Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences by Ernest F. Haussler, Richard J. Wood & Richard S. Paul 14th Edition

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Pedagogy and Hallmark Features

Applications: An abundance and variety of applications for the intended audience appear throughout the book so that students see frequently how the mathematics they are learning can be used. These applications cover such diverse areas as business, economics, biology, medicine, sociology, psychology, ecology, statistics, earth science, and archaeology. Many of these applications are drawn from literature and are documented by references, sometimes from the Web. In some, the background and context are given in order to stimulate interest. However, the text is self-contained, in the sense that it assumes no prior exposure to the concepts on which the applications are based. (See, for example, Chapter 15, Section 7, Example 2.)

Now Work Problem N: Throughout the text we have retained the popular Now Work Problem N feature. The idea is that after a worked example, students are directed toan end of section problem (labeled with a colored exercise number) that reinforces theideas of the worked example. This gives students an opportunity to practice what theyhave just learned. Because the majority of these keyed exercises are odd-numbered, studentscan immediately check their answer in the back of the book to assess their level ofunderstanding. The complete solutions to the odd-numbered exercises can be found inthe Student Solutions Manual.

Cautions: Cautionary warnings are presented in very much the same way an instructor would warn students in class of commonly-made errors. These appear in the margin, along with other explanatory notes and emphases.

Definitions, key concepts, and important rules and formulas: These are clearly stated and displayed as a way to make the navigation of the book that much easier for the student. (See, for example, the Definition of Derivative in Section 11.1)

Review material: Each chapter has a review section that contains a list of important terms and symbols, a chapter summary, and numerous review problems. In addition, key examples are referenced along with each group of important terms and symbols.

Inequalities and slack variables: In Section 1.2, when inequalities are introduced we point out that a _ b is equivalent to “there exists a non-negative number s such that a C s D b”. The idea is not deep but the pedagogical point is that slack variables, key to implementing the simplex algorithm in Chapter 7, should be familiar and not distract from the rather technical material in linear programming.

Absolute value: It is common to note that ja ¿� bj provides the distance from a to b. In Example 4e of Section 1.4 we point out that “x is less than _ units from _” translates as jx ¿� _j < _. In Section 1.4 this is but an exercise with the notation, as it should be, but the point here is that later (in Chapter 9) _ will be the mean and _ the standard deviation of a random variable. Again we have separated, in advance, a simple idea from a more advanced one. Of course Problem 12 of Problems 1.4, which asks the student to set up jf.x/ ¿� Lj < _, has a similar agenda for Chapter 10 on limits.

Early treatment of summation notation: This topic is necessary for study of the definite integral in Chapter 14 but it is useful long before that. Since it is a notation that is new to most students at this level, but no more than a notation, we get it out of the way in Chapter 1. By using it when convenient, before coverage of the definite integral, it is not a distraction from that challenging concept.

Section 1.6 on sequences: This section provides several pedagogical advantages. The very definition is stated in a fashion that paves the way for the more important and more basic definition of function in Chapter 2. In summing the terms of a sequence we are able to practice the use of summation notation introduced in the preceding section. The most obvious benefit though is that “sequences” allows us a better organization in the annuities section of Chapter 5. Both the present and the future values of an annuity are obtained by summing (finite) geometric sequences. Later in the text, sequences arise in the definition of the number e in Chapter 4, in Markov chains in Chapter 9, and in Newton’s method in Chapter 12, so that a helpful unifying reference is obtained.

Sum of an infinite sequence: In the course of summing the terms of a finite sequence, it is natural to raise the possibility of summing the terms of an infinite sequence. This is a nonthreatening environment in which to provide a first foray into the world of limits. We simply explain how certain infinite geometric sequences have well-defined sums and phrase the results in a way that creates a toehold for the introduction of limits in Chapter 10. These particular infinite sums enable us to introduce the idea of a perpetuity, first informally in the sequence section, and then again in more detail in a separate section in Chapter 5.

Section 2.8, Functions of Several Variables: The introduction to functions of several variables appears in Chapter 2 because it is a topic that should appear long before Calculus. Once we have done some calculus there are particular ways to use calculus in the study of functions of several variables, but these aspects should not be confused with the basics that we use throughout the book. For example, “a-sub-n-angle-r” and “s-sub-nangle- r” studied in the Mathematics of Finance, Chapter 5, are perfectly good functions of two variables and Linear Programming seeks to optimize linear functions of several variables subject to linear constraints.

Leontief’s input-output analysis in Section 6.7: In this section we have separated various aspects of the total problem. We begin by describing what we call the Leontief matrix A as an encoding of the input and output relationships between sectors of an economy. Since this matrix can often be assumed to be constant, for a substantial period of time, we begin by assuming that A is a given. The simpler problem is then to determine the production X which is required to meet an external demand D for an economy whose Leontief matrix is A. We provide a careful account of this as the solution of .I¿�A/X D D.

Since A can be assumed to be fixed while various demands D are investigated, there is some justification to compute .I ¿� A/¿�1 so that we have X D .I ¿� A/¿�1D. However, use of a matrix inverse should not be considered an essential part of the solution. Finally, we explain how the Leontief matrix can be found from a table of data that might be available to a planner.

Birthday probability in Section 8.4: This is a treatment of the classic problem of determining the probability that at least 2 of n people have their birthday on the same day. While this problem is given as an example in many texts, the recursive formula that we give for calculating the probability as a function of n is not a common feature. It is reasonable to include it in this book because recursively defined sequences appear explicitly in Section 1.6.

Markov Chains: We noticed that considerable simplification to the problem of finding steady state vectors is obtained by writing state vectors as columns rather than rows. This does necessitate that a transition matrix T D .tij¿ have tij D“probability that next state is i given that current state is j” but avoids several artificial transpositions.

Sign Charts for a function in Chapter 10: The sign charts that we introduced in the 12th edition now make their appearance in Chapter 10. Our point is that these charts can be made for any real-valued function of a real variable and their help in graphing a function begins prior to the introduction of derivatives. Of course we continue to exploit their use in Chapter 13 “Curve Sketching” where, for each function f, we advocate making a sign chart for each of f, f0, and f00, interpreted for f itself. When this is possible, the graph of the function becomes almost self-evident. We freely acknowledge that this is a blackboard technique used by many instructors, but it appears too rarely in textbooks.