, and limx → 3 h(x) = N, find limx → 3 (f (x)g(x) + h(x)).

Answer: limx → 3 (f (x)g(x) + h(x)) = [limx → 3 f (x)][limx → 3 g(x)] + limx → 3 h(x) = LM + N

In Section 6.1, some students may have difficulty interpreting the absolute value function. In Theorem 6.2, while I agree that the absolute value is appropriate to find the area of a region, I would suggest an alternative method. Find the x-coordinates of intersection between two functions, divide the domain into intervals according to these x-values, and identify the top (higher y-value) and bottom (lower y-value) functions in each interval. Then, the integrand is the top function minus the bottom function and does not use absolute values.

Some solutions may need to show more steps. In Example 4.2, the derivative is simply stated as x dx/dt = s ds/dt. Students may not automatically understand that the factor of 2 cancels and may need the preceding step 2x dx/dt = 2s ds/dt. Furthermore, it would be nice to have more explanation in Example 3.78 about how to apply the properties of logarithms and that the derivative of ln (sin x) is (1/sin x) cos x = cos x / sin x = cot x.

In Section 4.1, the problem-solving strategy states, “Using the chain rule, differentiate both sides of the equation found in step 3 with respect to the independent variable.” Because the topic is on related rates, where the independent variable is time, it is more precise to state “with respect to time.”

Clarity Rating: 3 out of 5

##### Q: The text is internally consistent in terms of terminology and framework

The text has overall consistency. Some suggestions are:
- To complement the text, use both set-builder and interval notation in Example 1.2.
- The table in Exercise 10 should have φ rather than x to be consistent with the function f (φ) = cos (πφ).
- In the definition for Equation 2.6, “x approaches a from the left” should have “a” in italics to denote as a variable. Similarly, for Equation 2.7, “approach the real number L” should have “L” in italics.
- The table in Exercise 36 should have a lowercase x rather than an uppercase X.
- In Section 3.5, the variable x should be italicized in the expression d/dx (cos x) = −sin x.

Consistency Rating: 4 out of 5

##### Q: The text is easily and readily divisible into smaller reading sections that can be assigned at different points within the course (i.e., enormous blocks of text without subheadings should be avoided). The text should not be overly self-referential, and should be easily reorganized and realigned with various subunits of a course without presenting much disruption to the reader.

Modularity Rating: 5 out of 5

##### Q: The topics in the text are presented in a logical, clear fashion

There are several instances where the organization of topics is confusing. The concepts in Section 1.1 may be reorganized. The subsection on representing functions may be more suited as an overall introduction to Section 1.1 rather than after explaining what is a function. Also, there is a disconnect between Figures 1.2-1.4 and the text immediate before the figures. The text uses the function f (x) = x2 as an example, but the figures present a general idea about functions (Figure 1.2) and specific examples of functions (Figures 1.3-1.4) unrelated to the example in the text. Perhaps the figures and text can be reorganized as:

Everyday example of a function (Table 1.1 and Figure 1.6-1.7).
Text explanation about what is a function, domain, and range.
Visualization of the concept of a function (Figure 1.2).
Visualization of ordered pairs (Figure 1.3).
Graphical representation of ordered pairs (Figure 1.4).
Graphical representation of ordered pairs related by a function (Figure 1.5).
Text example of the function f (x) = x2.
Graphical representation of f (x) = x2.

Piecewise-defined functions are complicated for students because the function changes with the domain. Students may be used to having the same function for the entire domain. I would suggest including a labelled graph to show how distinct functions are defined along the range of x-values.

Some questions use concepts that are covered later on in the text or not at all. Exercises 94 and 95 in Section 5.2 need the knowledge of definite integrals involving odd functions over a symmetric domain. This concept should be introduced in Section 5.2 rather than later on in Section 5.4. Example 1.2 assumes students know the fundamental rules for domain, for example, the argument within a square root is positive and division by zero is not allowed. If students do not have this knowledge, then Example 1.2 is difficult to understand.

Hyperbolic trigonometric functions are a staple for engineering math courses. The derivatives of hyperbolic trigonometric functions are found in the chapter about applications of integration. It may be more suitable to include these derivatives in the chapter about derivatives. Furthermore, Chapter 1 review can have a list of hyperbolic trigonometric functions in terms of exponential functions. Students may need this extra reminder because students may not have previous high school knowledge of hyperbolic trigonometric functions.

There are other minor issues about the order in which topics are presented:
- The indeterminate form 0/0 is stated in Section 2.3, but there are other indeterminate forms that students should know, i.e. 0·∞, ∞ − ∞, ±∞ / ±∞, 1∞, ∞0, 00, that are covered much later in Section 4.8.
- In Section 3.7, the text does explain why the derivative of functions with rational exponents is in the section about the derivative of inverse functions. However, it may be more straightforward to introduce the derivative rule in the section about the power law.
- Sections 4.3 Maxima and Minima and 4.5 Derivatives and the Shape of a Graph could be organized next to each other, rather than have the intervening section on the mean value theorem, because both sections are about curve sketching.
- It is not apparent why the Wolfram integral calculator is introduced as media in Section 6.2. This resource is more suitable when introduced as a tool to evaluate integrals in the integration chapter.
- Antiderivatives and initial-value problems could be introduced in the integration chapter rather than in the chapter about applications of derivatives.
- Limits at infinity and horizontal asymptotes are discussed in the chapter about applications of derivatives. Limits at infinity and horizontal asymptotes may be more related to the chapter about limits.

Organization Rating: 3 out of 5

##### Q: The text is free of significant interface issues, including navigation problems, distortion of images/charts, and any other display features that may distract or confuse the reader

The text was reviewed as a web page rather than as a PDF file. A problem with navigation is that when I minimize and re-open the web page, there is a jump to a location higher up on the page. When I click on the search bar and then click on the page, there is a jump to the bottom of the page. In either case, it was inconvenient to scroll back to the original location where I had been reading. Another problem is that the media (http://www.openstax.org/l/20_riemannsums) in Section 5.1 is inaccessible. The web page may have been removed. Overall, the text is mostly free of significant interface issues.

Interface Rating: 4 out of 5

##### Q: The text contains no grammatical errors

There are no major grammatical or spelling errors.

Grammar Rating: 5 out of 5

##### Q: The text is not culturally insensitive or offensive in any way. It should make use of examples that are inclusive of a variety of races, ethnicities, and backgrounds

The text is not insensitive or offensive but has limited diversity and inclusion regarding culture, gender, ethnicity, national origin, age, disability, sexual orientation, education, religion. Given the subject, mathematics, it may be difficult to incorporate extensive discussion of diversity and inclusion.

The theme of the text is largely American, with European history of mathematics as a background. Thinkers from outside of Europe formulated mathematical concepts relevant to calculus, but the text limits the discussion of the development of calculus from the European perspective of Newton and Leibniz. American locations are used throughout the textbook to set the context for discussing mathematical concepts, and the practice questions often make references to American themes. Imperial units of measurement are used frequently. Students in British Columbia and Canada may feel somewhat out of place when reading the text if the students do not have knowledge about America or imperial units.

Cultural Relevance Rating: 2 out of 5