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Algebra for the Sciences Textbook

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Algebra for the Sciences Textbook Algebra for the Sciences K.Oty B.Elliott August 6, 2018 i Preface This book was developed as an alternative to the traditional college algebra book currently used at many universities. The first edition was written while we were partially funded by a grant from the National Science Foundation and some of the projects in the second edition were written with the support of the Ok- lahoma IDeA Network of Biomedical Research Excellence (OK-INBRE) during the Summer of 2010. The motivation for this book was to find material for stu- dents who might not be continuing their studies in mathematics but who would need to apply mathematical skills to science courses. Additionally, we designed this book for elementary education majors to satisfy the algebra requirements as recommended by the National Council of Teachers in Mathematics (NCTM). We began the development of this book by having discussions with a variety of science faculty and asking what their expectations were for mathematical skills for students completing a college algebra course. It was enlightening to learn that many of the skills that our science colleagues expected of these stu- dents were not skills that were taught in a traditional college algebra course. The topics of estimation, unit conversions, geometry and extrapolation were the skills most often mentioned that were not covered in College Algebra. We have included these topics which makes our book quite different than a traditional algebra text. The other striking difference between our book and a traditional college al- gebra book is the use of demonstrations and projects. We wanted to motivate the students to study the mathematics by presenting a problem from a field of science that would use the particular mathematics. We have written these demonstrations so that no previous knowledge of the scientific principles in- volved are required from either the instructor or the student. By introducing the framework first we hope that this will motivate the students to study the mathematics and to address the perennial question “why do I need to learn this?” We include skill based exercises throughout the chapters so that students can check their progress. Instead of the traditional problem sets at the end of the chapter, we have compiled a list of projects that use the skills developed in the chapter. When possible, we have used real data sets and occasionally ask the student to collect the data themselves. The solutions to the projects are intended to be written reports using the mathematics and explaining the procedures and concepts. Typically we assign one or two projects per student. For some of the more involved projects that require data collection we might assign the project to a group. This book is used to teach a one semester course. We cover all of the material in that time. We do use class time for demonstrations, collaborative exercises, and for working with technology. The authors wish to acknowledge and recognize the significant work on the first edition by co-authors Dr. Bryan Clark and Dr. John McArthur. We also ii wish to thank Dr. Diane Dixon for contributing projects for the second edition. iii Technology Students will need to have access to either computer software or a graphing calculator that does linear, quadratic, and logarithmic or exponential regression. We have used both the statistical software JMP c , the TI-83 Plus graphing c calculator, and Mathematica in teaching this course. In teaching this course with a graphing calculator, it should be noted that starting in Chapter 4 we ask for the adjusted R2 value which the TI-83 Plus does not find. The questions can be modified appropriately but it is important to point out that comparing the R2 value between a linear and a quadratic fit, for example, is not an appropriate way to decide which one of the two curves fits the data better. We have included web addresses for projects that ask students to find addi- tional information before completing the project. However, with the fluidity of the world wide web, these addresses may change and the students might need to find the information at some other site. Supplements There is an instructor’s guide available to instructors who are using this book in teaching their course. It contains all the solutions to the problems and projects when possible. For the projects that rely on students collecting data or other information, we have included a grading guide in the instructor’s manual. Additionally, there is a list of equipment that is needed for the demonstrations and the projects. Appendices We have included three appendices with the book. The first, Appendix A, discusses basic scientific notation. Appendix B is a list of references and web sites for further information. The last appendix contains the answers to the odd problems. iv Contents 1 Estimation 1 1.1 Objectives: ............................... 1 1.2 Introduction: ............................. 1 1.3 Demonstration: ............................ 2 1.4 The Mathematics: .......................... 3 1.4.1 NumericalEstimation . 3 1.4.2 SamplingTechniques. 5 1.4.3 UnitConversions.. ... .... ... .... ... .... 7 1.5 Homework Projects: ......................... 12 2 Geometry 15 2.1 Objectives: ............................... 15 2.2 Motivation: .............................. 15 2.3 Introduction to Home Range and Territory: ............ 16 2.4 Demonstration: ............................ 18 2.5 The Mathematics: .......................... 19 2.5.1 Area.............................. 19 2.5.2 Perimeter ........................... 24 2.5.3 Volume ............................ 26 2.5.4 SurfaceArea ......................... 29 2.6 Homework Projects: ......................... 31 3 Lines 39 3.1 Objectives: ............................... 39 3.2 Motivation: .............................. 39 3.3 Introduction to Population Size Estimation: ............ 40 3.4 Demonstration: ............................ 42 3.5 The Mathematics: .......................... 43 3.5.1 Slope ............................. 43 3.5.2 TheEquationofaLine . 45 3.5.3 FindingSlopesandIntercepts . 47 3.5.4 Graphs ............................ 48 3.5.5 Functions ........................... 50 3.5.6 LinearRegression. 52 v vi CONTENTS 3.6 Extrapolation and Interpolation: ................... 56 3.7 Homework Projects: ......................... 58 4 Quadratics 67 4.1 Objectives: ............................... 67 4.2 Motivation: .............................. 67 4.3 Introduction to Kinematics: ..................... 68 4.4 Demonstration: ............................ 69 4.5 The Mathematics: .......................... 70 4.5.1 Vertex............................. 70 4.5.2 Intercepts ........................... 72 4.5.3 Graphs ............................ 74 4.5.4 Regression........................... 77 4.6 Homework Projects: ......................... 83 5 Exponentials 97 5.1 Objectives: ............................... 97 5.2 Motivation: .............................. 97 5.3 Introduction to Population Growth: ................. 98 5.4 Demonstration: ............................ 98 5.5 The Mathematics: ..........................100 5.5.1 IntegerExponents . 100 5.5.2 Graphs ............................105 5.6 Homework Projects: .........................107 6 Logarithms 115 6.1 Objectives: ...............................115 6.2 Motivation: ..............................115 6.3 Introduction to Sound: ........................116 6.4 Demonstration: ............................118 6.5 The Mathematics: ..........................119 6.5.1 Logarithms .......................... 119 6.5.2 CommonLogsandNaturalLogs . 122 6.5.3 Graphs ............................123 6.5.4 Equations........................... 125 6.5.5 RatesofGrowth . .... ... .... ... .... ... 127 6.6 Homework Projects: .........................130 7 Systems of Equations 139 7.1 Objectives: ...............................139 7.2 Motivation: ..............................139 7.3 Demonstration: ............................140 7.4 The Mathematics: ..........................141 7.4.1 TwoLinearEquations . 141 7.4.2 ThreeLinearEquations . 146 7.4.3 NonlinearSystems . 149 CONTENTS vii 7.5 Homework Projects: .........................153 A Scientific Notation 171 BReferencesandFurtherReadings 173 B.1 Chapter1 ............................... 173 B.1.1 BodyMassIndex. 173 B.1.2 EmpireStateBuilding . 173 B.1.3 Molarity............................ 173 B.2 Chapter2 ............................... 173 B.2.1 Bats .............................. 173 B.2.2 CMRMethod......................... 174 B.3 Chapter3 ............................... 174 B.3.1 Bodyweightversusranges. 174 B.3.2 TargetHeartRates. 174 B.4 Chapter4 ............................... 174 B.4.1 History ............................ 174 B.4.2 Reflectivityofwatersurfaces . 174 B.4.3 SpeciesDiversity . 175 B.4.4 PopulationDynamics . 175 B.5 Chapter5 ............................... 175 B.5.1 History ............................ 175 B.5.2 Population .......................... 175 B.5.3 WhoopingCranes . 175 B.6 Chapter6 ............................... 176 B.6.1 History ............................ 176 B.6.2 Titanium-44 .... ... .... ... .... ... .... 176 B.6.3 ShannonDiversityIndex. 176 B.7 Chapter7 ............................... 176 B.7.1 History ............................ 176 B.7.2 Population Growth versus Food Production . 176 B.7.3 Censusdata. .... ... .... ... .... ... .... 176 B.7.4 OxygenConsumption . 177 B.7.5 Bluebirds ........................... 177 C Answers to Odd Problems 179 C.1 Chapter1 ..............................
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