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Calculus (Book): Difference between revisions

Calculus (Book) (edit)

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| 1.22 || Calculation of the integral of a bounded monotonic function || 79
| 1.22 || Calculation of the integral of a bounded monotonic function || 79
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| 1.23 || Calculation of the integral \(\int_0^b x^p dx\) when \(p\) is a positive integer || 79
| 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> is a positive integer || 79
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| 1.24 || The basic properties of the integral || 80
| 1.24 || The basic properties of the integral || 80
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| 6.4 || The graph of the natural logarithm || 230
| 6.4 || The graph of the natural logarithm || 230
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| 6.5 || Consequences of the functional equation \(L(ab) = L(a) + L(b)\) || 230
| 6.5 || Consequences of the functional equation <math>L(ab) = L(a) + L(b)</math> || 230
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| 6.6 || Logarithms referred to any positive base \(b \ne 1\) || 232
| 6.6 || Logarithms referred to any positive base <math>b \ne 1</math> || 232
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| 6.7 || Differentiation and integration formulas involving logarithms || 233
| 6.7 || Differentiation and integration formulas involving logarithms || 233
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| 6.13 || Exponentials expressed as powers of e || 242
| 6.13 || Exponentials expressed as powers of e || 242
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| 6.14 || The definition of \(e^x\) for arbitrary real x || 244
| 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244
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| 6.15 || The definition of \(a^x\) for \(a > 0\) and x real || 245
| 6.15 || The definition of <math>a^x</math> for <math>a > 0</math> and x real || 245
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| 6.16 || Differentiation and integration formulas involving exponentials || 245
| 6.16 || Differentiation and integration formulas involving exponentials || 245
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| 7.13 || Exercises || 295
| 7.13 || Exercises || 295
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| 7.14 || The symbols \(+\inf\) and \(-\inf\). Extension of L'Hopital's rule || 296
| 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296
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| 7.15 || Infinite limits || 298
| 7.15 || Infinite limits || 298
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| 7.16 || The behavior of log\(x\) and \(e^x\) for large \(x\) || 300
| 7.16 || The behavior of log<math>x</math> and <math>e^x</math> for large <math>x</math> || 300
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| 7.17 || Exercises || 303
| 7.17 || Exercises || 303
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| 8.8 || Linear equations of second order with constant coefficients || 322
| 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
| 8.9 || Existence of solutions of the equation <math>y^{''} + by = 0</math> || 323
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| 8.10 || Reduction of the general equation to the special case \(y^{''} + by = 0\) || 324
| 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324
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| 8.11 || Uniqueness theorem for the equation \(y^{''} + by = 0\) || 324
| 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324
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| 8.12 || Complete solution of the equation \(y^{''} + by = 0\) || 326
| 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326
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| 8.13 || Complete solution of the equation \(y^{''} + ay^' + by = 0\) || 326
| 8.13 || Complete solution of the equation <math>y^{''} + ay^' + by = 0</math> || 326
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| 8.14 || Exercises || 328
| 8.14 || Exercises || 328
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| 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329
| 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
| 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^' + by = R</math> || 332
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| 8.17 || Exercises || 333
| 8.17 || Exercises || 333
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| 9.3 || The complex numbers as an extension of the real numbers || 360
| 9.3 || The complex numbers as an extension of the real numbers || 360
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| 9.4 || The imaginary unit \(i\) || 361
| 9.4 || The imaginary unit <math>i</math> || 361
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| 9.5 || Geometric interpretation. Modulus and argument || 362
| 9.5 || Geometric interpretation. Modulus and argument || 362
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| 12.2 || The vector space of n-tuples of real numbers || 446
| 12.2 || The vector space of n-tuples of real numbers || 446
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| 12.3 || Geometric interpretation for \(n \leq 3\) || 448
| 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448
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| 12.4 || Exercises || 450
| 12.4 || Exercises || 450
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| 12.15 || Exercises || 467
| 12.15 || Exercises || 467
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| 12.16 || The vector space \(V_N(C)\) of n-tuples of complex numbers || 468
| 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468
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| 12.17 || Exercises || 470
| 12.17 || Exercises || 470
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