Calculus: Early Transcendentals
Reviewed by James L. Bailey, Chair, British Columbia Committee on the Undergraduate Program in Mathematics and Statistics, College of the Rockies on 7/15/13
Comprehensiveness
The BCcupms Core Calculus Report (revised 2013):
In 2002 the British Columbia Committee on the Undergraduate Program in Mathematics and Statistics (BCcupms) accepted the Core Calculus Report. It was reviewed in 2007 and revised in 2013. This document has a list of core topics which all first year (two semester) Science Calculus courses must include and a list of additional topics, at least four of which must be chosen. Any text which is adopted for a first year Science Calculus course must be consistent with this report.
Core topics:
Limits, continuity, intermediate value theorem.
Limits are introduced in Section 2.3 where Definition 2.3 is the $\epsilon, \delta$-definition of a limit. The definition is used to show that \[\lim_{x\rightarrow2}x^2=4\] (Example 2.5, page 40). Properties of limits are stated in Theorem 2.7 (page 42). One sided limits are defined, together with an example, in Section 2.3.
Continuity is covered in Section 2.5. There is a problem with Figure 2.3(a) (the left half of the figure below is my attempt at reproducing it). The author states that ``a function $f$ is continuous if it is continuous at every point in its domain'' (page 53). It is claimed that Figure 2.3(a) is the graph of a discontinuous function, but it is not clear that the function is defined at the discontinuities, viz. $x=-1,0,\text{ and }2$. In fact, the function would be continuous if it were not defined at these values. Something like the right half of the figure would have made it clearer what function values were intended at the discontinuities ($x=-1$ and $x=1$).
image not available
Figure 2.3.a in the Text, Discontinuities.
There is no discussion of removable or jump discontinuities.
The Intermediate Value Theorem is found in Section 2.5 together with an application, using a binary search to approximate a zero of a function.
Differentiation
First and second derivatives with geometric and physical interpretations. The following are covered:
The derivative of a function is introduced at the bottom of page 32, as a summary of the procedure used to find the slope of the tangent to $\sqrt{625-x^2}$ at any point. Section 2.4 introduces the main notations, $y'=f'(x)$ and $\frac{dy}{dx}$, and has a discussion of places where a function does not have a derivative (corners and cusps). The dot notation, $\dot x$, is introduced on page 128.
Interpretations of the derivative: slope of tangent line; velocity, acceleration (velocity and acceleration are also discussed in Section 9.2 when discussing integration); rate of change in general.
The second derivative does not have its own section. It is first introduced with the second derivative test for extrema (Section 5.3) and concavity (Section 5.4).
Interpretations of the second derivative: concavity and acceleration.
Mean Value Theorem The Mean Value Theorem is treated in Section 6.5. The authors first prove Rolle's Theorem and then use that to prove the Mean Value Theorem.
Derivatives of the exponential and logarithm functions, exponential growth and decay. The derivatives of the exponential and logarithm functions are covered. On page 85 where the authors find the derivative of $\log_ax$ they show that $\log_ae=\frac{1}{\ln a}$. For no more work they could have derived the change of base formula, $\log_ax=\frac{\ln x}{\ln a}$ and then found the derivative of $\log_ax$ more economically. Exponential growth and decay is not covered, presumably because there are no differential equations.
Derivatives of trigonometric functions and their inverses. The derivatives of $\sin x$, $\cos x$, $\tan x$, and $\sec x$ are covered; $\cot x$ and $\csc x$ are left as exercises. On pages 75--76, in giving the usual geometric argument that \[\lim_{x\rightarrow0}\frac{\sin x}{x}=1,\] the authors argue that, with a little algebra, \[\frac{\cos x \sin x}{2} \leq \frac{x}{2}\Rightarrow \frac{\sin x}{x}\leq \frac{1}{\cos x}.\] They do not point out that we need $0\lt x\lt\frac{\pi}{2}$ in order to keep the various quantities positive and avoid problems with the inequalities. Because the argument is essentially geometric, and this is the restriction which is implied by the diagram, they may feel that it is unnecessary to point this out. The derivatives of the inverse trigonometric functions: the derivative of $\arcsin x$ is done, but the derivatives of $\arccos x$, $\arctan x$, and $\text{arccot}\, x$ are left as exercises. The derivative of $\text{arcsec}\, x$ is not discussed.
Differentiation rules (including chain rule, implicit differentiation) The authors start by deriving the power rule $\frac{d}{dx}x^n=nx^{n-1}$ for integer $n$ using an ad hoc argument which gives the first two terms of the Binomial Theorem (page 56); rational exponents are handled after they have covered implicit differentiation. The other rules (constant multiple, sum, product, quotient, and chain) are presented in order. Finding derivatives by implicit differentiation is covered, but finding the second derivative of an function defined implicitly is discussed only in the section on polar coordinates. Logarithmic differentiation is not covered. On page 67, where they show how to differentiate \[f(x)=\frac{x^2-1}{x\sqrt{x^2+1}}\] they say: "The last operation here is division, so to get started we need to use the quotient rule first," but there is no indication why this is important. It may be better to state the implied rule, that the differentiation rules are applied in the reverse order to that which is used when doing a calculation.
Linear approximations and Newton's Method Newton's Method is well covered but the section on Linear Approximations is a little thin. In particular, I would have liked to see problems such as "Use a linear approximation to estimate $\sqrt{10}$," and some problems which do not have a unique answer because the student has to make choices.
Optimization --- local and absolute extrema with applications Optimization is well covered with a large number of exercises.
Taylor polynomials and special Taylor series $\left(\sin x,\,\cos x,\,e^x,\, \frac{1}{1-x}\right)$, plus enough sequences and series to understand the radius of convergence; in particular, the concept of series and convergence, the ratio test, and how to find the radius of convergence. These are all covered. In addition, differentiation and integration of power series are covered and there is a proof of the Lagrange form of the remainder.
Curve Sketching. Chapter 5 covers curve sketching. Intercepts are not discussed. Horizontal and vertical asymptotes are discussed but the authors say that slant asymptotes "are somewhat more difficult to identify and we will ignore them." Even and odd symmetry is mentioned.
Integration
Definition of the definite integral and approximate integration. Both are covered. There is an example of using the limit of a Riemann sum to calculate an area, although the term ``Riemann sum'' is not used.
Areas of plane regions Covered.
Average value of a function. Covered by example although the general formula, \[f_{\text{avg}}=\frac{1}{b-a}\int_a^bf(x)\,dx\] is not given.
Fundamental Theorem of Calculus Both forms of the Fundamental Theorem are covered.
Integration techniques: substitution (including trig substitution), parts, partial fractions. The following integration techniques are covered:
$u$-substitutions. I have a problem with the authors' approach: the authors allow both $x$ and $u$ in the same integral. In the example on page 163 they have \begin{eqnarray*} \int x^3\sqrt{1-x^2}dx&=&\int x^3\sqrt u\frac{-2x}{-2x}dx\qquad u=1-x^2,\,du=-2x\,dx \\ &=&\int \frac{x^2}{-2}\sqrt u\frac{du}{dx}dx\\ &=&\int \frac{x^2}{-2}\sqrt u\, du\qquad x^2=1-u\\ &=&\int -\frac{1}{2}(1-u)\sqrt u\, du\\ \end{eqnarray*} The authors advise that it is necessary to "translate the given function so that it is written entirely in terms of u, with no x remaining in the expression" but I have found that students often miss this nicety.
powers of $\sin x$ and $\cos x$ are covered in Section 8.3, using examples only; it is not explicitly stated that for $\int\sin^nx\cos^mx\,dx$ use a $u$-substitution if one of $n$, $m$ is odd and the double angle formula if both are even. Powers of $\tan x$ and $\sec x$ are not covered. The use of reduction formulae is not discussed.
trigonometric substitutions are covered, but not systematically.
there is a section on integration by parts and tabular integration. There are no rules of thumb to help students decide when to use integration by parts. Reduction formulae are not discussed.
rational functions are covered, but only the easy cases: when the denominator is of the form $(ax+b)^n$, $(x-r)(x-s)$, or an irreducible quadratic $x^2+bx+c$.
Applications of integration. Applications of integration are in Chapter 9.
Areas between curves.
Distance, velocity, acceleration.
Content Accuracy
The diagrams are very good. I managed to spot six errors:
page 53, figure 2.3.a: It is not clear that the function is defined at the discontinuities (This has been discussed under Comprehensiveness).
Page 83: ``the limit $\displaystyle\left[\lim_{\Delta x\rightarrow0}\frac{a^{\Delta x}}{\Delta x}\right]$ varies directly with the value of a''. I think that ``varies directly'' usually means ``is directly proportional to.'' This is a quibbling point, and I don't know how it could have been worded better, certainly not "is a monotone increasing function of $a$."
page 83: ``figure p. 4.3.'' (``p.'' should not be there.)
page 111: ``vertical asymptote where the derivative is zero'' (should be ``where the denominator is zero'').
Page 163: $\int x^3\sqrt{1-x^2}$ should be $\int x^3\sqrt{1-x^2}\,dx$ (the ``$dx$'' is missing).
Page 200, Exercise 9.3.1: ``$dy$'' missing at the end of the second integral which should read $\int_1^4\left(1+\sqrt y\right)^2-\left(y-1\right)^2dy$.
Relevance/Longevity
There are no problems.
Clarity
The style of writing is clear, informal, almost chatty. The authors keep jargon to a minimum, perhaps to a fault. For example, the term ``Riemann sum'' is not used even though there is an example of calculating $\int_0^x3t\,dt$ using $n$ rectangles of equal width using the left endpoint approximation and letting $n\rightarrow\infty$. The style of the book is to work from the concrete to the abstract, from the particular to the general. For example, to introduce the idea of the derivative and to motivate the idea of a limit they have a long discussion about the slope of the tangent to the semicircle $y=\sqrt{625-x^2}$ at the point $(7,24)$. First working numerically, they calculate the slope of the secant lines between $x=7$ and $x=7.1$. Next they find the general formula \[\frac{\sqrt{625-(7+\Delta x)^2}-24}{\Delta x}\] and substitute $\Delta x=0.01$ to get a better approximation for the slope of the tangent. There is a link in Figure 2.1 to a \href{http://www.whitman.edu/mathematics/calculus/live/jsxgraph/secant_lines.html}{Sage worksheet} of the function $y=2x(1-x)$ in which one end of a secant line is fixed at $x=0.15$ but the other can be moved, so it is possible to watch the secant line approach the tangent line. Next they rationalize the previous expression to get \[\frac{-14-\Delta x}{\sqrt{625-(7-\Delta x)^2}+24}\] and argue that as $\Delta x\rightarrow0$ the slope of the secant line approaches the slope of the tangent line, $-\frac{7}{24}$. They point out that we are able check this answer because the line from the centre of a circle is perpendicular to the tangent, so their slopes must be negative reciprocals. They continue by doing the same calculation but with 7 replaced by $x$ to get the slope at an arbitrary point on the circle and then set $x=7$ to see that the new formula gives the earlier answer. They then replace $\sqrt{625-x^2}$ by $f(x)$ get the derivative of an arbitrary function as the limit of the difference quotient. The three approaches used in this example (numerical, graphical, and algebraic) together with the rather leisurely pace help the student understand this difficult concept. Unfortunately the authors are not consistent in the use of this three-pronged attack. For example, their approaches to the product rule and the chain rule are purely algebraic; their argument is rigorous but doesn't give the student any insight into what is happening. It would have been helpful to have some visual representation of these rules, perhaps something like the following figure:
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Visualizations for the Product and Chain Rules.
For the most part the authors do not state general results, but rather expect the student either to imitate the process in the examples or to invent their own formulae. This has the advantage of discouraging students from memorizing a formula and ``plug and chug'' in exercises.
Consistency
Sometimes the authors use terminology before it is discussed.
Modularity
There are no problems
Organization/Structure/Flow
I have a few minor concerns:
Page 54, Exercise 2.5.4: assumes that the natural logarithm $\ln x$ is known (the logarithm function, $\log x$, is defined on page 80 and the natural logartithm function, $\ln x$ on page 83).
Page 62: the authors assume that the meaning of $\sum$ is known when they introduce the notation for a product, $\Pi_{k=1}^{n}f_k$. They do not define sigma notation until page 150.
Page 87: Exercise 4.7.22 asks the reader to use implicit differentiation to find the derivative of $y=\log_ax$ but implicit differentiation is the topic of the next section. In fact, using implicit differentiation to find the derivative of $y=\ln x$ starts of the next section!
Page 91: $\mathbb R$ is used without any explanation.
Page 99: needs to be preceded by a discussion of even and odd functions. which are not defined until page 112.
Page 102: injective is used without definition.
Page 112, Exercise 5.5: students are asked to find intercepts even though they are not discussed in the text.
Interface
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. I have only a minor concern: some of the figures are marked with "(AP)" which points to Sage worksheet or an interactive applet. I found that not all the Sage worksheets opened correctly with Internet Explorer (for example Figure 2.1) although they work fine in Firefox and Google Chrome.
Grammatical Errors
I didn't notice any.
Cultural Relevance
The examples are similar to those found in any Calculus text.
CommentsThis review originated in the BC Open Textbook Collection and is licensed under CC BY-ND.