Dale Varberg Calculus Pdf

Varberg, Purcell, and Rigdon: Calculus 9e Review Questionnaire: May 13, 2003 Name: Joel W Robbin School: University of Wisconsin, Madison, WI 53706 Dept: Department of Mathematics Phone # 608 263 4698 Email: [email protected] Home Address: 2240 Rowley Avenue, Madison, WI 53726 Course Title: Calculus and Analytic Geometry Annual Enrollment: 3500 Text in Use: Varberg, Purcell, and Rigdon: Calculus, 8e. Previous Text Used: Thomes Finney 5e. May we mention your name in the preface of the text? Yes Are you considering writing a text? Yes. Please answer all applicable questions below. Additional comments are welcomed. Answers should be given based on your current or previous use of the book Varberg, Purcell, and Rigdon: Calculus, 8e. In preparing this report I made an informal survey of the instructors who used the text this year. In general the like the Varberg text, but there is no unanimity on minor complaints. In my remarks below, I will try to reflect the diversity of opinion I heard.
Topical Coverage 1. Comment on the overall scope and coverage of the book. What changes, if any, would you recommend to the organization of the topics? Why? The order of the topics does not correspond to our syllabus and we are contemplating a further change (series to the end). This has not been a major problem for us. I’ve attached a copy of our syllabus. 2. Please photocopy the Table of Contents. Indicate the chapters and sections you DO NOT cover by putting a line through them. Please fax to: Aja Shevelew (201) 236-7400. 1
3. Please list topics that you cover in your course THAT ARE NOT covered in this book. Do you recommend covering them in them in the new edition? If so, where would you the coverage, and how much material is required? Kepler’s laws (at least in the exercises). Many of us view this as the object of the second semester course as it uses analytic geometry, polar coordinates, and vectors to explain one of the great achievements of human history. 4. Chapter 1 and almost half of Chapter 2 are precalculus review. How much of these do you cover? Even if you don’t cover these, is it important to have them in the book for students’ reference? Are there topics in precalculus that can be omitted? I would leave most of this material in, but we don’t usually go over it in class. (The students should know this material already.) Section 1.2 is not appropriate for us since we de-emphasize calculators. 5. Should Sections 13.2 Vectors in the Plane: Geometric Approach, and 13.3 Vectors in the Plane: Algebraic Approach be combined into one? They are both short sections, but the approach to vectors is different. If the two sections were merged, what exercises are essential? For which kinds of exercises could we reduce the number, if any? I would merge them. I would give each geometric definition first and then immediately follow it with a theorem which says how to compute it. In particular, I would teach the boxed formula on page 573 as the definition of the dot product and the boxed formula on page 572 as a theorem. I avoid the notation ha, bi because I think it is not used in other subjects and because I like to emphasize the difference between vectors and points. I would box the first formula on page 574 and emphasize that these are two different notations for the same thing. I teach two and three dimensions at the same time. 6.Do you cover any part of Chapter 18 on Differential Equations? Which parts? If this coverage is minimal, is there any way to move a topic into an existing section or chapter? Would it be sufficient to have Chapter 18 on the web? Yes. But we do this material in the second semester. One respondent said he didn’t have time to treat the applications (e.g. electric circuits) in the chapter on on ODE. 7. Do you use the Technology Projects? If not, to save pages, would you prefer they be removed from the book? If so, do you use 2 per chapter or just 1? If you do use them, would it be sufficient to have these on a website for the book? Would it be better to keep them out of the book, and then have a “lab style” workbook for the Technology Projects that left room for students to fill in answers? Should these lab books include tutorials on the specific application used? For example, we could have lab books for Maple, Mathematica, and the TI 93 calculator that would include enough of a tutorial that students could learn the technology. Besides Maple, Mathematica, and the TI 93 calculator, what other technologies should we look at? I think that the course is too crowded to do extra projects, although one instructor has successfully assigned technology projects. Technology must be 2
used with caution because it encourages passivity, the exact antithesis of what is required. Ease of use is important – we don’t have time to teach the maple language. I use Maple occasionally in my lectures but only after I can reasonably ask the students to predict what will happen. (This semester I did a very successful demo on the day after an exam.) Maple and Mathematica have a learning curve which is unacceptable in the elementary course; I would use them for demos but would never ask (all) the students to use them. I have written a simple graphing calculator1 and with the equipment available at UW have used this with good effect in my lectures. With this aid I can illustrate the effect of changing the range in a parametric curve and how the linear and quadratic approximations approximate the function. The calculator can be used over the web and parses typed commands but (intentionally) does not do symbolic manipulation. A student who can reproduce the effect that I produce in lecture may learn not to write f (x) ≈ f (x) + f 0 (x)(x − a) where s/he should write f (x) ≈ f (a) + f 0 (a)(x − a). One could also use the calculator to check graphs obtained by hand use “sophisticated graphing” techniques. It is important to realize that there is educational value in learning these techniques even though the computer draws the graph better. In any technology which the student is required to use the following points are vital: (1) The user interface must be simple. Commands must be typed in a natural format, easily visible, and easily edited repeatedly. (2) We don’t have time to teach programming. (3) Students should not be asked to buy a calculator. (4) The technology should not do too much for the students. (5) Exercises should be designed so as to force the student to predict what should happen. I prefer web based technology.
Content 8. In your experience, are there specific sections where coverage needs to be expanded? Which sections? What would you like to see expanded? Less is more. The brevity of the exposition is to me the best feature of the book. 9. Are there sections where coverage can be reduced? Which sections, and how would you suggest reducing the coverage? Don’t introduce notations like R (see section 1.1) which are not used subsequently. Engineering profs frequently say that they want students to see more applications early on. My impression is that the students are overwhelmed by this. Better to have problems about water than about electricity: everyone understands water flowing into a tank, but most students do not know (yet) what a capacitor is. On the other hand, I can live with application problems so long as there is no attempt to teach the science and it is made clear that you don’t have to know the science to do the math problem. I prefer problems in geometry and elementary mechanics (F = ma) to problems that use concepts the students have not seen. 1 See
http://www.math.wisc.edu/ robbin/Jplot0/Jplot0.html
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One respondent said he didn’t do the economic applications and went light on the sophisticated graphing even though he likes both topics. I always treat ε − δ very lightly but most instructors do some problems where the students finds δ for a given ε. 10. Are there sections where the presentation can be clarified? Please indicate where the discussion could be clarified, and what precisely is lacking clarity currently. One respondent said he didn’t like the treatment of center of mass (see below). Also problems 44-47 on pages 127-8 of section 3.5 are related rates problems though this topic isn’t treated till later in section 3.9. This doesn’t bother me but one respondent complained about it. (It would be helpful to ask the students to do these problems again in section 3.9.) This respondent does recommend chain rule problems where the student is given some information about f at a and g at f (a) and asked to reason about (g ◦f )0 (a). (It is important in these problems not to specify f and g.) One respondent complained that the the power rule for differentiation is derived as a consequence of the chain rule, but that the reverse is not done in the analogous situation for substitution in integrals. It is always good to repeat ideas. One instructor I know teaches integrations by parts in the first semester because it is the product rule backwards, but I don’t advocate this. 11. Are there enough examples? Indicate where you have seen a need for more examples, and if possible, specify what examples you would like to see. 12. Should applications of the derivative and integral should be emphasized more? Less? Are there specific types of applications that you would like to see more/less of ? I like the current mix. 13. Do you like the placement of antiderivatives in Chapter 5? Or, does that come too quickly and would you prefer to see it delayed? If so, where would you like to see it? I have always disliked the notation for the indefinite integral but I suppose avoiding it would be too radical. I explain it by saying that the notation Z F (x) = f (x) dx + C is an abbreviation for the equation F 0 (x) = f (x). I don’t mind doing antiderivatives before definite integrals but the role of dummy variables in the latter should be emphasized. 14. What, if any, proofs can be deleted? Are there theorems that we leave unproved for which we might include a full or partial proof ? I think the balance here is good. 15. For the norm or magnitude of a vector u, is the “double bar” notation, e.g. kuk, better than the “single bar notation”, e.g. u ?
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I prefer kuk. Anything which helps distinguish vectors and scalars is good. Point out in a footnote that some books write u for kuk. 16. Parametric surfaces currently are covered only in Tech Proj 17.2. Do you cover parametric surfaces? If we were to include them, we would introduce them in section 16.6, then use them further in sections 17.5 - 17.7. Do you agree that we should do this, or not? (keep in mind our goal is to not add extraneous material) I would treat parametric curve early in the text so that students get used to describing curves in parametric form. I would emphasize that the graph y = f (x) is a special case, namely, x = t, y = f (t). I teach parametric arc length using parametric equations and emphasize that it is independent of the parameterization. I think there should be examples of parametric equations in chapter 1 e.g. x = cos t, y = sin t. In the same spirit I use parameterization of surfaces (rather than just the special case z = f (x, y)) to do surface integrals. The formula ° ° ° ∂R ∂R ° ° ° du dv dA = ° × ∂u ∂v ° is both easier to remember and more general than the special case q dA = 1 + fx2 + fy2 dx dy. You can also get the formula for the area element in polar coordinates this RR way. Again it is important to emphasize that integrals like S F · n dS are independent of the parameterization.
Exercises 17. Have you found the exercises to be sufficient in quantity? Specifically, are there a suitable number of basic drill exercises? Conceptual problems? Theoretical problems? Applications? Other? This is fine. 18. Is there a sufficient variety of exercises (type and level of challenge)? If not, what types of exercises are lacking (and where)? One respondent claimed that to many of the routine exercises told the students what steps to carry out to get the answer. (He cited the example of problems 11-28 on page 279.) His point is that the students know these things from reading the text. I am not sure I agree. Wording which is appropriate for a homework assignment is not necessarily the same as wording that is appropriate for an exam. 19. Are they suitably “graded” from easiest to most difficult? Have you ever encountered problems with making assignments because students hit hard problems too quickly? This is not something I have ever considered. Donald Knuth in his series of books are computer science gave each exercise a number indicating its difficulty. 5
I’m not sure I myself could do this in a reasonable way; many problems are trivial once you see the trick. Students tend to find a problem difficult if the algebra is complicated, even though the problem is conceptually simple. I don’t think it matters much, but a warning on the tricky problems is a good idea. (More for the instructors than the students. There is nothing more embarrassing then getting confused before a class of freshman.) 20. Generally, what suggestions do you have for the authors regarding exercises? Are there additional exercise types you would like to see universally? Please let the authors know exactly how you would like to see the exercises sets improved. I currently make students do at least one essay question on each exam. In this question they must state a definition or theorem correctly and possibly give a coherent proof. I give them a list of possible questions and answers and I’ve discovered that, with this preparation, they do about as well here as on other problems. The book could enforce this by including questions like the following in the “sample test questions”: 3. State and prove the product rule for differentiation. (Answer: Write what is on lines . . . on page . . .) 21. Currently, the book has “Additional Problems” sections at the end of each chapter (see Pg. 268, Section 5.10 or Pg. 158)). Do you ever utilize these problems? Is there a good reason to leave them as they are, or would you prefer that these problems be integrated into the appropriate end-of-section problem sets to enrich those? These are good because they don’t tell the student which method to apply. I like the current balance.
Design 22. Do you like the overall look/feel of the pages? Is there anything that jumps out at you that you would like to see changed? I like this fine. Try to keep the cost down. 23. Currently, the examples have no titles. Do you prefer this, or do you prefer that each example has a clear title, ie “Example 1: Critical Numbers Where the Derivative Does Not Exist”? Keep in mind that adding titles will increase the overall number of pages in the book, and answer within that context. I don’t think explicit titles are important. What is important is that every definition be immediately followed by an example of something which satisfies the definition and an example of something which doesn’t.
Supplements 24. Do you use any of the current book supplements? Which ones? I do not use any supplements. Some instructors think that solution manuals are bad – students should solve problems themselves. In the past we have used
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books where all the answers (but not the solutions) are in the back of the book, not just the the answers to the odd numbered problems. 25. Is there anything you would really like to have available to you and/or your students along with the book (feel free to think outside the box)? Over the years I have come to believe that the content of the calculus course is of much less importance than the process of learning calculus. Many students do not know how to listen to a lecture (especially if it is boring), most cannot read a math book (no matter how well written), and few can write coherently. These are the most important skills they learn in college. What the students need most is careful grading (feedback) and this is precisely what they don’t get in the lecture-discussion format used at UW. A publishing company who can successfully argue that their book is read by students would be doing something noteworthy, but I am unclear how to do that. In the future we may see textbooks written in hypertext. This could be useful in teaching reading skills. For example, the statement of a theorem having the word continuous in its statement could contain a hyperlink to the definition of continuity. Most undergraduates (even juniors and seniors) do not realize that they have to know the definition of something to prove anything about it.
Other 26. Overall, how would you describe the current book’s strengths and weaknesses? What I like best about the text is that there is very little superfluous material; the book is succinct and to the point. Other respondents said the same. This is important because (a) we don’t want students to be overwhelmed and (b) the book should not interfere with the individual instructor’s approach. I don’t like the use of the notation Dx for d/dx. It is nonstandard. I also don’t like ha, bi for the same reason. A warning should be attached to any notation which is introduced but not used heavily in the text. The students need to know what to remember. One instructor said that the book has too many formulas and uses too much formalism. He said students should learn methods, not formulas. As an example, he pointed to the formulas on the bottom of page 307. The picture on the next page illustrates the point better than the formulas: The center of mass is the center of mass of the centers of mass. He suggests the explanation that in certain situations (balancing experiments) a body behaves as if all its mass is concentrated at its center of mass. 27. Please make any other suggestions, not covered above, to the author regarding the revision. What would you plan on doing if you were revising this book? Treating differentials is not done well in any calculus book that I know, including this one. It is best to say that dy = f 0 (x) dx is an abbreviation for dy/dx = f 0 (x) and not otherwise make a big deal about it. One should emphasize the use of differential notation in deriving differential equations from
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Pn physical reasoning and the similarity in the notations i=1 f (xi )(∆x)i and Rb f (x) dx. The differential way of thinking (and Leibniz notation) is very ima portant since this is how students meet calculus in their science courses. This is also why everyone likes the easy O.D.E. in Chapter 5. P P7 4 Fix example 5 on page 224. The number of oranges is k=1 k 2 not k=1 k 2 . The math department may move infinite series to the third semester but some ideas (without proof) can be introduced earlier. I like to tell students as early as the first semester that lim
x→a
f (x) − Pn (x) =0 (x − a)n
where Pn (x) =
n X f (k) (a)
k!
k=0
(x − a)k
and give them problems like “find the polynomial P (x) of degree 2 such that P (1) = 3, P 0 (1) = 7, P 00 (1) = −5” or “find the polynomial P (x) of degree 2 which has the same derivatives of order ≤ 2 as sin x at x = 0”. One could put this earlier in the text and draw a few pictures illustrating and function and its linear and quadratic approximations. Average velocity is used to to motivate the derivative on page 102. On page 255 one should return to this theme by pointing out that the average velocity is the average of the velocity.
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