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