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THIRD SEMESTER CALCULUS

Student Supplement

EXPANDED FOURTH EDITION

by Professor Mimi Rasky

Table of Contents

 Preface  v

 Note to Instructors  v

 Preface to the 4TH Edition vii

 Acknowledgements viii


PART I  STUDY TIPS  & TRICKS  1

Chapter 1 Vectors and the Geometry of Space 3

1.1  Vectors in the Plane  4

1.2  Space Coordinates and Vectors in Space  13

1.3  The Dot Product of Two Vectors  15

1.4  The Cross Product of Two Vectors in Space  18

1.5  Lines and Planes in Space  20

1.6  Surfaces in Space  25

1.7  Cylindrical and Spherical Coordinates  26

Chapter 2 Vector-Valued Functions  36

2.1  Vector-Valued Functions  36

2.2  Differentiation and Integration of
       Vector-Valued Functions 56

2.3  Velocity and Acceleration  69

2.4  Tangent Vectors and Normal Vectors  73

2.5  Arc Length and Curvature  79

Chapter 3 Functions of Several Variables  85

3.1  Introduction to Functions of Several Variables  85

3.2  Limits and Continuity  89

3.3  Partial Derivatives  92

3.4  Differentials  94

3.5  Chain Rules for Functions of Several Variables  98

3.6  Directional Derivatives and Gradients 106

3.7  Tangent Planes and Normal Lines  108

3.8  Extrema of Functions of Two Variables  111

3.9  Lagrange Multipliers  123

Chapter 4 Multiple Integration  130

4.1  Iterated Integrals and Area in the Plane  130

4.2  Double Integrals and Volume  134

4.3  Change of Variables: Polar Coordinates  140

4.4  Center of Mass  148

4.5  Surface Area  151

4.6  Triple Integrals and Applications  158

4.7  Triple Integrals in Cylindrical and
       Spherical Coordinates  166

Chapter 5 Vector Analysis  177

5.1  Vector Fields  177

5.2  Line Integrals  182

5.3  Conservative Vector Fields and Independence of Path  193

5.4  Green’s Theorem in the Plane  205

5.5  Surface Integrals & Flux Integrals  222

5.6  The Divergence Theorem  232

5.7  Stokes’ Theorem  238

PART II TEST  BANK  241

 Quiz 1  243

 Quiz 2  244

 Quiz 3  251

 Practice Midterm 1  252

 Quiz 4  257

 Quiz 5  262

 Quiz 6  269

 Practice Midterm 2  271

 Quiz 7  278

 Quiz 8  279

 Quiz 9  287

 Practice Midterm 3  296

 Quiz 10  305

 Practice Midterm 4  313

 Quiz 11  321

 Practice FINAL Exam  329


PART III ANSWER KEYS  341

 Quiz 1 KEY  343

 Quiz 2 KEY  344

 Quiz 3 KEY  352

 Practice Midterm 1 KEY  353

 Quiz 4 KEY  360

 Quiz 5 KEY  370

 Quiz 6 KEY  379

 Practice Midterm 2 KEY  381

 Quiz 7 KEY  396

 Quiz 8 KEY  398

 Quiz 9 KEY  417

 Practice Midterm 3 KEY  433

 Quiz 10 KEY  454

 Practice Midterm 4 KEY  468

 Quiz 11 KEY  478

 Practice FINAL Exam KEY  487