| Dedication and Acknowledgments |
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xii | |
| Preface |
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xiii | |
| About the Authors |
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xiv | |
| Introduction: The Five-Step Program |
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xv | |
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STEP 1 Set Up Your Study Plan |
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1 What You Need to Know About the AP Calculus BC Exam |
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3 | (5) |
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1.1 What Is Covered on the AP Calculus BC Exam? |
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4 | (1) |
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1.2 What Is the Format of the AP Calculus BC Exam? |
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4 | (1) |
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1.3 What Are the Advanced Placement Exam Grades? |
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5 | (1) |
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How Is the AP Calculus BC Exam Grade Calculated? |
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5 | (1) |
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1.4 Which Graphing Calculators Are Allowed for the Exam? |
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6 | (2) |
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Calculators and Other Devices Not Allowed for the AP Calculus BC Exam |
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7 | (1) |
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Other Restrictions on Calculators |
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7 | (1) |
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8 | (9) |
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2.1 Three Approaches to Preparing for the AP Calculus BC Exam |
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8 | (2) |
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Overview of the Three Plans |
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8 | (2) |
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2.2 Calendar for Each Plan |
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10 | (7) |
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Summary of the Three Study Plans |
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13 | (4) |
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STEP 2 Determine Your Test Readiness |
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17 | (24) |
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21 | (1) |
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21 | (6) |
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3.3 Answers to Diagnostic Test |
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27 | (1) |
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3.4 Solutions to Diagnostic Test |
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28 | (10) |
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38 | (3) |
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38 | (1) |
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AP Calculus BC Diagnostic Exam |
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38 | (3) |
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STEP 3 Develop Strategies for Success |
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4 How to Approach Each Question Type |
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41 | (8) |
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4.1 The Multiple-Choice Questions |
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42 | (1) |
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4.2 The Free-Response Questions |
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42 | (1) |
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4.3 Using a Graphing Calculator |
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43 | (1) |
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44 | (5) |
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What Do I Need to Bring to the Exam? |
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44 | (1) |
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45 | (4) |
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STEP 4 Review the Knowledge You Need to Score High |
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49 | (26) |
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5.1 The Limit of a Function |
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50 | (7) |
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Definition and Properties of Limits |
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50 | (1) |
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50 | (2) |
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52 | (3) |
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55 | (2) |
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5.2 Limits Involving Infinities |
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57 | (7) |
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Infinite Limits (as x → a) |
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57 | (2) |
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Limits at Infinity (as x → ±∞) |
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59 | (2) |
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Horizontal and Vertical Asymptotes |
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61 | (3) |
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5.3 Continuity of a Function |
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64 | (3) |
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Continuity of a Function at a Number |
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64 | (1) |
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Continuity of a Function over an Interval |
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64 | (1) |
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64 | (3) |
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67 | (2) |
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69 | (1) |
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5.6 Cumulative Review Problems |
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70 | (1) |
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5.7 Solutions to Practice Problems |
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70 | (3) |
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5.8 Solutions to Cumulative Review Problems |
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73 | (2) |
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75 | (28) |
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6.1 Derivatives of Algebraic Functions |
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76 | (6) |
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Definition of the Derivative of a Function |
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76 | (3) |
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79 | (1) |
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The Sum, Difference, Product, and Quotient Rules |
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80 | (1) |
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81 | (1) |
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6.2 Derivatives of Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions |
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82 | (5) |
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Derivatives of Trigonometric Functions |
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82 | (2) |
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Derivatives of Inverse Trigonometric Functions |
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84 | (1) |
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Derivatives of Exponential and Logarithmic Functions |
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85 | (2) |
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6.3 Implicit Differentiation |
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87 | (3) |
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Procedure for Implicit Differentiation |
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87 | (3) |
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6.4 Approximating a Derivative |
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90 | (2) |
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6.5 Derivatives of Inverse Functions |
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92 | (2) |
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6.6 Higher Order Derivatives |
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94 | (1) |
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L'Hopital's Rule for Indeterminate Forms |
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95 | (1) |
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95 | (2) |
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97 | (1) |
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6.9 Cumulative Review Problems |
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98 | (1) |
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6.10 Solutions to Practice Problems |
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98 | (3) |
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6.11 Solutions to Cumulative Review Problems |
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101 | (2) |
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7 Graphs of Functions and Derivatives |
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103 | (46) |
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7.1 Rolle's Theorem, Mean Value Theorem, and Extreme Value Theorem |
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103 | (5) |
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104 | (1) |
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104 | (3) |
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107 | (1) |
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7.2 Determining the Behavior of Functions |
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108 | (12) |
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Test for Increasing and Decreasing Functions |
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108 | (3) |
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First Derivative Test and Second Derivative Test for Relative Extrema |
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111 | (3) |
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Test for Concavity and Points of Inflection |
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114 | (6) |
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7.3 Sketching the Graphs of Functions |
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120 | (3) |
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Graphing without Calculators |
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120 | (1) |
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Graphing with Calculators |
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121 | (2) |
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7.4 Graphs of Derivatives |
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123 | (5) |
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7.5 Parametric, Polar, and Vector Representations |
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128 | (5) |
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128 | (1) |
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129 | (1) |
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129 | (1) |
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130 | (1) |
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131 | (1) |
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132 | (1) |
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133 | (4) |
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137 | (2) |
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7.8 Cumulative Review Problems |
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139 | (1) |
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7.9 Solutions to Practice Problems |
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140 | (7) |
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7.10 Solutions to Cumulative Review Problems |
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147 | (2) |
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8 Applications of Derivatives |
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149 | (25) |
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149 | (6) |
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General Procedure for Solving Related Rate Problems |
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149 | (1) |
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Common Related Rate Problems |
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150 | (1) |
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Inverted Cone (Water Tank) Problem |
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151 | (1) |
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152 | (1) |
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Angle of Elevation Problem |
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153 | (2) |
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8.2 Applied Maximum and Minimum Problems |
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155 | (5) |
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General Procedure for Solving Applied Maximum and Minimum Problems |
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155 | (1) |
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155 | (1) |
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156 | (3) |
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159 | (1) |
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160 | (1) |
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161 | (2) |
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8.5 Cumulative Review Problems |
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163 | (1) |
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8.6 Solutions to Practice Problems |
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164 | (7) |
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8.7 Solutions to Cumulative Review Problems |
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171 | (3) |
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9 More Applications of Derivatives |
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174 | (33) |
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9.1 Tangent and Normal Lines |
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174 | (9) |
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174 | (6) |
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180 | (3) |
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9.2 Linear Approximations |
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183 | (3) |
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Tangent Line Approximation (or Linear Approximation) |
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183 | (2) |
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Estimating the nth Root of a Number |
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185 | (1) |
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Estimating the Value of a Trigonometric Function of an Angle |
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185 | (1) |
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186 | (4) |
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Instantaneous Velocity and Acceleration |
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186 | (2) |
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188 | (1) |
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188 | (2) |
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9.4 Parametric, Polar, and Vector Derivatives |
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190 | (5) |
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Derivatives of Parametric Equations |
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190 | (1) |
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Position, Speed, and Acceleration |
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191 | (1) |
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Derivatives of Polar Equations |
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191 | (1) |
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Velocity and Acceleration of Vector Functions |
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192 | (3) |
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195 | (1) |
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196 | (2) |
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9.7 Cumulative Review Problems |
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198 | (1) |
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9.8 Solutions to Practice Problems |
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199 | (5) |
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9.9 Solutions to Cumulative Review Problems |
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204 | (3) |
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Big Idea 3 Integrals and the Fundamental Theorems of Calculus |
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207 | (24) |
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10.1 Evaluating Basic Integrals |
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208 | (5) |
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Antiderivatives and Integration Formulas |
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208 | (2) |
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210 | (3) |
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10.2 Integration by U-Substitution |
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213 | (8) |
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The U-Substitution Method |
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213 | (1) |
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U-Substitution and Algebraic Functions |
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213 | (2) |
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U-Substitution and Trigonometric Functions |
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215 | (1) |
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U-Substitution and Inverse Trigonometric Functions |
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216 | (2) |
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U-Substitution and Logarithmic and Exponential Functions |
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218 | (3) |
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10.3 Techniques of Integration |
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221 | (2) |
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221 | (1) |
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Integration by Partial Fractions |
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222 | (1) |
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223 | (1) |
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224 | (1) |
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10.6 Cumulative Review Problems |
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225 | (1) |
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10.7 Solutions to Practice Problems |
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226 | (3) |
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10.8 Solutions to Cumulative Review Problems |
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229 | (2) |
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231 | (26) |
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11.1 Riemann Sums and Definite Integrals |
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232 | (5) |
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Sigma Notation or Summation Notation |
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232 | (1) |
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Definition of a Riemann Sum |
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233 | (1) |
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Definition of a Definite Integral |
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234 | (1) |
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Properties of Definite Integrals |
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235 | (2) |
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11.2 Fundamental Theorems of Calculus |
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237 | (4) |
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First Fundamental Theorem of Calculus |
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237 | (1) |
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Second Fundamental Theorem of Calculus |
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238 | (3) |
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11.3 Evaluating Definite Integrals |
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241 | (5) |
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Definite Integrals Involving Algebraic Functions |
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241 | (1) |
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Definite Integrals Involving Absolute Value |
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242 | (1) |
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Definite Integrals Involving Trigonometric, Logarithmic, and Exponential Functions |
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243 | (2) |
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Definite Integrals Involving Odd and Even Functions |
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245 | (1) |
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246 | (2) |
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Infinite Intervals of Integration |
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246 | (1) |
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247 | (1) |
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248 | (1) |
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249 | (1) |
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11.7 Cumulative Review Problems |
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250 | (1) |
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11.8 Solutions to Practice Problems |
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251 | (3) |
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11.9 Solutions to Cumulative Review Problems |
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254 | (3) |
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12 Areas, Volumes, and Arc Lengths |
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257 | (52) |
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12.1 The Function F(x) = ∫xa f(t)dt |
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258 | (4) |
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12.2 Approximating the Area Under a Curve |
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262 | (5) |
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Rectangular Approximations |
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262 | (4) |
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Trapezoidal Approximations |
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266 | (1) |
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12.3 Area and Definite Integrals |
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267 | (9) |
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267 | (5) |
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272 | (4) |
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12.4 Volumes and Definite Integrals |
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276 | (13) |
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Solids with Known Cross Sections |
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276 | (4) |
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280 | (5) |
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285 | (4) |
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12.5 Integration of Parametric, Polar, and Vector Curves |
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289 | (3) |
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Area, Arc Length, and Surface Area for Parametric Curves |
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289 | (1) |
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Area and Arc Length for Polar Curves |
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290 | (1) |
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Integration of a Vector-Valued Function |
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291 | (1) |
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292 | (3) |
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295 | (1) |
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12.8 Cumulative Review Problems |
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296 | (1) |
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12.9 Solutions to Practice Problems |
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297 | (8) |
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12.10 Solutions to Cumulative Review Problems |
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305 | (4) |
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13 More Applications of Definite Integrals |
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309 | (37) |
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13.1 Average Value of a Function |
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310 | (3) |
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Mean Value Theorem for Integrals |
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310 | (1) |
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Average Value of a Function on [ a, b] |
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311 | (2) |
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13.2 Distance Traveled Problems |
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313 | (3) |
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13.3 Definite Integral as Accumulated Change |
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316 | (3) |
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316 | (1) |
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317 | (1) |
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318 | (1) |
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318 | (1) |
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13.4 Differential Equations |
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319 | (5) |
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Exponential Growth/Decay Problems |
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319 | (2) |
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Separable Differential Equations |
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321 | (3) |
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324 | (4) |
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13.6 Logistic Differential Equations |
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328 | (2) |
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330 | (2) |
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Approximating Solutions of Differential Equations by Euler's Method |
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330 | (2) |
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332 | (2) |
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334 | (2) |
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13.10 Cumulative Review Problems |
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336 | (1) |
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13.11 Solutions to Practice Problems |
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337 | (6) |
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13.12 Solutions to Cumulative Review Problems |
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343 | (3) |
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346 | (27) |
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14.1 Sequences and Series |
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347 | (1) |
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347 | (1) |
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348 | (2) |
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348 | (1) |
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348 | (1) |
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348 | (1) |
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349 | (1) |
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350 | (4) |
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350 | (1) |
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350 | (1) |
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351 | (1) |
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352 | (1) |
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352 | (1) |
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353 | (1) |
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354 | (3) |
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354 | (1) |
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Absolute and Conditional Convergence |
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355 | (2) |
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357 | (1) |
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Radius and Interval of Convergence |
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357 | (1) |
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358 | (1) |
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Taylor Series and MacLaurin Series |
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358 | (1) |
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359 | (1) |
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14.7 Operations on Series |
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359 | (3) |
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359 | (1) |
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Differentiation and Integration |
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360 | (1) |
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361 | (1) |
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362 | (2) |
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364 | (1) |
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14.10 Cumulative Review Problems |
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365 | (1) |
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14.11 Solutions to Practice Problems |
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365 | (4) |
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14.12 Solutions to Cumulative Review Problems |
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369 | (4) |
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STEP 5 Build Your Test-Taking Confidence |
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AP Calculus BC Practice Exam 1 |
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373 | (30) |
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AP Calculus BC Practice Exam 2 |
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403 | (30) |
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433 | (8) |
| Bibliography |
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441 | (2) |
| Websites |
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443 | |
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