Calculus: Early Transcendentals (Stewart's Calculus Series)
by James Stewart
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Book Summary Information
Author: James StewartBrand: NA
Edition: Hardcover
Audio: English (Unknown); English (Original Language); English (Published)
Published: 2007-06-07
ISBN: 0495011665
Number of pages: 1336
Publisher: Brooks Cole
Accessories:
- Study Guide for Stewart's Multivariable Calculus, 6th
- Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 6th
Book Reviews of Calculus: Early Transcendentals (Stewart's Calculus Series)
Book Review: Stewart's Calculus
Summary: 1 Stars
In this review, I am using as a reference the edition from 2008 of James Stewart's Single Variable Calculus, Early Transcendentals, 6E, Volume I: Chapters 1-6. My review focuses on Chapter 2: Limits and Derivatives. This review has turned out to also be a review in contrast, of the edition from 2001 of Thomas' Calculus, Early Transcendentals, 10th edition. A recent edition of that book is Thomas' Calculus Early Transcendentals (12th Edition). The rating given is for Stewart's book. I would rate this 10th edition of Thomas' book at 5 stars.
Stewart's Calculus textbook is widely used across the United States to introduce students to higher mathematics. Its impact is far-reaching. It has the power to influence a student's self-confidence, and implicitly thereby, to encourage or discourage further studies in mathematics. A textbook is the map to the territory of the subject. If the map cannot be understood and followed, the subject cannot be learned. Because college mathematics departments are locked into a learning sequence that insists upon calculus before abstract algebra, even though this algebra doesn't rely upon the concepts of calculus, books like Stewart's Calculus have the power to support, hinder, or even stop, a student's progress in college level mathematics.
The fundamental mathematical concepts of calculus are: Limit, Continuity, Derivative, and Integral. The first three are introduced in chapter two, the last in chapter five, of Stewarts Calculus, 6th edition. How well does he do?
Not very well. Stewart seems to have no idea what needs to be said and what can be left for some other time. His book is cluttered and bloated with unnecessary "help" that can only get in the way and obscure the core purpose of learning. It's impossible to critique this book with adequate supporting detail in a relatively short review. In general: He seems to have no intuitive sense of how to present ideas clearly.
When introducing the concepts of Limit, Continuity, and Derivative, he loses the point of what he is supposed to be teaching and gets caught up almost immediately in giving unnecessarily complex examples rather than covering the topics efficiently and with clarity, then moving on. When he introduces the formal epsilon-delta definition of limit, he botches the opportunity to make it clear and helpful. He should have either not bothered or done a better job.
Here's an example of how Stewart doesn't think through what he's trying to teach. On page 119 he's just introduced the definition of a function continuous at a point. He's going to define discontinuous at a point, and this is what he writes [I'll use angled brackets to indicate italic, and asterisks for bold]:
"If <f> is defined near <a> (in other words, <f> is defined on an open interval containing <a>, except perhaps at <a>), we say that <f> is *discontinuous at a* (or <f> has a *discontinuity* at <a>) if <f> is not continuous at <a>."
Ignoring the stylistic and cognitive clutter of the parenthetical insertions, this definition, abstractly, is of the logical form {if p then (if q then r)}, where r denotes the clause where the terminology *discontinuous at a* is introduced; but notice that isn't how he structures it. Stewart says, in effect, "if p, then r if q" which is, of course, logically equivalent, so he's not technically wrong, but stylistically he fails to communicate well. The sentence reads as you go along as if it's going to tell you that if <f> is defined near <a> that it is *discontinuous at a* -- and it is only after you reach the end of the sentence you see what is actually being said. This stylistic infelicity (to speak kindly of it) most likely arose from blindly following the convention of defining mathematical terms in the form: We call such and such an object an O if [condition C]. But in this instance it could lead to reader confusion and frustration.
Here's how the same idea is introduced in the 10th edition (2001) of Thomas' Calculus, page 125:
"If a function <f> is not continuous at a point <c>, we say that <f> is *discontinuous* at <c> and <c> is a *point of discontinuity* of <f>. Note that <c> need not be in the domain of <f>."
Which is clearer? Going further in comparing these two books on the topic of continuity, Thomas' Calculus first defines *continuity at a point* (both at an interior point and at a left or right endpoint), then after introducing discontinuity as quoted above, the next paragraph gives the additional terminology of *right-continuous* and *left-continous*, and after explaining how this additional terminology is used in relation to continuity at end points, we are told that: "A function is continuous at an interior point <c> of its domain if and only if it is both right-continuous and left-continuous at <c> (Figure 1.46)." The figure is well designed and helpful.
This is followed by two very short examples, and then they mention the three conditions of the Continuity Test. There is another example and then they show a collection of six graphs that represent types of discontinuity and they quickly mention terminology such as *jump continuity*, *infinite discontinuity*, and *oscillating discontinuity*. They are building up to where they can mention algebraic combinations of continuous functions on the next page and after that the composition of continuous functions, and following that, the Intermediate Value Theorem. So they introduce more terminology and tell us on page 127:
"A function is *continuous on an interval* if and only if it is continuous at every point of an interval. A *continuous function* is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval. For example, y=1/x is not continuous on [-1,1] (Figure 1.51)."
This is all presented in a style that is very efficient and clean. The authors are aware of where they are headed and don't go off the path or linger along the way. Stewart, on the other hand, is a tourist. He finds everything of equal interest and seems to have no purpose but to linger and wander about. He actually touches on a few topics in this section that Thomas' book avoids, but he does it at the expense of clarity. He presents the continuity test immediately after defining how a function is continuous at a point, but then he doesn't reach the definition of a function continuous from the right or left (which Thomas' book gave at the same time as defining continuity at a point) until after he's given a page worth of examples and talked about discontinuities.
To define *continuous on an interval* Stewart writes, on page 121:
"A function <f> is *continuous on an interval* if it is continuous at every number in the interval. (If <f> is defined only on one side of an endpoint of the interval, we understand <continuous> at the endpoint to mean <continuous from the right> or <continuous from the left>."
Notice that *continuous function* is not mentioned in this definition. In fact, he never formally defines it. It almost shows up implicitly in a theorem statement about rational functions on page 122: "Any rational function is continuous wherever it is defined; that is, it is continuous on its domain." He starts to use the term explicitly, without giving its meaning, within the proof of the related theorem that any polynomial function is continuous everywhere, i.e. on the Real numbers. The sentence in the proof is: "This equation is precisely the statement that the function f(x)=x^m is a continuous function." (x^m signifies x with exponent m.) If you look up "continuity of a function" in the index, you're referred to page 119, which defines continuity at a number.
Now, let's compare how the two books state the Intermediate Value Theorem.
Here's Stewart:
"Suppose that <f> is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) is not equal to f(b). Then there exists a number <c> in (a,b) such that f(c)=N." He uses the inequality sign in the text.
Here's Thomas' Calculus:
"A function y=f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). In other words, if k is any value between f(a) and f(b), then k=f(c) for some c in [a,b]." They use y[subscript 0] where I wrote k.
Thomas' book includes a graph within the highlighted box giving the theorem and labels the theorem: The Intermediate Value Theorem for Continuous Functions. Stewart gives the necessary specification of continuity within the statement but not in the name of the theorem. Afterwards he gives two graphs. Both books refer soon after stating the theorem (Stewart takes longer) to the necessity of continuity as a condition of the theorem's truth. Stewart says: "It is important that the function <f> in Theorem 10 be continuous. The Intermediate Value Theorem is not true in general for discontinuous functions (see Exercise 44)." In Thomas' book we read: "The continuity of <f> on the interval is essential to Theorem 10. If <f> is discontinuous at even a point of the interval, the theorem's conclusion may fail, as it does for the function graphed in Figure 1.55." Which of these appears more helpful?
At this point, Stewart moves backward in relation to Thomas' Calculus and discusses limits at infinity and horizontal asymptotes (in Thomas' book these are discussed before continuity). Stewart then proceeds to tangents, velocities, and rates of change. Both books are heading towards a definition of the derivative, and the tangent is an important stop along the way. What's interesting, though, is how this is achieved in the two books.
Stewart, suddenly in a hurry, tosses the reader into the deeper end without preparation:
"If a curve C has equation y=f(x) and we want to find the tangent line to C at the point P(a, f(a)), then we consider a nearby point Q(x, f(x)), where x is not equal to a, and compute the slope of the secant line PQ: m[subscript PQ]= f(x)-f(a) / x-a . Then we let Q approach a number m. If m[subscript PQ] approaches a number m, then we define <tangent t> to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P. See Figure 1.)"
Not stopping to catch his breath, he then gives the definition of the *tangent line* using the limit as x approaches a for the quotient given above. In an example, he introduces the alternative form using f(a+h)-f(a) / h with the limit as h approaches zero. This quotient is then used in his discussion of velocity, then also in the definition of the *derivative of a function at a number a*, where he introduces the notation f'(a). He then backtracks and rewrites this limit f'(a) using the quotient form of f(x)-f(a) / x-a with the limit as x approaches a, after which he remarks that: "The tangent line to y=f(x) at (a,f(a)) is the line through (a, f(a)) whose slope is equal to f'(a), the derivative of f at a." Notice that this is the derivative at a point, not the general derivative of the function.
He then switches to rates of change and the *increment* of x, using the [delta]x notation. Of course, the [delta]x / [delta]y quotient notation is only temporary and is soon altered to dy/dx once the general derivative of a function is introduced in the final section of the chapter. By the end of the chapter he has gone back to the f(a+h)-f(a) / h quotient form, swapped out the constant a for the variable x and used the new form to define the *derivative of f*, after which he defines *differentiable at a* and *differentiable on an open interval*, and also introduces higher derivatives, and proves the theorem that: "If f is differentiable at a, then f is continuous at a."
Chapter 3, which I won't discuss (nor any further chapters) concerns differentiation rules. Chapter 4 is on applications of differentiation. The last two chapters of this particular volume are on integration. Integration techniques are not covered. I am referring to the edition from 2008 of James Stewart's Single Variable Calculus, Early Transcendentals, 6E, Volume I: Chapters 1-6.
Thomas' Calculus, 10th edition, ends the chapter on limits and continuity with a discussion on tangent lines. They are not hasty but they don't dawdle either. They discuss the concept of tangent lines, as usual, with clarity, and then when they introduce the formal definition of *slope* and *tangent line*, unlike Stewart, they immediately use the quotient form f(t+h)-f(t) / h and take the limit as h approaches zero, provided the limit exists. (They use x[subscript 0] where I wrote t. Likewise in what follows.) They give some examples, then remark that the quotient used above is called the *difference quotient of f at t with increment h*, and that if it has a limit as h approaches zero, that the limit is called the *derivative of f at t*. They don't bother to write f'(t), which, to my mind, is good sense, because the differentiability of a function is not yet generally defined, at least explicitly, so f'(x) is meaningless, and therefore, f'(t) would, at this point, be an anomalous use of function notation. The notation f'(x) is properly introduced along with the definition of *derivative* of a function, at the beginning of their next chapter.
This particular edition of Thomas' Calculus discusses limits and continuity in their chapter 1, derivatives in chapter 2, and applications of differentiation in 3. As in Stewart, the integral and applications of integration follow in two more chapters.
Students have no choice which textbook their professor assigns. If you're stuck with Stewart and having troubles, take a look at Thomas' Calculus. It may help.
Summary of Calculus: Early Transcendentals (Stewart's Calculus Series)
Success in your calculus course starts here! James Stewart's CALCULUS texts are world-wide best-sellers for a reason: they are clear, accurate, and filled with relevant, real-world examples. With CALCULUS: EARLY TRANCENDENTALS, Sixth Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an appreciation for the intrinsic beauty of the subject. His patient examples and built-in learning aids will help you build your mathematical confidence and achieve your goals in the course!Mathematics Books
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