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.
The text is already subdivided into smaller reading sections.
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
Q: Are there any other comments you would like to make about this book, for example, its appropriateness in a Canadian context or specific updates you think need to be made?
The book is average compared to the hardcopy and online textbooks for calculus that I have read. The language used in this book is more complex than the level for students to self-study without an instructor. There are many graphical elements to illustrate mathematical concepts and many exercise questions to promote learning. Students can build a strong foundation in calculus using this book.