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Mathematics
Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences
Calculus: Applications and Technology 3rd edition, CD-ROM by Edmond C. Tomastik, Hubert Hohn (Brooks/Cole Publishing Company) Calculus: Applications and Technology is designed to be used in a one- or two-semester calculus course aimed at students majoring in business, management, economics, or the life or social sciences. The text is written for a student with two years of high school algebra. A wide range of topics is included, giving the instructor considerable flexibility in designing a course.
Since the text uses technology as a major tool, the reader is required to use a computer or a graphing calculator. The Student's Suite CD with the text, gives all the details, in user friendly terms, needed to use the technology in conjunction with the text. This text, together with the accompanying Student's Suite CD, constitutes a completely organized, self-contained, user-friendly set of material, even for students without any knowledge of computers or graphing calculators.
The writing of this text has been guided by four basic principles, all of which are consistent with the call by national mathematics organizations for reform in calculus teaching and learning.
- The Rule of Four: Where appropriate, every topic should be presented graphically, numerically, algebraically, and verbally.
- Technology: Incorporate technology into the calculus instruction.
- The Way of Archimedes: Formal definitions and procedures should evolve from the investigation of practical problems.
- Teaching Method: Teach calculus using the investigative, exploratory approach.
The Rule of Four. By always bringing graphical and numerical, as well as algebraic, viewpoints to bear on each topic, the text presents a conceptual understanding of the calculus that is deep and useful in accommodating diverse applications. Some-times a problem is done algebraically, then supported numerically and/or graphically (with a grapher). Sometimes a problem is done numerically and/or graphically (with a grapher), then confirmed algebraically. Other times a problem is done numerically or graphically because the algebra is too time-consuming or impossible.
Technology. Technology permits more time to be spent on concepts, problem solving, and applications. The technology is used to assist the student to think about
the geometric and numerical meaning of the calculus, without undermining the algebraic aspects. In this process, a balanced approach is presented. I point out clearly that the computer or graphing calculator might not give the whole story, motivating the need to learn the calculus. On the other hand, I also stress common situations in which exact solutions are impossible, requiring an approximation technique using the technology. Thus, I stress that the graphers are just another needed tool, along with the calculus, if we are to solve a variety of problems in the applications.
Applications and the Way of Archimedes. The text is written for users of mathematics. Thus, applications play a central role and are woven into the development of the material. Practical problems are always investigated first, then used to motivate, to maintain interest, and to use as a basis for developing definitions and procedures. Here too, technology plays a natural role, allowing the forbidding and time-consuming difficulties associated with real applications to be overcome.
The Investigative, Exploratory Approach. The text also emphasizes an investigative and exploratory approach to teaching. Whenever practical, the text gives students the opportunity to explore and discover for themselves the basic calculus concepts. Again, technology plays an important role. For example, using their graphers, students discover for themselves the derivatives of x2, x3, and x4 and then generalize to x n. They also discover the derivatives of In x and e x. None of this is realistically possible without technology.
Student response in the classroom has been exciting. My students enjoy using their computers or graphing calculators and feel engaged and part of the learning process. I find students much more receptive to answering questions about their observations and more ready to ask questions.
A particularly effective technique is to take 15 or 20 minutes of class time and have students work in small groups to do an exploration or make a discovery. By walking around the classroom and talking with each group, the instructor can elicit lively discussions, even from students who do not normally speak. After such a minilab the whole class is ready to discuss the insights that were gained.
Fully in sync with current goals in teaching and learning mathematics, every section in the text includes a more challenging exercise set that encourages exploration, investigation, critical thinking, writing, and verbalization.
Interactive Illustrations. The Student's Suite CD with interactive illustrations is now included with each text. These interactive illustrations provide the student and instructor with wonderful demonstrations of many of the important ideas in the calculus. These demonstrations and explorations are highlighted in the text at appropriate times. They provide an extraordinary means of obtaining deep and clear insights into the important concepts. We are extremely excited to present these in this format. They are one more important example of the use of technology and fit perfectly into the investigative and exploratory approach.
Chapter 1. Section 1.0 contains some examples that clearly indicate instances when the technology fails to tell the whole story and therefore motivates the need to learn the calculus. This failure of the technology to give adequate information is complemented elsewhere by examples in which our current mathematical knowledge is inadequate to find the exact values of critical points, requiring us to use some approximation technique on our computers or graphing calculators. This theme of needing both mathematical analysis and technology to solve important problems continues
throughout the text. Section 1.1 begins with functions; the second section contains applications of linear and nonlinear functions in business and economics, including an introduction to the theory of the firm. Next is a section on exponential functions, followed by the algebra of functions, and finally logarithmic functions.
Chapter 2. This chapter consists entirely of fitting curves to data using least squares. It includes linear, quadratic, cubic, quartic, power, exponential, logarithmic, and logistic regression.
Chapter 3. Chapter 3 begins the study of calculus. Section 3.1 introduces limits intuitively, lending support with many geometric and numerical examples. Section 3.2 covers average and instantaneous rates of change. Section 3.3 is on the derivative. In this section, technology is used to find the derivative. From the limit definition of derivative This is confirmed algebraically in Chapter 4. Section 3.4 covers local linearity and introduces marginal analysis.
Chapter 4. Section 4.1 begins the chapter with some rules for derivatives. In this section we also discover the derivatives of a number of functions using technology. In the same way we find the derivative. This is an exciting and innovative way for students to find these derivatives. Now that the derivatives of 1n x and ex are known, these functions can be used in conjunction with the product and quotient rules found in Section 4.2, making this material more interesting and compelling. Section 4.3 covers the chain rule, and Section 4.4 derives the derivatives of the exponential and logarithmic functions in the standard fashion. Section 4.5 is on elasticity of demand, and Section 4.6 is on the management of renewable natural resources.
Chapter 5. Graphing and curve sketching are begun in this chapter. Section 5.1 describes the importance of the first derivative in graphing. We show clearly that the technology can fail to give a complete picture of the graph of a function, demonstrating the need for the calculus. We also consider examples in which the exact values of the critical points cannot be determined and thus need to resort to using an approximation technique on our computers or graphing calculators. Section 5.2 presents the second derivative, its connection with concavity, and its use in graphing. Section 5.3 covers limits at infinity, Section 5.4 covers additional curve sketching, and Section 5.5 covers absolute extrema. Section 5.6 includes optimization and modeling. Section 5.7 covers the logistic model. Section 5.8 covers implicit differentiation and related rates. Extensive applications are given, including Laffer curves used in tax policy, population growth, radioactive decay, and the logistic equation with derived estimates of the limiting human population of the earth.
Chapter 6. Sections 6.1 and 6.2 present antiderivatives and substitution, respectively. Section 6.3 lays the groundwork for the definite integral by considering left-and right-hand Riemann sums. Here again technology plays a vital role. Students can easily graph the rectangles associated with these Riemann sums and see graphically and numerically what happens as n — oo. Sections 6.4, 6.5, and 6.6 cover the definite integral, the fundamental theorem of calculus, and area between two curves, respectively. Section 6.7 presents a number of additional applications of the integral, including average value, density, consumer's and producer's surplus, Lorentz's curves, and money flow.
Chapter 7. This chapter contains material on integration by parts, integration using tables, numerical integration, and improper integrals.
Chapter 8. Section 8.1 presents an introduction to functions of several variables, including cost and revenue curves, Cobb-Douglas production functions, and level curves. Section 8.2 then introduces partial derivatives with applications that include competitive and complementary demand relations. Section 8.3 gives the second derivative test for functions of several variables and applied application on optimization. Section 8.4 is on Lagrange multipliers and carefully avoids algebraic complications. The tangent plane approximations is presented in Section 8.5. Sec-tions 8.6, on double integrals, covers double integrals over general domains, Riemann sums, and applications to average value and density. A program is given for the graphing calculator to compute Riemann sums over rectangular regions.
Chapter 9. This chapter covers an introduction to the trigonometric functions. Section 9.1 starts with angles, and Sections 9.2, 9.3, and 9.4 cover the sine and cosine functions, including differentiation and integration. Section 9.5 covers the remaining trigonometric functions. Notice that these sections include extensive business applications, including models by Samuelson and Phillips. Notice in Section 9.3 that the derivatives of sin x and cos x are found by using technology and that technology is used throughout this chapter.
Chapter 10. This chapter covers Taylor polynomials and infinite series. Sections 10.1, 10.2, and 10.7 constitute a subchapter on Taylor polynomials. Section 10.7 is written so that the reader can go from Section 10.2 directly to Section 10.7. Section 10.1 introduces Taylor polynomials, and Section 10.2 considers the errors in Taylor polynomial approximation. The graphers are used extensively to compare the Taylor polynomial with the approximated function. Section 10.7 looks at Taylor series, in which the interval of convergence is found analytically in the simpler cases while graphing experiments cover the more difficult cases. Section 10.3 introduces infinite sequences, and Sections 10.4, 10.5, and 10.6 are on infinite series and includes a variety of test for convergence and divergence.
Chapter 11. This chapter is on probability. Section 11.1 is a brief review of discrete probability. Section 11.2 then considers continuous probability density functions, and Section 11.3 presents the expected value and variance of these functions. Section 11.4 covers the normal distribution, arguably the most important probability density function.
Chapter 12. This chapter is a brief introduction to differential equations and includes the technique of separation of variables, approximate solutions using Euler's method, some qualitative analysis, and mathematical problems involving the harvesting of a renewable natural resource. The graphing calculator is used to graph approximate solutions and to do some experimentation.
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