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| 6.2 || Motivation for the definition of the natural logarithm as an integral || 227
| 6.2 || Motivation for the definition of the natural logarithm as an integral || 227
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| 6.3 || The definition of the logarithm. Basic properties || 226
| 6.3 || The definition of the logarithm. Basic properties || 229
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| 6.4 || The graph of the natural logarithm || 226
| 6.4 || The graph of the natural logarithm || 230
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| 6.5 || Consequences of the functional equation L(ab) = L(a) + L(b) || 226
| 6.5 || Consequences of the functional equation \(L(ab) = L(a) + L(b)\) || 230
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| 6.6 || Logarithms referred to any positive base \(b \ne 1\) || 226
| 6.6 || Logarithms referred to any positive base \(b \ne 1\) || 232
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| 6.7 || Introduction || 226
| 6.7 || Differentiation and integration formulas involving logarithms || 233
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| 6.8 || Introduction || 226
| 6.8 || Logarithmic differentiation || 235
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| 6.9 || Introduction || 226
| 6.9 || Exercises || 236
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| 6.10 || Introduction || 226
| 6.10 || Polynomial approximations to the logarithm || 236
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| 6.11 || Introduction || 226
| 6.11 || Exercises || 242
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| 6.12 || Introduction || 226
| 6.12 || The exponential function || 242
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| 6.13 || Introduction || 226
| 6.13 || Exponentials expressed as powers of e || 242
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| 6.14 || Introduction || 226
| 6.14 || The definition of \(e^x\) for arbitrary real x || 244
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| 6.15 || Introduction || 226
| 6.15 || The definition of \(a^x\) for \(a > 0\) and x real || 245
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| 6.16 || Introduction || 226
| 6.16 || Differentiation and integration formulas involving exponentials || 245
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| 6.17 || Introduction || 226
| 6.17 || Exercises || 248
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| 6.18 || Introduction || 226
| 6.18 || The hyperbolic functions || 251
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| 6.19 || Introduction || 226
| 6.19 || Exercises || 251
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| 6.20 || Introduction || 226
| 6.20 || Derivatives of inverse functions || 252
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| 6.21 || Introduction || 226
| 6.21 || Inverses of the trigonometric functions || 253
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| 6.22 || Introduction || 226
| 6.22 || Exercises || 256
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| 6.23 || Introduction || 226
| 6.23 || Integration by partial fractions || 258
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| 6.24 || Introduction || 226
| 6.24 || Integrals which can be transformed into integrals of rational functions || 264
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| 6.25 || Exercises || 267
| 6.25 || Exercises || 267
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| 6.26 || Miscellaneous review exercises || 268
| 6.26 || Miscellaneous review exercises || 268
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! colspan="3" | Chapter 10: Segments, Rays, and Lines
! colspan="3" | 7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS
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| 1 || Segments || 229
| 7.1 || Introduction || 272
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| 2 || Rays || 231
| 7.2 || The Taylor polynomials generated by a function || 273
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| 3 || Lines || 236
| 7.3 || Calculus of Taylor polynomials || 275
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| 4 || Ordinary equation for a line || 246
| 7.4 || Exercises || 278
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! colspan="3" | Chapter 11: Trigonometry
| 7.5 || Taylor's formula with remainder || 278
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| 1 || Radian measure || 249
| 7.6 || Estimates for the error in Taylor's formula || 280
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| 2 || Sine and cosine || 252
| 7.7 || Other forms of the remainder in Taylor's formula || 283
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| 3 || The graphs || 264
| 7.8 || Exercises || 284
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| 4 || The tangent || 266
| 7.9 || Further remarks on the error in Taylor's formula. The o-notation || 286
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| 5 || Addition formulas || 272
| 7.10 || Applications to indeterminate forms || 289
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| 6 || Rotations || 277
| 7.11 || Exercises || 290
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! colspan="3" | Chapter 12: Some Analytic Geometry
| 7.12 || L'Hopital's rule for the indeterminate form 0/0 || 292
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| 1 || The straight line again || 281
| 7.13 || Exercises || 295
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| 2 || The parabola || 291
| 7.14 || The symbols \(+\inf\) and \(-\inf\). Extension of L'Hopital's rule || 296
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| 3 || The ellipse || 297
| 7.15 || Infinite limits || 298
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| 4 || The hyperbola || 300
| 7.16 || The behavior of log\(x\) and \(e^x\) for large \(x\) || 300
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| 5 || Rotation of hyperbolas || 305
| 7.17 || Exercises || 303
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! colspan="3" | 8. INTRODUCTION TO DIFFERENTIAL EQUATIONS
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| 8.1 || Introduction || 305
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| 8.2 || Terminology and notation || 306
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| 8.3 || A first-order differential equation for the exponential function || 307
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| 8.4 || First-order linear differential equations || 308
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| 8.5 || Exercises || 311
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| 8.6 || Some physical problems leading to first-order linear differential equations || 313
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| 8.7 || Exercises || 319
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| 8.8 || Linear equations of second order with constant coefficients || 322
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| 8.9 || Existence of solutions of the equation \(y^{''} + by = 0\) || 323
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| 8.10 || Reduction of the general equation to the special case \(y^{''} + by = 0\) || 324
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| 8.11 || Uniqueness theorem for the equation \(y^{''} + by = 0\) || 324
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| 8.12 || Complete solution of the equation \(y^{''} + by = 0\) || 326
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| 8.13 || Complete solution of the equation \(y^{''} + ay^' + by = 0\) || 326
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| 8.14 || Exercises || 328
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| 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329
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| 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation \(y^{''} + ay^' + by = R\) || 332
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| 8.17 || Exercises || 333
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| 8.18 || Examples of physical problems leading to linear second-order equations with constant coefficients || 334
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| 8.19 || Exercises || 339
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| 8.20 || Remarks concerning nonlinear differential equations || 339
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| 8.21 || Integral curves and direction fields || 341
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| 8.22 || Exercises || 344
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| 8.23 || First-order separable equations || 345
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| 8.24 || Exercises || 347
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| 8.25 || Homogeneous first-order equations || 347
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| 8.26 || Exercises || 350
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| 8.27 || Some geometrical and physical problems leading to first-order equations || 351
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| 8.28 || Miscellaneous review exercises || 355
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! colspan="3" | 9. COMPLEX NUMBERS
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| 9.1 || Historical introduction || 358
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| 9.2 || Definitions and field properties || 358
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| 9.3 || The complex numbers as an extension of the real numbers || 360
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| 9.4 || The imaginary unit \(i\) || 361
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| 9.5 || Geometric interpretation. Modulus and argument || 362
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| 9.6 || Exercises || 365
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| 9.7 || Complex exponentials || 366
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| 9.8 || Complex-valued functions || 368
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| 9.9 || Examples of differentiation and integration formulas || 369
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| 9.10 || Exercises || 371
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! colspan="3" | PART IV: MISCELLANEOUS
! colspan="3" | PART IV: MISCELLANEOUS
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