Calculus: Single Variable, Student Study and Solutions Companion

Brian Erf Blank is an associate professor of mathematics at Washington University in St. Louis. He received his Ph.D. in 1980 at Cornell University, with Anthony Knapp as advisor. His 20th century work involved harmonic analysis. Steven George Krantz is an American scholar, mathematician, and writer. He has authored more than 235 research papers and more than 125 books. Additionally, Krantz has edited journals such as the Notices of the American Mathematical Society and The Journal of Geometric Analysis.

1.1 Number Systems  1

1.2 Planar Coordinates and Graphing in the Pl ne  4

1.3 Lines and Their Slopes  10

1.4 Functions and Their Graphs  16

1.5 Combining Functions  21

1.6 Trigonometry 26

2 Limits 35

2.1 The Concept of Limit  35

2.2 Limit Theorems  41

2.3 Continuity 46

2.4 In nite Limits and Asymptotes  52

2.5 Limits of Sequences  57

2.6 Exponential Functions and Logarithms  60

3 The Derivative 68

3.1 Rates of Change and Tangent Lines  68

3.2 The Derivative  74

3.3 Rules for Dierentiation  80

3.4 Dierentiation of Some Basic Functions  88

3.5 The Chain Rule  93

3.6 Derivatives of Inverse Functions  99

3.7 Higher Derivatives  104

3.8 Implicit Dierentiation  109

3.9 Dierentials and Approximation of Functions  117

3.10 Other Transcendental Functions  122

4 Applications of the Derivative 130

4.1 Related Rates 130

4.2 The Mean Value Theorem  133

4.3 Maxima and Minima of Functions  139

4.4 Applied Maximum-Minimum Problems  145

4.5 Concavity  151

4.6 Graphing Functions  156

4.7 L'H^opital's Rule  162

4.8 The Newton-Raphson Method  168

4.9 Antidierentiation and Applications  172

5 The Integral 182

5.1 Introduction to Integration|The Area Problem  182

5.2 The Riemann Integral  187

5.3 Properties of the Integral  192

5.4 The Fundamental Theorem of Calculus  197

5.5 A Calculus Approach to the Logarithm and Exponential Function 202

5.6 Integration by Substitution  208

5.7 More on the Calculation of Area  216

5.8 Numerical Techniques of Integration  223

6 Techniques of Integration 239

6.1 Integration by Parts  239

6.2 Powers and Products of Trigonometric Functions 252

6.3 Trigonometric Substitution 264

6.4 Partial Fractions|Linear Factors 276

6.5 Partial Fractions|Irreducible Quadratic Factors  285

6.6 Improper Integrals|Unbounded Integrands  294

6.7 Improper Integrals|Unbounded Intervals  304

7 Applications of the Integral 325

7.1 Volumes  325

7.2 Arc Length and Surface Area  332

7.3 The Average Value of a Function 337

7.4 Center of Mass  341

7.5 Work  346

7.6 First Order Dierential Equations{Separable Equations  350

7.7 First Order Dierential Equations{Linear Equations  358

8 In nite Series 371

8.1 Series  371

8.2 The Divergence Test and the Integral Test  377

8.3 The Comparison Tests  382

8.4 Alternating Series  385

8.5 The Root and Ratio Tests  388

8.6 Introduction to Power Series  392

8.7 Representing Functions by Power Series  403

8.8 Taylor Series  411

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