The Humongous Book of Calculus Problems (Humongous Books) 9781592575121, 2006930703, 1592575129

Table of contents :
Contents iii
Introduction ix
Chapter 1: Linear Equations and Inequalities 1
Linear Geometry 2
Linear Inequalities and Interval Notation 5
Absolute Value Equations and Inequalities 8
Systems of Equations and Inequalities 11
Chapter 2: Polynomials 15
Exponential and Radical Expressions 16
Operations on Polynomial Expressions 18
Factoring Polynomials 21
Solving Quadratic Equations 23
Chapter 3: Rational Expressions 27
Adding and Subtracting Rational Expressions 28
Multiplying and Dividing Rational Expressions 30
Solving Rational Equations 33
Polynomial and Rational Inequalities 35
Chapter 4: Functions 41
Combining Functions 42
Graphing Function Transformations 45
Inverse Functions 50
Asymptotes of Rational Functions 53
Chapter 5: Logarithmic and Exponential Functions 57
Exploring Exponential and Logarithmic Functions 58
Natural Exponential and Logarithmic Functions 62
Properties of Logarithms 63
Solving Exponential and Logarithmic Equations 66
Chapter 6: Conic Sections 69
Parabolas 70
Circles 76
Ellipses 79
Hyperbolas 85
Chapter 7: Fundamentals of Trigonometry 91
Measuring Angles 92
Angle Relationships 93
Evaluating Trigonometric Functions 95
Inverse Trigonometric Functions 102
Chapter 8: Trigonometric Graphs, Identities, and Equations 105
Graphing Trigonometric Transformations 106
Applying Trigonometric Identities 110
Solving Trigonometric Equations 115
Chapter 9: Investigating Limits 123
Evaluating One-Sided and General Limits Graphically 124
Limits and Infinity 129
Formal Definition of the Limit 134
Chapter 10: Evaluating Limits 137
Substitution Method 138
Factoring Method 141
Conjugate Method 146
Special Limit Theorems 149
Chapter 11: Continuity and the Difference Quotient 151
Continuity 152
Types of Discontinuity 153
The Difference Quotient 163
Differentiability 166
Chapter 12: Basic Differentiation Methods 169
Trigonometric, Logarithmic, and Exponential Derivatives 170
The Power Rule 172
The Product and Quotient Rules 175
The Chain Rule 179
Chapter 13: Derivatives and Function Graphs 187
Critical Numbers 188
Signs of the First Derivative 191
Signs of the Second Derivative 197
Function and Derivative Graphs 202
Chapter 14: Basic Applications of Differentiation 205
Equations of Tangent Lines 206
The Extreme Value Theorem 211
Newton’s Method 214
L’Hôpital’s Rule 218
Chapter 15: Advanced Applications of Differentiation 223
The Mean Value and Rolle’s Theorems 224
Rectilinear Motion 229
Related Rates 233
Optimization 240
Chapter 16: Additional Differentiation Techniques 247
Implicit Differentiation 248
Logarithmic Differentiation 255
Differentiating Inverse Trigonometric Functions 260
Differentiating Inverse Functions 262
Chapter 17: Approximating Area 269
Informal Riemann Sums 270
Trapezoidal Rule 281
Simpson’s Rule 289
Formal Riemann Sums 291
Chapter 18: Integration 297
Power Rule for Integration 298
Integrating Trigonometric and Exponential Functions 301
The Fundamental Theorem of Calculus 303
Substitution of Variables 313
Chapter 19: Applications of the Fundamental Theorem 319
Calculating the Area Between Two Curves 320
The Mean Value Theorem for Integration 326
Accumulation Functions and Accumulated Change 334
Chapter 20: Integrating Rational Expressions 343
Separation 344
Long Division 347
Applying Inverse Trigonometric Functions 350
Completing the Square 353
Partial Fractions 357
Chapter 21: Advanced Integration Techniques 363
Integration by Parts 364
Trigonometric Substitution 368
Improper Integrals 383
Chapter 22: Cross-Sectional and Rotational Volume 389
Volume of a Solid with Known Cross-Sections 390
Disc Method 397
Washer Method 406
Shell Method 417
Chapter 23: Advanced Applications of Definite Integrals 423
Arc Length 424
Surface Area 427
Centroids 432
Chapter 24: Parametric and Polar Equations 443
Parametric Equations 444
Polar Coordinates 448
Graphing Polar Curves 451
Applications of Parametric and Polar Differentiation 456
Applications of Parametric and Polar Integration 462
Chapter 25: Differential Equations 467
Separation of Variables 468
Exponential Growth and Decay 473
Linear Approximations 480
Slope Fields 482
Euler’s Method 488
Chapter 26: Basic Sequences and Series 495
Sequences and Convergence 496
Series and Basic Convergence Tests 498
Telescoping Series and p-Series 502
Geometric Series 505
The Integral Test 507
Chapter 27: Additional Infinite Series Convergence Tests 511
Comparison Test 512
Limit Comparison Test 514
Ratio Test 517
Root Test 520
Alternating Series Test and Absolute Convergence 524
Chapter 28: Advanced Infinite Series 529
Power Series 530
Taylor and Maclaurin Series 538
Appendix A: Impo rtant Graphs to memorize and Graph Transformations 545
Appendix B: The Unit Circle 551
Appendix C: Trigonometric Identities 553
Appendix D: Derivative Fo rmulas 555
Appendix E: Anti-Derivative Formulas 557
Index 559