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Fundamentals of Calculus 1/ Chapters 1-217/ All Chapters
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TEST BANKFUNDAMENTALS OF CALCULUS 1/Chapter 1-217/ Full
Book
,Table of Contents
1.1 Functions ........................................ 2
1.2 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Trig Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Solving Trig Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Solving Trig Equations with Calculators, Part I . . . . . . . . . . . . . . . . . . .
.8
1.6 Solving Trig Equations with Calculators, Part II ..............
..... 9
1.7 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Logarithm Functions .................................. 11
1.9 Exponential And Logarithm Equations . . . . . . . . . . . . . . . . . . 12
1.10 Common Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Limits 15
2.1 Tangent Lines And Rates Of Change . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 The Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 One-Sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Limit Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Infinite Limits ...................................... 28
2.7 Limits at Infinity, Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Limits at Infinity, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.10 The Definition of the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Derivatives 35
3.1 The Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Interpretation of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Product and Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Derivatives of Trig Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
,3.6 Derivatives of Exponentials & Logarithms . . . . . . . . . . . . . . . . . . 45
,3.7 Derivatives of Inverse Trig Functions . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8 Derivatives of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . .47
3.9 Chain Rule ....................................... 48
3.10 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.11 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.12 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.13 Logarithmic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Derivative Applications 56
4.1 Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
4.2 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Minimum and Maximum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Finding Absolute Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62
4.5 The Shape of a Graph, Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 The Shape of a Graph, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.7 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.8 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.9 More Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.10 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
4.11 Linear Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.12 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.13 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.14 Business Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Integrals 77
5.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Computing Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Substitution Rule for Indefinite Integrals . . . . . . . . . . . . . . . . . . 81
5.4 More Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.5 Area Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Definition of the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . .85
,5.7 Computing Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.8 Substitution Rule for Definite Integrals . . . . . . . . . . . . . . . . . . 89
6 Applications of Integrals 90
6.1 Average Function Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
6.2 Area Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Volume with Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4 Volume with Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.5 More Volume Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.6 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ……….97
7 Integration Techniques 98
7.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101
7.2 Integrals Involving Trig Functions . . . . . . . . . . . . . . . . . . . . . . . . 102
7.3 Trig Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103
7.4 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
7.5 Integrals Involving Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
7.6 Integrals Involving Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
7.7 Integration Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107
7.8 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
7.9 Comparison Test for Improper Integrals . . . . . . . . . . . . . . . . . 109
7.10 Approximating Definite Integrals . . . . . . . . . . . . . . . . . . . . . . .110
8 More Applications of Integrals 111
8.1 Arc Length ....................................... 112
8.2 Surface Area ...................................... 113
8.3 Center Of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.4 Hydrostatic Pressure and Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.5 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9 Parametric and Polar 118
9.1 Parametric Equations and Curves .........................
.119
9.2 Tangents with Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . 121
,9.3 Area with Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.4 Arc Length with Parametric Equations . . . . . . . . . . . . . . . . . . 123
9.5 Surface Area with Parametric Equations . . . . . . . . . . . . . . . . . . . . . .124
9.6 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
9.7 Tangents with Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 127
9.8 Area with Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128
9.9 Arc Length with Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 129
9.10 Surface Area with Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . .130
9.11 Arc Length and Surface Area Revisited . . . . . . . . . . . . . . . . . . . . . . .131
10 Series and Sequences 132
10.1 Sequences ....................................... 133
10.2 More on Sequences .................................. 134
10.3 Series - Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.4 Convergence/Divergence of Series . . . . . . . . . . . . . . . . . . . . . . . . . .136
10.5 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.6 Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
10.7 Comparison & Limit Comparison Test . . . . . . . . . . . . . . . . . . . . . . . .139
10.8 Alternating Series Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10.9 Absolute Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.10 Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.11 Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10.12 Strategy for Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.13 Estimating the Value of a Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .145
10.14 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10.15 Power Series and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.16 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.17 Applications of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.18 Binomial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
11 Vectors 151
,11.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
11.2 Vector Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153
11.3 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.4 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12 3D Space 156
12.1 The 3-D Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.2 Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
12.3 Equations of Planes .................................. 159
12.4 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
12.5 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . 161
12.6 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162
12.7 Calculus with Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . 163
12.8 Tangent, Normal and Binormal Vectors . . . . . . . . . . . . . . . . . . . . . . 164
12.9 Arc Length with Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
12.10 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.11 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
12.12 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.13 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
13 Partial Derivatives 170
13.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
13.3 Interpretations of Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 173
13.4 Higher Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
13.5 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
13.6 Chain Rule ....................................... 176
13.7 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
14 Applications of Partial Derivatives 179
14.1 Tangent Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
14.2 Gradient Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
14.3 Relative Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
,14.4 Relative Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
14.5 Lagrange Multipliers .................................. 184
15 Multiple Integrals 185
15.1 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .186
15.2 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187
15.3 Double Integrals over General Regions ....................
.188
15.4 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 190
15.5 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
15.6 Triple Integrals in Cylindrical Coordinates ....................
.192
15.7 Triple Integrals in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . .
.193
15.8 Change of Variables .................................. 194
15.9 Surface Area ...................................... 195
15.10 Area and Volume Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
16 Line Integrals 197
16.1 Vector Fields ...................................... 198
16.2 Line Integrals - Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
16.3 Line Integrals - Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
16.4 Line Integrals of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
16.5 Fundamental Theorem for Line Integrals . . . . . . . . . . . . . . . . . . . . . .206
16.6 Conservative Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207
16.7 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
17 Surface Integrals 210
17.1 Curl and Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
17.2 Parametric Surfaces .................................. 212
17.3 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
17.4 Surface Integrals of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . .214
17.5 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .215
,17.6 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Index 218
,
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