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Calculus for Scientists and Engineers

William Briggs, Lyle Cochran, and Bernard Gillett
Publication Date: 
Number of Pages: 
[Reviewed by
Peter Olszewski
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William Briggs, Lyle Cochran, and Bernard Gillett have written, in my opinion, a successful Calculus text. The true success of this text is that it reflects how today’s college Calculus students learn: beginning with the exercises and referencing back to the worked out examples within the text.

The book has well thought out examples that will be clear to the student. Many pictures are given through out the text to aid students’ understanding of the concepts. While reading the text, it was as though I was sitting in a Calculus class and my instructor was talking to me. In addition, there was a lot of handholding, but not overbearing handholding. The text is to the point! I fully enjoyed the pure mathematical examples with the well thought out pictures; they show the years of teaching experience of each author.

One of the first things I noticed in Chapter 1 was the way domain and range were treated from a graphical perspective, by taking any point on the curves and mapping it back to the x and y axes. The use of color is also helpful. In addition, I liked how the authors introduced the concept of secant lines early. In most other texts, they aren’t introduced until the following chapter, which is typically about limits.

What I also really enjoyed about Chapter 1 was the review of trigonometry in Section 1.3. Many of my calculus students have forgotten the basic concepts of trigonometry, so it is a wonderful idea to have a review available, to be used at the instructors’ choice.

As the title states, this text is for Scientists and Engineers, so it is fitting for secant lines to be quickly connected to velocity. The motivation for considering the slope of secant lines is very well done and the diagrams on page 41 are excellent. This is what the students need to see.

Jumping to the middle of the book, I strongly believe students are overwhelmed when it comes down to sequences and series. In most other texts I’ve read, sequences and series and all related topics are contained in one big chapter. Having these concepts broken into two chapters is more digestible for the students.

Of course, there are places where the book can be improved. I only mention a few.

In Chapter 1, the authors discuss domains and ranges of functions but not for compositions of functions. I would like to see examples of these, with graphics to support the solutions. In addition, having examples of finding domains and ranges of piecewise defined functions and domains and ranges of transformations of functions would further enhance this section. It has been my experience that students need extra guidance in finding domains and ranges.

After Theorem 2.2, I would recommend stating another theorem about using direct substitution of a polynomial and rational function followed by examples. In addition, I would like to see more examples following Example 6 on “other techniques” for finding limits analytically.

Many times, students forget simple algebraic concepts needed for Calculus. In the exercise set for Section 2.3, I believe the authors should limit the amount of problems where the limits of functions are tending towards a number a. I believe it would be more useful for students to see the direct numerical results as this is what they are more likely to see in their careers.

For Section 2.4, as an aid to help students understand the logic of when functions tend to zero and to infinity, it’s useful to have an informal statement that 1/large tends to 0 and 1/small tends to infinity. This will help students quickly recall these two critical facts.

In Section 2.6, I feel as though there should be an example using the Intermediate Value Theorem involving a function before proceeding to the financial application.

The section on higher order derivatives in Section 3.2 is out of place. I strongly believe higher order derivatives need a section all their own, as there are many examples students need to see.

My suggestion for Sections 3.5 and 3.6 is that they be flipped. If the Chain Rule is learned first, many Chain Rule applications problems can be woven into the section on derivatives as rates of change.

I believe the statement of the Second Derivative Test in Section 4.2 will confuse students. Instead of saying, “If f′′(c) = 0, the test is inconclusive; f may have a local maximum, local minimum, or neither at c” I would suggest, “In such a case, the First Derivative Test can be used to determine if f is a local maximum, minimum, or neither.”

While Example 3 is excellent in Section 4.3, I believe another example is needed before Example 3 involving a rational function that contains both vertical and horizontal asymptotes and a hole in the graph. In addition, Section 2.5 should be inserted in Chapter 4 before this section since the important connection to limits at infinity can now be made with the summary of curve sketching.

I was hoping to see more examples in Sections 4.5–4.7. I also feel as though Section 4.7, L’Hôpital’s Rule, is out of place with the rest of the text. I believe this section should be either contained in Section 7.6 or be a new section before Section 7.6.

Reading further into the text, I believe there could be many more applications presented. For example, in Chapter 13, there are too many proofs in the homework set for dot products and the section on cross products ends very quickly with not enough applications. Section 13.7 is much better as the authors give many more applications. I especially enjoyed reading through Examples 5 and 6 in this section. So many times, students are told to ignore friction and air resistance. Here is a problem were the angle must be found to adjust the flight of a ball.

Moving to Section 13.9, once again, I believe there are too many proofs and not enough applied problems. After all, part of the title of the book is “for Scientists and Engineers”!

In the past, when I have taught engineering students, they wish to see how the mathematics they learn, whether it be calculus or matrices, will be used. Perhaps having student projects at the end of each chapter or section, specifically for the Sciences and Engineers could further enhance the text. The teaching and writing style of this text is excellent but I believe more applications would further student’s motivation, creativity, and problem solving skills.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at Webpage: Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.

1. Functions

1.1 Review of functions

1.2 Representing functions

1.3 Trigonometric functions and their inverses



2. Limits

2.1 The idea of limits

2.2 Definitions of limits

2.3 Techniques for computing limits

2.4 Infinite limits

2.5 Limits at infinity

2.6 Continuity

2.7 Precise definitions of limits



3. Derivatives

3.1 Introducing the derivative

3.2 Rules of differentiation

3.3 The product and quotient rules

3.4 Derivatives of trigonometric functions

3.5 Derivatives as rates of change

3.6 The Chain Rule

3.7 Implicit differentiation

3.8 Derivatives of inverse trigonometric functions

3.9 Related rates



4. Applications of the Derivative

4.1 Maxima and minima

4.2 What derivatives tell us

4.3 Graphing functions

4.4 Optimization problems

4.5 Linear approximation and differentials

4.6 Mean Value Theorem

4.7 L'Hôpital's Rule

4.8 Newton's method

4.9 Antiderivatives



5. Integration

5.1 Approximating areas under curves

5.2 Definite integrals

5.3 Fundamental Theorem of Calculus

5.4 Working with integrals

5.5 Substitution rule



6. Applications of Integration

6.1 Velocity and net change

6.2 Regions between curves

6.3 Volume by slicing

6.4 Volume by shells

6.5 Length of curves

6.6 Surface area

6.7 Physical applications

6.8 Hyperbolic functions



7. Logarithmic and Exponential Functions

7.1 Inverse functions

7.2 The natural logarithm and exponential functions

7.3 Logarithmic and exponential functions with general bases

7.4 Exponential models

7.5 Inverse trigonometric functions

7.6 L'Hôpital's rule and growth rates of functions



8. Integration Techniques

8.1 Basic approaches

8.2 Integration by parts

8.3 Trigonometric integrals

8.4 Trigonometric substitutions

8.5 Partial fractions

8.6 Other integration strategies

8.7 Numerical integration

8.8 Improper integrals



9. Differential Equations

9.1 Basic ideas

9.2 Direction fields and Euler's method

9.3 Separable differential equations

9.4 Special first-order differential equations

9.5 Modeling with differential equations



10. Sequences and Infinite Series

10.1 An overview

10.2 Sequences

10.3 Infinite series

10.4 The Divergence and Integral Tests

10.5 The Ratio, Root, and Comparison Tests

10.6 Alternating series



11. Power Series

11.1 Approximating functions with polynomials

11.2 Properties of power series

11.3 Taylor series

11.4 Working with Taylor series



12. Parametric and Polar Curves

12.1 Parametric equations

12.2 Polar coordinates

12.3 Calculus in polar coordinates

12.4 Conic sections



13. Vectors and Vector-Valued Functions

13.1 Vectors in the plane

13.2 Vectors in three dimensions

13.3 Dot products

13.4 Cross products

13.5 Lines and curves in space

13.6 Calculus of vector-valued functions

13.7 Motion in space

13.8 Length of curves

13.9 Curvature and normal vectors



14. Functions of Several Variables

14.1 Planes and surfaces

14.2 Graphs and level curves

14.3 Limits and continuity

14.4 Partial derivatives

14.5 The Chain Rule

14.6 Directional derivatives and the gradient

14.7 Tangent planes and linear approximation

14.8 Maximum/minimum problems

14.9 Lagrange multipliers



15. Multiple Integration

15.1 Double integrals over rectangular regions

15.2 Double integrals over general regions

15.3 Double integrals in polar coordinates

15.4 Triple integrals

15.5 Triple integrals in cylindrical and spherical coordinates

15.6 Integrals for mass calculations

15.7 Change of variables in multiple integrals



16. Vector Calculus

16.1 Vector fields

16.2 Line integrals

16.3 Conservative vector fields

16.4 Green's theorem

16.5. Divergence and curl

16.6 Surface integrals

16.6 Stokes' theorem

16.8 Divergence theorem