A SHORT COURSE IN MATHEMATICS
CONTENTS PROLOGUE: CALCULATOR ARITHMETIC TYPING MATHEMATICAL EXPRESSIONS ROUNDING PARENTHESES AND GROUPING MINUS SIGNS SPECIAL NUMBERS ? AND E CHAIN CALCULATIONS SCIENTIFIC NOTATION ANOTHER LOOK: ORDER OF OPERATIONS CHAPTER P SUMMARY CHAPTER P REVIEW EXERCISES FUNCTIONS 1 1 FUNCTIONS GIVEN BY FORMULAS FUNCTIONS OF ONE VARIABLE FUNCTIONS OF SEVERAL VARIABLES ANOTHER LOOK: DEFINITION OF A FUNCTION 1 2 FUNCTIONS GIVEN BY TABLES READING TABLES OF VALUES FILLING GAPS BY AVERAGING AVERAGE RATES OF CHANGE SPOTTING TRENDS ANOTHER LOOK: MORE ON AVERAGE RATES OF CHANGE 1 3 FUNCTIONS GIVEN BY GRAPHS READING GRAPHS CONCAVITY AND RATES OF CHANGE INFLECTION POINTS DRAWING A GRAPH ANOTHER LOOK: SECANT LINES 1 4 FUNCTIONS GIVEN BY WORDS COMPARING FORMULAS AND WORDS GETTING FORMULAS FROM WORDS PROPORTION ANOTHER LOOK: GEOMETRIC CONSTRUCTIONS CHAPTER 1 SUMMARY CHAPTER 1 REVIEW EXERCISES GRAPHICAL AND TABULAR ANALYSIS 2 1 TABLES AND TRENDS GETTING TABLES FROM FORMULAS OPTIMIZING WITH TABLES OF VALUES ANOTHER LOOK: LIMITS 2 2 GRAPHS HAND-DRAWN GRAPHS FROM FORMULAS GRAPHING WITH THE CALCULATOR TRACING THE GRAPH CHOOSING A VIEWING WINDOW IN PRACTICAL SETTINGS GETTING LIMITING VALUES FROM GRAPHS ANOTHER LOOK: SHIFTING AND STRETCHING 2 3 SOLVING LINEAR EQUATIONS THE BASIC OPERATIONS REVERSING THE ROLES OF VARIABLES ANOTHER LOOK: EQUATIONS THAT ARE LINEAR IN ONE VARIABLE 2 4 SOLVING NONLINEAR EQUATIONS THE CROSSING-GRAPHS METHOD THE SINGLE-GRAPH METHOD ANOTHER LOOK: SOLVING NONLINEAR EQUATIONS BY FACTORING 2 5 OPTIMIZATION OPTIMIZING AT PEAKS AND VALLEYS OPTIMIZING AT ENDPOINTS ANOTHER LOOK: OPTIMIZING WITH PARABOLAS CHAPTER 2 SUMMARY CHAPTER 2 REVIEW EXERCISES STRAIGHT LINES AND LINEAR FUNCTIONS 3 1 THE GEOMETRY OF LINES CHARACTERIZATIONS OF STRAIGHT LINES THE SLOPE OF A LINE GETTING SLOPE FROM POINTS ANOTHER LOOK: PLANE GEOMETRY: PARALLEL AND PERPENDICULAR LINES 3 2 LINEAR FUNCTIONS CONSTANT RATES OF CHANGE LINEAR FUNCTIONS AND STRAIGHT LINES LINEAR EQUATIONS FROM DATA ANOTHER LOOK: FORMS OF EQUATIONS 3 3 MODELING DATA WITH LINEAR FUNCTIONS TESTING DATA FOR LINEARITY LINEAR MODELS GRAPHING DISCRETE DATA ANOTHER LOOK: UNEVENLY SPACED DATA 3 4 LINEAR REGRESSION THE REGRESSION LINE USES OF THE REGRESSION LINE: SLOPE AND TRENDS ANOTHER LOOK: LINEAR REGRESSION FORMULA, THE ERROR 3 5 SYSTEMS OF EQUATIONS GRAPHICAL SOLUTIONS OF SYSTEMS OF EQUATIONS ALGEBRAIC SOLUTIONS ANOTHER LOOK: SOLUTION BY ELIMINATION, MATRICES CHAPTER 3 SUMMARY CHAPTER 3 REVIEW EXERCISES EXPONENTIAL FUNCTIONS 4 1 EXPONENTIAL GROWTH AND DECAY EXPONENTIAL GROWTH EXPONENTIAL DECAY CONSTANT PROPORTIONAL CHANGE UNIT CONVERSION EXPONENTIAL FUNCTIONS AND DAILY EXPERIENCE ANOTHER LOOK: ELEMENTARY PROPERTIES OF EXPONENTS 4 2 MODELING EXPONENTIAL DATA RECOGNIZING EXPONENTIAL DATA CONSTRUCTING AN EXPONENTIAL MODEL GROWTH AND DECAY FACTOR UNITS IN EXPONENTIAL MODELING ANOTHER LOOK: TESTING UNEVENLY SPACED DATA 4 3 MODELING NEARLY EXPONENTIAL DATA EXPONENTIAL REGRESSION ANOTHER LOOK: NONLINEAR MODELS 4 4 LOGARITHMIC FUNCTIONS THE RICHTER SCALE HOW THE COMMON LOGARITHM WORKS THE LOGARITHM AS THE INVERSE OF THE EXPONENTIAL FUNCTION ANOTHER LOOK: SOLVING EXPONENTIAL EQUATIONS 4 5 CONNECTING EXPONENTIAL AND LINEAR DATA FROM LINEAR TO EXPONENTIAL DATA FROM EXPONENTIAL TO LINEAR DATA: THE LOGARITHM EXPONENTIAL REGRESSION USING THE NATURAL LOGARITHM ANOTHER LOOK: APPLYING THE LAWS OF LOGARITHMS CHAPTER 4 SUMMARY CHAPTER 4 REVIEW EXERCISES A SURVEY OF OTHER COMMON FUNCTIONS 5 1 POWER FUNCTIONS HOMOGENEITY PROPERTY OF POWER FUNCTIONS COMPARING EXPONENTIAL AND POWER FUNCTIONS ANOTHER LOOK: HOMOGENEITY, MORE ON POWER AND EXPONENTIAL FUNCTIONS 5 2 MODELING DATA WITH POWER FUNCTIONS THE CONNECTION BETWEEN POWER DATA AND LINEAR DATA GETTING A POWER MODEL FROM DATA ALMOST POWER DATA GRAPHING ON A LOGARITHMIC SCALE: COMMON VERSUS NATURAL LOGARITHMS ANOTHER LOOK: MORE ON POWER REGRESSION 5 3 COMBINING AND DECOMPOSING FUNCTIONS SUMS, PRODUCTS, AND LIMITING VALUES COMPOSITION OF FUNCTIONS PIECEWISE-DEFINED FUNCTIONS ANOTHER LOOK: CHAOTIC BEHAVIOR 5 4 QUADRATIC FUNCTIONS AND PARABOLAS LINEAR RATES OF CHANGE AND SECOND-ORDER DIFFERENCES QUADRATIC REGRESSION ALMOST QUADRATIC DATA THE QUADRATIC FORMULA ANOTHER LOOK: COMPLETING THE SQUARE, COMPLEX NUMBERS 5 5 HIGHER-DEGREE POLYNOMIALS AND RATIONAL FUNCTIONS HIGHER-DEGREE POLYNOMIALS AND THEIR ROOTS CUBIC AND OTHER POLYNOMIAL MODELS RATIONAL FUNCTIONS ANOTHER LOOK: FACTORING POLYNOMIALS, BEHAVIOR AT INFINITY CHAPTER 5 SUMMARY CHAPTER 5 REVIEW EXERCISES THE GEOMETRY OF RIGHT TRIANGLES 6 1 SIDES OF RIGHT TRIANGLES THE THEOREM OF PYTHAGORAS AREA OF A TRIANGLE ANOTHER LOOK: A DEMONSTRATION OF THE PYTHAGOREAN THEOREM 6 2 ANGLES DEGREE AND RADIAN MEASURE SLICES OF PIE SIMILAR TRIANGLES ANOTHER LOOK: MORE ON SIMILAR TRIANGLES 6 3 RIGHT TRIANGLE TRIGONOMETRY SINE, COSINE, AND TANGENT FUNCTIONS SECANT, COSECANT, AND COTANGENT FUNCTIONS ANOTHER LOOK: SPECIAL TRIANGLES CHAPTER 6 SUMMARY CHAPTER 6 REVIEW EXERCISES PERIODIC FUNCTIONS 7 1 TRIGONOMETRIC FUNCTIONS AS PERIODIC FUNCTIONS EXTENDING THE TRIGONOMETRIC FUNCTIONS QUADRANTS PERIOD AND AMPLITUDE OF SINE AND COSINE FUNCTIONS ADJUSTING PERIOD AND AMPLITUDE ANOTHER LOOK: REFERENCE ANGLES 7 2 MODELING WITH PERIODIC FUNCTIONS PERIOD AND AMPLITUDE FOR PERIODIC PHENOMENA SOUND WAVES USING THE SINE FUNCTION TO MODEL PERIODIC PHENOMENA ANOTHER LOOK: OTHER PERIODIC FUNCTIONS 7 3 MODELING PERIODIC DATA VERTICAL AND HORIZONTAL SHIFTING SINE MODELS BY ESTIMATION SINE REGRESSION ANOTHER LOOK: HOW CALCULATORS HANDLE TRIGONOMETRIC FUNCTIONS CHAPTER 7 SUMMARY CHAPTER 7 REVIEW EXERCISES RELATIONSHIPS AMONG TRIGONOMETRIC FUNCTIONS 8 1 TRIGONOMETRIC IDENTITIES BASIC IDENTITIES AREA AND SUM FORMULAS DOUBLE- AND HALF-ANGLE FORMULAS ANOTHER LOOK: CLEVER TRIANGLES 8 2 LAWS OF SINES AND COSINES LAW OF SINES LAW OF COSINES ANOTHER LOOK: A CURIOUS RELATIONSHIP 8 3 INVERSE TRIGONOMETRIC FUNCTIONS DEFINING INVERSE TRIGONOMETRIC FUNCTIONS MAKING PICTURES TO REPRESENT INVERSE TRIGONOMETRIC FUNCTIONS THE INVERSE TANGENT FUNCTION ANOTHER LOOK: MACHIN’S FORMULA 8 4 COMPLEX NUMBERS AND DE MOIVRE'S THEOREM COMPLEX NUMBERS THE COMPLEX EXPONENTIAL FUNCTION AND THE EULER FORMULA DE MOIVRE'S THEOREM ANOTHER LOOK: POLAR COORDINATES CHAPTER 8 SUMMARY CHAPTER 8 REVIEW EXERCISES RATES OF CHANGE 9 1 VELOCITY GETTING VELOCITY FROM DIRECTED DISTANCE CONSTANT VELOCITY MEANS LINEAR DIRECTED DISTANCE WHEN DISTANCE IS GIVEN BY A FORMULA 9 2 RATES OF CHANGE FOR OTHER FUNCTIONS EXAMPLES OF RATES OF CHANGE PROPERTIES THAT ALL RATES OF CHANGE SHARE WHEN THE RATE OF CHANGE IS ZERO WHEN THE RATE OF CHANGE IS CONSTANT 9 3 ESTIMATING RATES OF CHANGE RATES OF CHANGE FOR TABULATED DATA RATES OF CHANGE FOR FUNCTIONS GIVEN BY FORMULAS 9 4 EQUATIONS OF CHANGE: LINEAR AND EXPONENTIAL FUNCTIONS EQUATION OF CHANGE FOR LINEAR FUNCTIONS INSTANTANEOUS RATES OF CHANGE EQUATION OF CHANGE FOR EXPONENTIAL FUNCTIONS WHY EQUATIONS OF CHANGE? 9 5 EQUATIONS OF CHANGE: GRAPHICAL SOLUTIONS EQUILIBRIUM SOLUTIONS SKETCHING GRAPHS CHAPTER 9 SUMMARY CHAPTER 9 REVIEW EXERCISES 10 MATHEMATICS OF POPULATION ECOLOGY 10 1 POPULATION DYNAMICS: EXPONENTIAL GROWTH DOUBLING TIMES EMPIRICAL DATA DISCRETE GENERATIONS 10 2 POPULATION DYNAMICS: LOGISTIC GROWTH LOGISTIC GROWTH MODEL APPLICATION TO HARVESTING RENEWABLE RESOURCES LOGISTIC EQUATION OF CHANGE INTEGRAL FORM OF THE LOGISTIC EQUATION THE VALUE OF THE LOGISTIC MODEL 10 3 POPULATION STRUCTURE: SURVIVORSHIP CURVES LIFE TABLES AND SURVIVORSHIP CURVES CLASSIFYING SURVIVORSHIP CURVES USEFULNESS OF SURVIVORSHIP CURVES CHAPTER 10 SUMMARY CHAPTER 10 REVIEW EXERCISES