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How To Study with DWFK Precalculus

Chapter 9: Discrete Mathematics

In this chapter we have assembled a variety of topics from discrete mathematics, some of which have appeared in traditional precalculus courses (sequences and series, the Binomial Theorem), some of which are common in modern precalculus courses (probability, mathematical induction), and some of which have been touted for years as important for all students, but which have been hard to squeeze into the curriculum (conditional probability, statistics, and data analysis).

A new national emphasis on "quantitative literacy" argues powerfully for the inclusion of these topics, even at the cost of omitting some traditional syllabus standards. The hope has been that modern technology can free up some time by re-establishing our pedagogical priorities. We have tried to suggest that possibility throughout this book, even while including some arguably optional topics due to the necessity of adhering to state mandates.

Some teachers will assume (perhaps incorrectly) that their students have already seen these discrete topics in their algebra courses. Other teachers will skip this chapter even though they realize that this course is potentially the last opportunity for their students to study these topics during the years of their formal education. If you do wind up skipping this chapter in your precalculus course, you might consider familiarizing yourself with the material on your own. You may not use it in calculus, but you will see it in a multitude of other settings.

Section 9.1 Section 9.4 Section 9.7

Section 9.2 Section 9.5

Section 9.3 Section 9.6

Section 9.1 Basic Combinatorics

Objectives
You will be able to use the multiplication principle of counting, permutations, or combinations to count the number of ways that a described task can be done.

Key Ideas

Combination Multiplication principle of counting

Continuum n-set

Discrete mathematics Permutation

Explanatory variable Response variable

Study Tips
You might have heard the English word "discreet" (tactful), but few will know what "discrete" means, especially in its mathematical sense. We have tried to explain that briefly in the opening paragraph of this section.

It might seem that counting would be a pretty easy topic after the other topics in this book, but you will soon discover otherwise. The "alert" on page 674 is right on target. The best way for you to learn how to apply the formulas properly is to practice. We have tried to include enough exercises to allow for that, but additional examples can be found in the textbook ancillaries—and, indeed, in any textbook on finite math, discrete math, or probability.

Technology Tips
The nPr and nCr selections on graphing calculators have made counting topics more accessible and interesting to all students. When we are looking for the number of 5-card poker hands, the answer 2,598,960 is much more satisfying than (the equally correct) . We urge you to use your calculator all the time in this chapter.
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Section 9.2 The Binomial Theorem

Objectives
You will be able to expand an integral power of a binomial using the Binomial Theorem or Pascal's triangle. You will also be able to find the coefficient of a given term of a binomial expansion.

Key Ideas

Binomial coefficient
Binomial Theorem
Pascal's triangle

Study Tips
This section is short, but important. Calculus students will need to know the Binomial Theorem in order to prove the basic formula for the derivative of xⁿ (and perhaps for other things as well). We hope that the connection to Pascal's triangle will make it interesting for all students.

Technology Tips
The command "N nCr seq(X,X,0,N)" will produce the Nth row of Pascal's triangle on the home screen. Here it is producing row 5:

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Section 9.3 Probability

Objectives
You will be able to identify a sample space and calculate probabilities and conditional probabilities of events in sample spaces with equally likely or unequally likely outcomes.

Key Ideas

Binomial distribution Probability distribution

Conditional probability Probability function

Equally likely outcomes Probability of an event

Event Sample space

Independent event Tree diagram

Multiplication principle of probability Venn diagram

Study Tips
Most students who have studied algebra from modern textbooks will have seen probability problems involving equally likely outcomes. We extend this knowledge to probability functions for sample spaces with unequally likely outcomes. We also include subsections on conditional probability and the binomial distribution, which have begun to appear in state requirements for senior mathematics courses.

Technology Tips
If your calculator will do binomial distributions (see margin note on page 728), you can graph them fairly easily. Store the outcomes (0 through N) in L1 and store the probabilities (binompdf(N, p)) in L2. For example, if a 70% free throw shooter shoots 10 independent free throws, we produce the graph of the probability distribution as shown below:

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Section 9.4 Sequences and Series

Objectives
You will be able to express arithmetic and geometric sequences explicitly and recursively, including in sigma notation. You will be able to use basic summation formulas to find the sums of finite series or of convergent infinite geometric series.

Key Ideas

Arithmetic sequence Infinite series

Convergent series Partial sum (of a series)

Divergent series Recursively-defined sequence

Fibonacci sequence Sequence

Geometric sequence Sum of an infinite series

Index of summation Summation notation

Study Tips
This section, although properly in the Discrete Mathematics chapter, is an important section for all future calculus students to study. An understanding of convergent series and a familiarity with sigma notation are crucial for understanding the definite integral, one of the two foundational concepts encountered in the first year of calculus. As the examples and exercises indicate, though, there are also plenty of immediate applications.

Technology Tips
This section is already well larded with technology tips. Wary of introducing a new learning curve to climb this close to the end of the year, we have deliberately chosen not to rely on the calculator's less-familiar sequence mode (although we demonstrate its potential in Example 6). That is why we show other ways of generating sequences and graphs wherever possible.

As a general rule of good practice, we recommend that you never express your written answers in "calculator" notation. For example, you should not write as "sum(seq(K, K, 1, 10))"—even if that is how you wind up computing it.
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Section 9.5 Mathematical Induction

Objectives
You will be able to understand proofs by mathematical induction and will be able to produce mathematical induction proofs for simple propositions.

Key Ideas

Anchor Inductive step

Inductive hypothesis Mathematical induction

Study Tips
Mathematical induction is most assuredly not a topic that students will need for a first-year calculus course, and only the future mathematics majors are very likely to use it thereafter. It entered the precalculus curriculum as a means for proving summation formulas like the one in Example 2, necessary for computing limits of certain Riemann sums in calculus. Modern calculus textbooks put a greater emphasis on Riemann sums, but (ironically) the actual summing is rarely done with formulas, because technology does it better—making clear what the complicated formulas used to make unnecessarily muddy.

Nonetheless, showing you some simple (and, we hope, interesting) examples of mathematical induction will extend your understanding of proof, which is a worthwhile goal of the course. That said, we admit that classes that are pressed for time might have no choice but to skip this section.

Technology Tips
It is not necessary to run out and buy a Tower of Hanoi game to understand this section, although the game is certainly available if you want to do so. If you have access to an on-line computer, you can visit a number of web sites that simulate the game (with graphics ranging from primitive to classy). Here are a few that were available at the time these tips were written:

http://www.cut-the-knot.com/recurrence/hanoi.html
http://www.mazeworks.com/hanoi/index.htm
http://www.lhs.berkeley.edu/java/tower.html
http://javaboutique.internet.com/Tower/
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Section 9.6 Statistics and Data (Graphical)

Objectives
You will be able to display categorical and quantitative data in various kinds of graphs and will be able to interpret data that are graphically presented.

Key Ideas

Back-to-back stemplots Line graph

Bar chart Picture graph

Categorical variable Pie chart

Circle graph Quantitative variable

Frequency distribution Stem-and-leaf plot

Frequency table Stemplot

Histogram Time plot

Study Tips
As data analysis in the computer age comes to rely more and more on various graphical representations, it becomes increasingly necessary for educated citizens to understand how to use and interpret them. This section gives a quick look at the most common graphs, using data that we hope will interest you and numbers as current as we can get them. You do not need to know statistics in order to understand calculus, but it is often useful, and frequently essential, for understanding everything else in the mathematical, physical, biological, and social sciences.

Technology Tips
This section relies heavily on graphing calculators for producing statistical plots, so it is filled with technology tips already. You should be using your graphing calculator all the time while studying the connections between data and graphs.

Once you start doing stat plots on your calculator, you will occasionally encounter a "DIM MISMATCH" error when you try to graph a function. This usually means that you have unintentionally left a stat plot on, and that the lists have changed since you set it up.
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Section 9.7 Statistics and Data (Algebraic)

Objectives
You will know how and when to use various measures of center and variation to describe quantitative data, including mean, median, mode, the five-number summary, boxplots, variance, and standard deviation. You will understand the significance of the normal distribution and know how to use it to analyze data where appropriate.

Key Ideas

Boxplot (Box-and-whisker plot) Median Parameter Standard deviation

Descriptive statistics Mode Population Statistic

Five-number summary Modified boxplot Quartiles Symmetric

Inferential statistics Normal (Gaussian) curve Range Variance

Interquartile range (IQR) Normal distribution Sample

Mean Outlier Skewed (left or right)

Study Tips
The Study Tips from the previous section related to the importance of statistics apply equally well to this section. Although we introduce many new terms here, not all of them will be unfamiliar to today's students.

As in the previous section, we have tried to use recent, interesting data for the examples. We have admittedly packed a lot of material into yhis section, but our intent was to give as thorough an overview as possible without going into any concept too deeply.

Technology Tips
The ability to do significant number-crunching on hand-held devices is a great breakthrough for teaching statistics to beginners. We hope you will feel free to use your calculator all the time while studying this section.

The statistical capabilities of most graphing calculators go well beyond the few mentioned in this brief overview. Indeed, the fact that those capabilities are already in your hands is one more reason why we need to include more statistical topics in the curriculum.