Table Of Content

CK-12 F OUNDATION Pre-Calculus Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) Almukkahal Fiori Fortgang Landers Vigil To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook mate- rials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. 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Printed: August 11, 2011 Authors Raja Almukkahal, Nick Fiori, Art Fortgang, Mara Landers, Melissa Vigil i www.ck12.org Contents 1 Analyzing Conic Sections 1 1.1 Introduction to Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Circles and Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.5 General Algebraic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2 Analyzing Functions 63 2.1 Identifying Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2 Minimums and Maximums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.3 Increasing and decreasing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.4 End Behavior of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.5 Function families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.6 Transformations of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.7 Operations and Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.8 Headline text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.9 Functions and Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3 Analyzing Polynomial and Rational Functions 137 3.1 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.2 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.3 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 3.4 Analyzing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.5 Polynomial and Rational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.6 Finding Real Zeros of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 3.7 Approximating Real Zeros of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . 213 3.8 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 www.ck12.org ii 4 Analyzing Exponential and Logarithmic Functions 223 4.1 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 4.2 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 4.3 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 4.4 Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 4.5 Exponential and Logarithmic Models and Equations . . . . . . . . . . . . . . . . . . . . . . 265 4.6 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.7 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 4.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 5 Polar Equations and Complex Numbers 300 5.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.2 Polar-Cartesian Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 5.3 Systems of Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 5.4 Imaginary and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 5.5 Operations on Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 5.6 Trigonometric Form of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 5.7 Product and Quotient Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 5.8 Powers and Roots of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 6 Vectors 351 6.1 Vectors in a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 6.2 Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 6.3 Dot Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 6.4 Cross Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 6.5 Planes in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 6.6 Vector Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 6.7 Vector Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 6.8 Applications of Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 7 Introduction to Calculus 414 7.1 Limits (An Intuitive Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 7.2 Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 7.3 Tangent Lines and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 8 Sequences, Series, and Mathematical Induction 435 iii www.ck12.org 8.1 Recursive and Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 8.2 Summation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 8.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 8.4 Mathematical Induction, Factors, and Inequalities . . . . . . . . . . . . . . . . . . . . . . . 454 8.5 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 8.6 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 www.ck12.org iv Chapter 1 Analyzing Conic Sections 1.1 Introduction to Conic Sections Learning Objectives • Consider the results when two simple mathematical objects are intersected. • Be comfortable working with an infinite two-sided cone. • Know the basic types of figures that result from intersecting a plane and a cone. • Know some of the history of the study of conic sections. Introduction: Intersections of Figures Some of the best mathematical shapes come from intersecting two other important shapes. Two spheres intersect to form a circle: Two planes intersect to form a line: 1 www.ck12.org While simple and beautiful, for these two examples of intersections there isn’t much else to investigate. For the spheres, no matter how we put them together, their intersection is always either nothing, a point (when they just touch), a circle of various sizes, or a sphere if they happen to be exactly the same size and coincide. Once we’ve exhausted this list, the inquiry is over. Planes are even simpler: the intersection of two distinct planes is either nothing (if the planes are parallel) or a line. But some intersections yield more complex results. For instance a plane can intersect with a cube in numerous ways. Below a plane intersects a cube to form an equilateral triangle. Here a plane intersects a cube and forms a regular hexagon. Review Questions 1. Describe all the types of shapes that can be produced by the intersection of a plane and a cube. 2. What is the side-length of the regular hexagon that is produced in the above diagram when the cube www.ck12.org 2 has side-length 1? Review Answers 1. Square, rectangle, pentagon, others. √ 2. length = 2 2 Intersections with Cones Oneclassofintersectionsisofparticularinterest: Theintersectionofaplaneandacone. Theseintersections arecalledconic sectionsandthefirstpersonknowntohavestudiedthemextensivelyistheAncientGreek mathematician Menaechmus in the 3rd Century B. C. E. Part of his interest in the conic sections came from his work on a classic Greek problem called “doubling the cube,” and we will describe this problem and Menaechmus’ approach that uses conic sections in section three. The intersections of the cone and the plane are so rich that the resulting shapes have continued to be of interest and generate new ideas from Menaechmus’ time until the present. Before we really delve into what we mean by a plane and a cone, we can look at an intuitive example. Suppose by a cone we just mean an ice-cream cone. And by a plane we mean a piece of paper. Well, if you sliced through an upright ice-cream cone with a horizontal piece of paper you would find that the two objects intersect at a circle. That’s very nice. But unlike the intersection of two spheres, which also resulted in a circle, that’s not all we get. If we tilt the paper (or the ice cream cone) things start to get tricky. Firstthingsfirst,webettermakesureweknowwhatwemeanby“plane”and“cone”. Let’susethesimplest definitions possible. So by “plane”, we mean the infinitely thin flat geometric object that extends forever in all directions. Even though infinity is a tricky concept, this plane is in some sense simpler than one that ends arbitrarily. There is no boundary to think about with the infinite plane. And what do we mean by a cone? An ice-cream cone is a good start. In fact, it’s very similar to how the ancient Greeks defined a cone, as a right triangle rotated about one if its legs. 3 www.ck12.org Like we do with the infinite geometric plane, we want to idealize this object a bit too. As with the plane, to avoid having to deal with a boundary, let’s suppose it continues infinitely in the direction of its open end. But the Greek mathematician Apollonius noticed that it helps even more to have it go to infinity in the other direction. This way, a cone can be thought of as an infinite collection of lines, and since geometric lines go on forever in both directions, a cone also extends to infinity in both directions. Here is a picture of what we will call a cone in this chapter (remember it extends to infinity in both directions). A cone can be formally defined as a three-dimensional collection of lines, all forming an equal angle with a central line or axis. In the above picture, the central line is vertical. Review Questions 3. What other mathematical objects can be generated by a collection of lines? Review Answers 3. Cylinder (infinite), plane, three-sided infinite pyramids, others. www.ck12.org 4