Pre-Calculus Lesson Module Posted on MathAmazement.com

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MathAmazement.com has just completed a Pre-Calculus module that consists of twelve chapters, each with 5-12 sections. Each section contain sample problems along with solutions so that students can learn without needing to purchase a textbook.

MathAmazement.com has just completed a Pre-Calculus module that consists of twelve chapters, each with 5-12 sections. Each section contain sample problems along with solutions so that students can learn without needing to purchase a textbook.

The first chapter, called Chapter P, is on high school algebra, which is a prerequisite for the module. Topics include real numbers, exponents, scientific notation, radicals and rational expressions, polynomials, rational expressions, linear equations and inequalities and quadratic equations.

The next chapter (Chapter 1) is on graphs, functions, and models. Topics in this chapter include lines and slopes, distance and midpoint formulas, circles, and various topics concerning functions, such has the definition of a function, properties of functions, graphing functions, transformations of and combinations of functions, and modeling with functions.

Chapter 2 is concerned with polynomial and rational functions and complex numbers. Topics include the definition of complex numbers, complex arithmetic, quadratic functions, polynomial functions and their graphs, remainder and factor theorems for polynomials, zeros of polynomials, rational functions and their graphs, and polynomial and rational inequalities.

Chapter 3 concerns exponential and logarithmic functions in various bases, including simple exponential and logarithmic equations and modeling with these functions.

Chapter 4 concerns trigonometric functions. We start with a section on angles and their measure, followed by definitions of the six basic trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) in terms of both the unit circle and right triangles. Then we define the inverses of these functions. We also tabulate and graph all 12 of these functions and describe some of their propoerties. The chapter concludes with a section on applications of trigonometric functions.

Chapter 5 is on analytic trigonometry. A host of trigonometric identites are given and verified, such as sum and difference formulas, double-angle and half-angle formulas, and sum-to-product and product-to-sum formulas. The chapter concludes with a section on how to solve various trigonometric equations.

Chapter 6 is on additional topics in trigonometry, including the Law of Sines, the Law of Cosines, polar coordinates, polar forms of complex numbers, De Moivre's Theorem, vectors, and the dot product.

Chapter 7 concerns systems of equations and inequalities. Systems of equations in two and three variables are solved using row-reduction techniques. We also discuss partial fractions, systems of nonlinear equations in two variables, and systems of inequalities.

Chapter 8 is on matrices and determinants. The first section is concerned with basic properties of matrices, including how to add, subtract, and multiply them. The second section is on determinants and inverses of square matrices. The third section is on how to compute determinants and inverses as well as how to solve systems of linear equations by means of row reduction. The fourth section is on Cramer's Rule, and how to apply it to solve systems of linear equations. The last section is on linear coordinate transformations and the matrices associated with them.

Chapter 9 is on conic sections and analytic geometry. The three basic conic sections (the ellipse, the hyperbola, and the parabola) are all described in detail, including how to graph them and how to represent them by equations in both rectangular and polar coordinate as well as how their equations are modified by rotations. There is also a section on parametric equations.

Chapter 10 is on sequences, induction, and probability. The first section defines and describes general sequences, series, and summation notation. The next section describes arithmetic sequences and series, and the following section describes geometric sequences and series. The fourth section describes how to prove various theorems by means of mathematical induction. The fifth section states and proves the binomial theorem. The sixth section is on combinatorics, including permutations and combinations. The last section is an introduction to probability theory.

Chapter 11, the final chapter, is an introduction to calculus. The first section defines limits of functions. The second section defines the derivative and describes how to compute derivatives of simple functions, including polynomials. The third section is on the product rule, including applications. The fourth section is on the quotient rule and how to use it to compute derivatives of rational functions. The last section is on physical applications of derivatives, in particular, the law of falling bodies.

The author of this website is David Terr, a doctorate graduate in Mathematics from UC Berkeley.

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