Precalculus 1.3

[Malcolm Lozada]

Inmathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.[1]

For students to succeed at finding the derivatives and antiderivatives with calculus, they will need facility with algebraic expressions, particularly in modification and transformation of such expressions. Leonhard Euler wrote the first precalculus book in 1748 called Introductio in analysin infinitorum (Latin: Introduction to the Analysis of the Infinite), which "was meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of differential and integral calculus."[2] He began with the fundamental concepts of variables and functions. His innovation is noted for its use of exponentiation to introduce the transcendental functions. The general logarithm, to an arbitrary positive base, Euler presents as the inverse of an exponential function.

Precalculus prepares students for calculus somewhat differently from the way that pre-algebra prepares students for algebra. While pre-algebra often has extensive coverage of basic algebraic concepts, precalculus courses might see only small amounts of calculus concepts, if at all, and often involves covering algebraic topics that might not have been given attention in earlier algebra courses. Some precalculus courses might differ with others in terms of content. For example, an honors-level course might spend more time on conic sections, Euclidean vectors, and other topics needed for calculus, used in fields such as medicine or engineering. A college preparatory/regular class might focus on topics used in business-related careers, such as matrices, or power functions.

A standard course considers functions, function composition, and inverse functions, often in connection with sets and real numbers. In particular, polynomials and rational functions are developed. Algebraic skills are exercised with trigonometric functions and trigonometric identities. The binomial theorem, polar coordinates, parametric equations, and the limits of sequences and series are other common topics of precalculus. Sometimes the mathematical induction method of proof for propositions dependent upon a natural number may be demonstrated, but generally coursework involves exercises rather than theory.

From the preface, "These are notes for a course in precalculus, as it is taught at New York City College of Technology - CUNY (where it is offered under the course number MAT 1375). Our approach is calculator based. For this, we will use the currently standard TI-84 calculator, and in particular, many of the examples will be explained and solved with it. However, we want to point out that there are also many other calculators that are suitable for the purpose of this course and many of these alternatives have similar functionalities as the calculator that we have chosen to use. An introduction to the TI-84 calculator together with the most common applications needed for this course is provided in appendix A. In the future we may expand on this by providing introductions to other calculators or computer algebra systems."

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.

When I took precalculus, we learned about polynomials and how to factor them, we learned about trigonometry and lots of great and useful identities there, and we learned about matrices. They didn't call it linear algebra, they just called it matrices. We learned how to multiply them, we learned a whole bunch of ridiculous complicated methods to calculate determinants. We learned to solve systems of linear equations using Cramer's Rule, which is the mathematical equivalent of scrubbing a floor with a toothbrush. Never once did we actually learn what matrices were, how they would be useful, or why we would want to know any of this. I found the whole exercise to be tedious and pointless and quickly braindumped all of the matrix stuff as soon as the class was over.

Now, 20 years, a degree in math, a degree in physics, a career as a software engineer and data scientist, using linear algebra in building neural networks and many other places later, I am teaching a precalculus class, and I still have no idea what the point of teaching about matrices in it is. Like maybe some basic matrix/vector multiplication would be useful, applying them to systems of linear equations, but why do we force our students to painfully calculate determinants by hand when literally no one does this and the only reason to study these algorithms is if you are implementing a computer program to calculate them.

Don't get me wrong, I am a huge proponent of learning fundamentals and practicing working through problems by hand before you give it to your calculator. It is important to understand the concepts behind what you are doing and how they applies to the world around you so you can use them to solve problems in applications and they aren't just pure abstractions. Except that doesn't really work for matrices because we never bother to explain what they are or how to apply them, even less so for determinants.

Linear algebra is a very useful field used quite a lot in physics, engineering, computer science, and any number of other areas. And it has nothing to do with calculus. When I took a linear algebra class, I had to relearn all of these fundamental operations anyway, except there weren't any lengthy exercises in calculating determinants by hand because no one ever actually does that outside of a precalculus class. I have students asking me why we are learning this and I honestly don't have a good answer to give them. Why is it there?

Vector algebra is a standard 3rd-semester calculus topic (e.g., see OpenStax Calculus 3, Ch. 2-3). This includes calculations of the dot product, cross product, and related values. Standard applications include calculating results of work, force, and business productions.

While this can be done purely algebraically (e.g., OpenStax does so), it's natural to express these operations in matrix notation, so calculus books may have some sections devoted to showing it in that form. Example on my bookshelf: this appears in Stein and Barcellos, Calculus and Analytic Geometry (5E, 1992), Sec. 12.5-12.7. Common exercises of the determinant and cross product include computing the areas of parallelograms, volumes of parallelipipeds, possibly crystallography, and torque. (In fact, calculation of parallelipiped volume from the end of the calculus sequence is one of the key half-dozen standard questions we use for assessment of our graduating math majors every year in my community college department.)

So presumably matrices appearing in a precalculus class are partly meant to set the stage for this use in calculus. Related, the precalculus course has covered solving linear equations (possibly by substitution and elimination), and it would be natural to show yet another option, the "next step", so to speak. And many of the students who take calculus will need linear algebra, so why not give them an introductory taste of all that content at the next level? (without calling the course something unwieldy like Preparation for Calculus, Linear Algebra, and Related Mid-Undergraduate Mathematical Topics, of course).

In addition, matrices also appear in the standard undergraduate discrete mathematics and data structures course texts as the basic way to represent general relations, graphs, and weighted graphs (and hence used in implementations for Euler circuits, Dijkstra's algorithm, Prim's and Kruskal's algorithms, etc.). Note this is a context in which matrices are not being used to represent linear transformations, so it's a case that highlights that's not a priori an essential part of the meaning of a matrix. In my own work, this is my own customary usage of matrices, for example.

So that's my understanding of the legacy of why those subjects have been included. If someone thinks that's poorly justified, or those topics have been squeezed out of those courses in recent years, then I could imagine coherent arguments for that.

The College Board made curriculum decisions for their new AP Precalculus course that align with sentiments you express. The course is divided into four units, where unit four is titled Functions Involving Parameters, Vectors, and Matrices. But unit 4 is not assessed on the exam. Here is an excerpt from their Course and Exam Description document:

Units 1, 2, and 3 topics comprise the content and conceptualunderstandings in which colleges and universities typically expectstudents to be proficient, in order to qualify for college creditand/or placement. Therefore, these topics are included on the AP Exam.Unit 4 consists of topics that teachers may include based on state orlocal requirements.

It is incorrect to say linear algebra has nothing to do with calculus: linear algebra is all over the place in multivariable calculus. Maybe you meant to say that linear algebra has nothing to do with single-variable calculus.

I too saw some matrix algebra (in the $2 \times 2$ case) in my high school precalculus course. I never thought much about why anyone decided to put it there. Possibly it was included to expose students to some matrix work so the concept of a matrix would not be completely new if/when the students get to multivariable calculus. Or the person designing the curriculum knew that matrices are a mathematical tool that also comes up in later coursework not relying on calculus, e.g., some "math for business/economics majors" courses that I've seen have a unit on matrices: consider the Leontief input/output model.

3a8082e126