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Calculus/Print Version Calculus/Print version Wikibooks.org March 24, 2011. Contents 1 TABLEOF CONTENTS 1 1.1 PRECALCULUS1 ....................................... 1 1.2 LIMITS2 ........................................... 2 1.3 DIFFERENTIATION3 .................................... 3 1.4 INTEGRATION4 ....................................... 4 1.5 PARAMETRICAND POLAR EQUATIONS5 ......................... 8 1.6 SEQUENCES AND SERIES6 ................................. 9 1.7 MULTIVARIABLE AND DIFFERENTIAL CALCULUS7 ................... 10 1.8 EXTENSIONS8 ........................................ 11 1.9 APPENDIX .......................................... 11 1.10 EXERCISE SOLUTIONS ................................... 11 1.11 REFERENCES9 ........................................ 11 1.12 ACKNOWLEDGEMENTS AND FURTHER READING10 ................... 12 2 INTRODUCTION 13 2.1 WHAT IS CALCULUS?.................................... 13 2.2 WHY LEARN CALCULUS?.................................. 14 2.3 WHAT IS INVOLVED IN LEARNING CALCULUS? ..................... 14 2.4 WHAT YOU SHOULD KNOW BEFORE USING THIS TEXT . 14 2.5 SCOPE ............................................ 15 3 PRECALCULUS 17 4 ALGEBRA 19 4.1 RULES OF ARITHMETIC AND ALGEBRA .......................... 19 4.2 INTERVAL NOTATION .................................... 20 4.3 EXPONENTS AND RADICALS ................................ 21 4.4 FACTORING AND ROOTS .................................. 22 4.5 SIMPLIFYING RATIONAL EXPRESSIONS .......................... 23 4.6 FORMULAS OF MULTIPLICATION OF POLYNOMIALS . 23 1 Chapter 1 on page 1 2 Chapter 1.1 on page 1 3 Chapter 1.2 on page 2 4 Chapter 28 on page 179 5 Chapter 1.4.3 on page 7 6 Chapter 1.5.2 on page 9 7 Chapter 1.6.2 on page 9 8 Chapter 1.7 on page 11 9 Chapter 1.10 on page 11 10 Chapter 68 on page 427 III Contents 5 FUNCTIONS 25 5.1 CLASSICAL UNDERSTANDING OF FUNCTIONS ...................... 25 5.2 NOTATION .......................................... 26 5.3 MODERN UNDERSTANDING OF FUNCTIONS ....................... 26 5.4 REMARKS .......................................... 27 5.5 THE VERTICAL LINE TEST ................................. 28 5.6 IMPORTANT FUNCTIONS .................................. 29 5.7 EXAMPLEFUNCTIONS ................................... 29 5.8 MANIPULATING FUNCTIONS ............................... 30 5.9 DOMAINAND RANGE ................................... 36 6 GRAPHING LINEAR FUNCTIONS 43 6.1 EXAMPLE .......................................... 44 6.2 SLOPE-INTERCEPT FORM ................................. 44 6.3 POINT-SLOPE FORM .................................... 44 6.4 CALCULATING SLOPE .................................... 44 6.5 TWO-POINTFORM ..................................... 45 6.6 EXERCISES .......................................... 45 6.7 ALGEBRA ........................................... 45 6.8 FUNCTIONS ......................................... 47 6.9 GRAPHING .......................................... 49 7 LIMITS 51 7.1 INTUITIVE LOOK ...................................... 51 7.2 INFORMAL DEFINITION OF A LIMIT ............................ 53 7.3 LIMIT RULES ........................................ 53 7.4 FINDING LIMITS ...................................... 58 7.5 USING LIMIT NOTATION TO DESCRIBE ASYMPTOTES . 63 7.6 KEY APPLICATION OF LIMITS ............................... 64 7.7 EXTERNALLINKS ...................................... 65 8 FINITE LIMITS 67 8.1 INFORMAL FINITE LIMITS ................................. 67 8.2 ONE-SIDED LIMITS .................................... 68 9 INFINITE LIMITS 69 9.1 INFORMAL INFINITE LIMITS ................................ 69 9.2 LIMITS AT INFINITY OF RATIONAL FUNCTIONS ..................... 70 10 CONTINUITY 73 10.1 DEFINING CONTINUITY .................................. 73 10.2 DISCONTINUITIES ..................................... 74 10.3 ONE-SIDED CONTINUITY ................................. 75 10.4 INTERMEDIATE VALUE THEOREM ............................ 76 11 FORMAL DEFINITION OF THE LIMIT 79 11.1 FORMAL DEFINITION OF THE LIMIT AT INFINITY .................... 81 11.2 EXAMPLES .......................................... 81 11.3 FORMAL DEFINITION OF A LIMIT BEING INFINITY ................... 86 IV Contents 11.4 EXERCISES .......................................... 87 11.5 LIMITS WITH GRAPHS ................................... 87 11.6 BASIC LIMIT EXERCISES .................................. 87 11.7 ONE SIDED LIMITS ..................................... 87 11.8 TWO SIDED LIMITS .................................... 87 11.9 LIMITS TO INFINITY .................................... 88 11.10 LIMITS OF PIECE FUNCTIONS ............................... 88 11.11 LIMITS USING L’HÔPITAL’S RULE ............................. 89 12 DIFFERENTIATION 91 12.1 WHAT IS DIFFERENTIATION? ............................... 91 12.2 THE DEFINITION OF SLOPE ................................ 91 12.3 THE RATE OF CHANGEOFA FUNCTION AT A POINT . 97 12.4 THE DEFINITION OF THE DERIVATIVE .......................... 97 12.5 DIFFERENTIATION RULES ................................. 100 13 PRODUCT AND QUOTIENT RULES 105 13.1 PRODUCT RULE ....................................... 105 13.2 APPLICATION, PROOF OF THE POWER RULE ....................... 106 13.3 QUOTIENT RULE ...................................... 107 13.4 EXAMPLES .......................................... 107 14 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 109 15 CHAIN RULE 113 15.1 EXAMPLES .......................................... 113 15.2 CHAIN RULE IN PHYSICS .................................. 116 15.3 CHAIN RULE IN CHEMISTRY ................................ 120 15.4 POLYNOMIALS ....................................... 123 15.5 REFERENCES ........................................ 125 15.6 EXTERNALLINKS ...................................... 125 16 HIGH ORDER DERIVATIVES 127 16.1 NOTATION .......................................... 127 16.2 EXAMPLES .......................................... 128 17 IMPLICIT DIFFERENTIATION 129 17.1 EXPLICIT DIFFERENTIATION ................................ 129 17.2 IMPLICIT DIFFERENTIATION ................................ 129 17.3 USES ............................................. 130 17.4 IMPLICIT DIFFERENTIATION ............................... 130 17.5 INVERSE TRIGONOMETRIC FUNCTIONS . 132 18 DERIVATIVES OF EXPONENTIALAND LOGARITHM FUNCTIONS 135 18.1 EXPONENTIAL FUNCTION ................................. 135 18.2 LOGARITHM FUNCTION .................................. 136 18.3 LOGARITHMIC DIFFERENTIATION ............................ 137 18.4 EXERCISES .......................................... 140 18.5 FIND THE DERIVATIVE BY DEFINITION . 140 18.6 PROVE DIFFERENTIATION RULES ............................. 140 V Contents 18.7 FIND THE DERIVATIVE BY RULES ............................ 140 18.8 MORE DIFFERENTIATION ................................. 142 18.9 IMPLICIT DIFFERENTIATION ............................... 143 18.10 LOGARITHMIC DIFFERENTIATION ............................ 143 18.11 EQUATION OF TANGENT LINE ............................... 143 18.12 HIGHER ORDER DERIVATIVES .............................. 143 18.13 RELATIVE EXTREMA .................................... 144 18.14 RANGEOF FUNCTION ................................... 144 18.15 ABSOLUTE EXTREMA .................................... 144 18.16 DETERMINE INTERVALS OF CHANGE ........................... 144 18.17 DETERMINE INTERVALS OF CONCAVITY . 144 18.18 WORD PROBLEMS ..................................... 145 18.19 GRAPHING FUNCTIONS .................................. 145 19 APPLICATIONS OF DERIVATIVES 147 20 EXTREMAAND POINTS OF INFLECTION 149 20.1 THE EXTREMUM TEST ................................... 151 21 NEWTON’S METHOD 153 21.1 EXAMPLES .......................................... 154 21.2 NOTES ............................................ 155 21.3 SEE ALSO .......................................... 155 22 RELATED RATES 157 22.1 INTRODUCTION ....................................... 157 22.2 RELATED RATES ....................................... 157 22.3 COMMON APPLICATIONS ................................. 157 22.4 EXAMPLES .......................................... 158 22.5 EXERCISES .......................................... 162 23 KINEMATICS 165 23.1 INTRODUCTION ....................................... 165 23.2 DIFFERENTIATION ..................................... 165 23.3 INTEGRATION ........................................ 166 24 OPTIMIZATION 167 24.1 INTRODUCTION ....................................... 167 24.2 EXAMPLES .......................................... 167 25 EXTREME VALUE THEOREM 171 25.1 FIRST DERIVATIVE TEST .................................. 172 25.2 SECOND DERIVATIVE TEST ................................ 173 26 ROLLE’S THEOREM 175 26.1 EXAMPLES .......................................... 175 27 MEAN VALUE THEOREM 177 27.1 EXERCISES .......................................... 177 VI Contents 28 INTEGRATION 179 28.1 INTEGRATION11 ....................................... 179 28.2 INFINITE SUMS ....................................... 182 28.3 EXACT INTEGRALSAS LIMITS OF SUMS . 182 28.4 DERIVATIVE RULESANDTHE SUBSTITUTION RULE .................. 183 28.5 RECOGNIZING DERIVATIVES AND REVERSING DERIVATIVE RULES . 183 28.6 INTEGRATION BY SUBSTITUTION ............................. 184 28.7 INTEGRATION BY PARTS .................................. 187 28.8 INTEGRATION BY PARTS .................................. 187 28.9 INTEGRATION BY COMPLEXIFYING ............................ 190 28.10 RATIONAL FUNCTIONSBY PARTIAL FRACTIONAL DECOMPOSITION . 191 28.11 TRIGONOMETRIC SUBSTITUTIONS ...........................
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