| Sequences | |
| Basic definitions and theorems | p. 1 |
| Examples and exercises on general notions | p. 3 |
| Representation of a number by sequences | p. 5 |
| Evaluation of N ([varepsilon]) | p. 7 |
| Sequences given in the form n[subscript n+1] = f(u[subscript n]) | p. 8 |
| Methods for the evaluation of limits | p. 9 |
| Functions of a Single Variable | |
| Definition and notation | p. 12 |
| The elementary functions | p. 13 |
| Domain of definition | p. 16 |
| Even and odd functions | p. 16 |
| Rational functions | p. 17 |
| Logarithmic functions | p. 18 |
| Trigonometric functions | p. 18 |
| Hyperbolic functions | p. 19 |
| Inverse functions | p. 19 |
| The inverse trigonometric functions | p. 20 |
| The inverse hyperbolic functions | p. 21 |
| Composite functions | p. 21 |
| Periodic functions | p. 22 |
| Limit of a Function | |
| Definitions and general exercises | p. 23 |
| Evaluation of limits | p. 26 |
| Continuity | p. 30 |
| Differential Calculus for Functions of a Single Variable | |
| The notion of derivative and its physical and geometric interpretation | p. 36 |
| Evaluating derivatives | p. 38 |
| Evaluating derivatives of explicit functions | p. 42 |
| Differentiation of implicit functions | p. 45 |
| Parametric differentiation | p. 46 |
| Special cases in calculating derivatives | p. 46 |
| Higher derivatives | p. 49 |
| Calculation of y[superscript (n)] | p. 51 |
| Graphical differentiation | p. 53 |
| Various examples | p. 55 |
| Fundamental Theorems of the Differential Calculus | |
| The theorems of Rolle, Lagrange, and Cauchy | p. 57 |
| Taylor's and Maclaurin's formulas | p. 60 |
| Indeterminate forms: L'Hopital's rule | p. 65 |
| Applications of Differential Calculus | |
| Rate of change | p. 68 |
| Locating intervals in which a function increases or decreases | p. 69 |
| Minima and maxima | p. 71 |
| Concavity: points of inflection | p. 85 |
| Asymptotes | p. 88 |
| Curve tracing | p. 93 |
| Graphs in polar coordinates | p. 101 |
| Parametric equations | p. 105 |
| Tangent and normal | p. 107 |
| The order of contact | p. 113 |
| Osculating circle, radius of curvature | p. 114 |
| Evolute and involute | p. 118 |
| Solution of equations by Newton's approximation method | p. 120 |
| The Differential | |
| Definition of the differential | p. 125 |
| The invariance of the form of the differential | p. 125 |
| The differential as the principal part of the increment of the function: application to approximate calculations | p. 129 |
| Higher order differentials | p. 131 |
| The Indefinite Integral | |
| Definition and basic properties | p. 134 |
| Immediate integrals | p. 136 |
| The method of substitution | p. 140 |
| Integration by parts | p. 144 |
| Integrals of rational functions | p. 151 |
| Irrational integrals | p. 157 |
| Trigonometric integrals | p. 165 |
| Integrals of exponential and hyperbolic functions | p. 169 |
| Miscellaneous integrals | p. 170 |
| The Definite Integral | |
| Definition | p. 172 |
| Basic properties of the definite integral | p. 174 |
| Evaluation of the definite integral from its definition | p. 176 |
| Estimation of definite integrals | p. 179 |
| The mean value theorem of integral calculus | p. 180 |
| Integrals with variable limits | p. 181 |
| Evaluation of definite integrals | p. 182 |
| Changing the variable of integration | p. 184 |
| Approximate integration | p. 187 |
| Improper integrals | p. 191 |
| Miscellaneous problems | p. 195 |
| Applications of the Definite Integral | |
| Computation of plane areas | p. 200 |
| Computation of are length | p. 204 |
| Computation of volumes | p. 207 |
| Area of a surface of revolution | p. 214 |
| Moment of mass: centroids | p. 215 |
| Pappus' theorems | p. 219 |
| Moment of inertia | p. 220 |
| Physics problems | p. 222 |
| Infinite Series | |
| The general notion of a number series | p. 227 |
| Convergence of series with positive terms | p. 228 |
| Convergence of series with positive and negative terms | p. 235 |
| Arithmetic operations on series | p. 237 |
| Series of functions | p. 240 |
| Power series: radius of convergence | p. 248 |
| Taylor's and Maclaurin's series: operations on power series | p. 250 |
| Applications of Taylor's and Maclaurin's expansions | p. 259 |
| Various Problems | p. 265 |
| Solutions, Hints, Answers | p. 273 |
| Index | p. 451 |
| List of Greek Letters | p. 457 |
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