Precalculus with geometry and trigometry 1rdNft8IxRXJmN6HUjzYxmodRwg48tph pdf

Precalculus with Geometry and Trigonometry by Avinash Sathaye, Professor of Mathematics Department of Mathematics, University of Kentucky ¯ Aryabhat a This book may be freely downloaded for personal use from the author’s web site www.msc.uky.edu/sohum/ma110/text/ma110 fa08.pdf Any commercial use must be preauthorized by the author Send an email to sathaye@uky.edu for inquiries November 23, 2009 Partially supported by NSF grant thru AMSP(Appalachian Math Science Partnership) ii Introduction This book on Precalculus with Geometry and Trigonometry should be treated as simply an enhanced version of our book on College Algebra Most of the topics that appear here have already been discussed in the Algebra book and often the text here is a verbatim copy of the text in the other book We expect the student to already have a strong Algebraic background and thus the algebraic techniques presented here are more a refresher course than a first introduction We also expect the student to be heading for higher level mathematics courses and try to supply the necessary connections and motivations for future use Here is what is new in this book • In contrast with the Algebra book, we make a more extensive use of complex numbers We use Euler’s representation of complex numbers as well as the Argand diagrams extensively Even though these are described and shown to be useful, we not yet have tools to prove these techniques properly They should be used as motivation and as an easy method to remember the trigonometric results • We have supplied a brief introduction to matrices and determinants The idea is to supply motivation for further study and a feeling for the Linear Algebra • In the appendix, we give a more formal introduction to the structure of real numbers While this is not necessary for calculations in this course, it is vital for understanding the finer concepts of Calculus which will be introduced in higher courses • We have also included an appendix discussing summation of series - both finite and infinite, as well as a discussion of power series While details of convergence are left out, this should generate familiarity with future techniques and a better feeling for the otherwise mysterious trigonometric and exponential functions Contents Review of Basic Tools 1.1 Underlying field of numbers 1.1.1 Working with Complex Numbers 1.2 Indeterminates, variables, parameters 1.3 Basics of Polynomials 1.3.1 Rational functions 1.4 Working with polynomials 1.5 Examples of polynomial operations Example Polynomial operations Example Collecting coefficients Example Using algebra for arithmetic Example The Binomial Theorem Example Substituting in a polynomial Example Completing the square Solving linear equations 2.1 What is a solution? 2.2 One linear equation in one variable 2.3 Several linear equations in one variable 2.4 Two or more equations in two variables 2.5 Several equations in several variables 2.6 Solving linear equations efficiently Example Manipulation of equations Example Cramer’s Rule Example Exceptions to Cramer’s Rule Example Cramer’s Rule with many variables The division algorithm and applications 3.1 Division algorithm in integers Example GCD calculation in Integers ¯ 3.2 Aryabhat a algorithm: Efficient Euclidean algorithm Example Kuttaka or Chinese Remainder Problem 1 12 13 17 17 21 21 22 22 23 27 28 31 32 34 35 36 37 38 38 39 41 42 45 45 48 49 52 CONTENTS 3.3 3.4 3.5 Example More Kuttaka problems Example Disguised problems Division algorithm in polynomials Repeated Division The GCD and LCM of two polynomials ¯ Example Aryabhat a algorithm for polynomials Example Efficient division by a linear polynomial Example Division by a quadratic polynomial Introduction to analytic geometry 4.1 Coordinate systems 4.2 Geometry: Distance formulas 4.2.1 Connection with complex numbers 4.3 Change of coordinates on a line 4.4 Change of coordinates in the plane 4.5 General change of coordinates 4.5.1 Description of Isometies 4.5.2 Complex numbers and plane transformations 4.5.3 Examples of complex transformations 4.5.4 Examples of changes of coordinates: 53 54 56 60 62 64 67 69 73 73 75 77 78 79 81 81 82 82 83 Equations of lines in the plane 5.1 Parametric equations of lines Examples Parametric equations of lines 5.2 Meaning of the parameter t: Examples Special points on parametric lines 5.3 Comparison with the usual equation of a line 5.4 Examples of equations of lines Example Points equidistant from two given points Example Right angle triangles 87 87 89 91 93 95 100 102 103 Special study of Linear and Quadratic Polynomials 6.1 Linear Polynomials 6.2 Factored Quadratic Polynomial Interval notation Intervals on real line 6.3 The General Quadratic Polynomial 6.4 Examples of Quadratic polynomials 105 105 107 107 110 113 115 115 117 120 124 Functions 7.1 Plane algebraic curves 7.2 What is a function? 7.3 Modeling a function 7.3.1 Inverse Functions CONTENTS The 8.1 8.2 8.3 8.4 Circle Circle Basics Parametric form of a circle Application to Pythagorean Triples Pythagorean Triples Generation of Examples of equations of a circle Example Intersection of two circles Example 1a Complex intersection of two circles Example Line joining through the intersection of two circles Example Circle through three given points Example Exceptions to a circle through three points Example Smallest circle with a given center meeting a given line Example Circle with a given center and tangent to a given line Example The distance between a point and a line Example Half plane defined by a line Trigonometry 9.1 Trigonometric parameterization of a circle Definition Trigonometric Functions 9.2 Basic Formulas for the Trigonometric Functions 9.3 Connection with the usual Trigonometric Functions 9.4 Important formulas 9.5 Using trigonometry 9.6 Proofs I 9.7 Proofs II 10 Looking closely at a function 10.1 Introductory examples Parabola Analysis near its points Circle Analysis near its points 10.2 Analyzing a general curve y = f (x) near a point (a, f (a)) 10.3 The slope of the tangent, calculation of the derivative 10.4 Derivatives of more complicated functions 10.5 General power and chain rules 10.6 Using the derivatives for approximation Linear Approximation Examples 129 129 131 133 133 137 138 140 141 142 142 143 144 145 147 149 149 153 157 161 162 174 180 181 187 187 187 191 192 194 198 199 202 203 11 Root finding 209 11.1 Newton’s Method 209 11.2 Limitations of the Newton’s Method 211 12 Appendix √ 12.1 An analysis of as a real number 12.2 Idea of a Real Number 12.3 Summation of series 12.3.1 A basic formula 12.3.2 Using the basic formula 12.4 On the exponential and logarithmic functions 12.5 Infinite series and their use 12.6 Inverse functions by series 12.7 Decimal expansion of a Rational Number 12.8 Matrices and determinants: a quick introduction CONTENTS 213 213 215 217 218 218 222 225 233 238 241 Chapter Review of Basic Tools 1.1 Underlying field of numbers Mathematics may be described as the science of manipulating numbers The process of using mathematics to analyze the physical universe often consists of representing events by a set of numbers, converting the laws of physical change into mathematical functions and equations and predicting or verifying the physical events by evaluating the functions or solving the equations We begin by describing various types of numbers in use During thousands of years, mathematics has developed many systems of numbers Even when some of these appear to be counterintuitive or artificial, they have proved to be increasingly useful in developing advanced solution techniques To be useful, our numbers must have a few fundamental properties We should be able to perform the four basic operations of algebra: • addition • subtraction • multiplication and • division (except by 0) and produce well defined numbers as answers Any set of numbers having all these properties is said to be a field of numbers (or constants) Depending on our intended use, we work with different fields of numbers Here is a description of fields of numbers that we typically use Rational numbers Q The most natural idea of numbers comes from simple counting 1, 2, 3, · · · and these form the set of natural numbers often denoted by IN CHAPTER REVIEW OF BASIC TOOLS These are not yet good enough to make a field since subtraction like − is undefined To fix the subtraction property, we can add in the zero and negative numbers This produces the set of integers, ZZ = {0, ±1, ±2, · · · } These are still deficient because the division does not work You cannot divide by and get an integer back where m, n are integers The natural next step is to introduce the fractions m n and n = You probably remember the explanation in terms of picking up parts; thus 38 -th of a pizza is three of the eight slices of one pizza We now have a natural field at hand, the so-called field of rational numbers Q ={ m |m, n ∈ ZZ, n = 0} n Real numbers ℜ One familiar way of thinking of numbers is as decimal numbers, say something like 2.34567 which is nothing but a rational number 234567 whose denominator is a power of 10 100,000 This simple sounding idea took several hundred years to develop and be accepted, because the idea of a negative count is hard to imagine If we think of a number representing money owned, then a negative number can easily be thought of as money owed! The concept of negative numbers and zero was developed and expanded in India where a negative number is called r.n.a which also means debt! The word used for a positive number, is similarly dhana which means wealth The point is that even though the idea of certain numbers sounds unrealistic or impossible, one should keep an open mind and accept and use them as needed They can be useful and somebody may find a good interpretation for them some day Indeed, the number 10 can and is often replaced by other convenient numbers The computer scientists often prefer in place of 10, leading to the binary numbers, or they also use or 16 in other contexts, leading to octal or hexadecimal numbers In Number Theory, it is customary to replace 10 by some prime p and study the resulting p-adic numbers It is also possible and sometimes convenient to choose a product of suitable numbers, rather than power of a single one It is interesting to consider numbers of the form a1 + a2 a3 + + ··· 2! 3! where a1 is an arbitrary integer, while ≤ a2 < 2, ≤ a3 < and so on Thus, you can verify that: 17 1 =1+ + + + + 2! 3! 4! 5! 6! 1.1 UNDERLYING FIELD OF NUMBERS Thus, rational numbers whose denominators are powers of 10 are called decimal numbers A calculator, especially a primitive one, deals exclusively with such numbers One quickly realizes that even something as simple as 13 runs into problems if we insist on using only decimal numbers It has successive approximations 0.3, 0.33, 0.333, · · · but no finite decimal will ever give the exact value 13 While this is somewhat disturbing, we have the choice of writing 0.3 with the understanding that it is the decimal number obtained by repeating the digit under the bar indefinitely can be written as a repeating It can be shown that any rational number m n decimal which consists of an infinite decimal number with a certain group of digits repeating indefinitely from some point on For example, = 0.142857142857 · · · = 0.142857 For a detailed explanation of why all rationals can be written as repeating decimals, please see the appendix Thus, we realize that if we are willing to handle infinite decimals, we have all the rational numbers and as a bonus we get a whole lot of new numbers whose decimal expansions don’t repeat What we get? We get the so-called field of real numbers Because we are so familiar with the decimals in everyday life, these feel natural and easy, except for the fact that almost always we are dealing with approximations and issues of limiting values Thus for example, there cannot be any difference between the decimal 0.1 and the decimal 0.09999 · · · = 0.09 even though they appear different! The ideas that the sequence of longer and longer decimals 0.09, 0.099, 0.0999, · · · gets closer and closer to the decimal 0.1 and we can precisely formulate a meaning for the infinite decimal as a limit equal to 0.1 are natural, but rather sophisticated While these were informally being used all the time, they were formalized only in the last four hundred years The subject of Calculus pushes them to their natural logical limit You might recall that the set of real numbers can be represented as points of the number line by the following procedure: and it can be thought of as represented by · 12114 We may call this the factorial system This idea has the advantage of keeping all rationals as finite expressions, but is clearly not as easy to use as the decimal system! CHAPTER REVIEW OF BASIC TOOLS • Mark some convenient point as the origin associated with zero • Mark some other convenient point as the unit point, identified with We define its distance from the origin to be the unit distance or simply the unit • Then every decimal number gets an appropriate position on the line with its distance from the origin set equal to its absolute value and is placed on the correct side of the zero depending on its sign Thus the number gets marked in the same direction as the unit point but at twice the unit distance The number −3 gets marked on the opposite side of the unit point and at three times the unit distance In general, a positive number x is on the same side of the unit point at distance equal x times the unit distance and the point −x is on the other side at the same distance See some illustrated points below –3 –1 3.125 4.25 It seems apparent that the real numbers must fill up the whole number line and we should not need any further numbers Actually, this “feeling” is misleading, since you can get the same feeling by plotting lots of rational points on the number line and see it visibly fill up! It takes an algebraic manipulation and some √ clever argument to show that the positive square root of 2, usually denoted as cannot be a rational number and yet deserves its rightful position on the number line If you have not thought about it, it is a very good challenge to prove that there are no positive integers m, n which will satisfy: √ 2= m in other words m2 = 2n2 n It can, however, be “proved” that in some sense the real numbers (or the set of all decimal numbers, allowing infinite decimals) fill up the number line The precise proof of this fact will be presented in a good first course in Calculus It would seem natural to be satisfied that we have found all the numbers that need to be found and can stop at ℜ However, we now show that there is something more interesting to find! In the appendix, you will find a precise argument to show how on the real number line √ can be given a proper place ... by dividing both the right hand side (RHS) of the equation and the LHS by a Thus ax/a = x and b/a is simply b/a If a = and b = 0, then the equation is inconsistent and has no solution If both... 33 + · · · + 31000 and v = Divide u by v and find the division q and the remainder r The person with a calculator might be working a long time, just to determine what u is and may get overflow... should keep an open mind and accept and use them as needed They can be useful and somebody may find a good interpretation for them some day Indeed, the number 10 can and is often replaced by