contents
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1 functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1 the concept of a function . . . . . . . . . . . . . . . . . . . . 2
1.2 trigonometric functions . . . . . . . . . . . . . . . . . . . . . 12
1.3 ınverse trigonometric functions . . . . . . . . . . . . . . . . . 19
1.4 logarithmic, exponential and hyperbolic functions . . . . . . 26
2 limits and continuity 35
2.1 ıntuitive treatment and definitions . . . . . . . . . . . . . . . 35
2.1.1 ıntroductory examples . . . . . . . . . . . . . . . . . . 35
2.1.2 limit: formal definitions . . . . . . . . . . . . . . . . 41
2.1.3 continuity: formal definitions . . . . . . . . . . . . . 43
2.1.4 continuity examples . . . . . . . . . . . . . . . . . . . 48
2.2 linear function approximations . . . . . . . . . . . . . . . . . 61
2.3 limits and sequences . . . . . . . . . . . . . . . . . . . . . . . 72
2.4 properties of continuous functions . . . . . . . . . . . . . . . 84
2.5 limits and ınfinity . . . . . . . . . . . . . . . . . . . . . . . . 94
3 differentiation 99
3.1 the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2 the chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.3 differentiation of ınverse functions . . . . . . . . . . . . . . . 118
3.4 ımplicit differentiation . . . . . . . . . . . . . . . . . . . . . . 130
3.5 higher order derivatives . . . . . . . . . . . . . . . . . . . . . 137
4 applications of differentiation 146
4.1 mathematical applications . . . . . . . . . . . . . . . . . . . . 146
4.2 antidifferentiation . . . . . . . . . . . . . . . . . . . . . . . . 157
4.3 linear first order differential equations . . . . . . . . . . . . 164
4.4 linear second order homogeneous differential equations . . . 169
4.5 linear non-homogeneous second order differential equations 179
5 the definite ıntegral 183
5.1 area approximation . . . . . . . . . . . . . . . . . . . . . . . 183
5.2 the definite ıntegral . . . . . . . . . . . . . . . . . . . . . . . 192
5.3 ıntegration by substitution . . . . . . . . . . . . . . . . . . . . 210
5.4 ıntegration by parts . . . . . . . . . . . . . . . . . . . . . . . 216
5.5 logarithmic, exponential and hyperbolic functions . . . . . . 230
5.6 the riemann ıntegral . . . . . . . . . . . . . . . . . . . . . . 242
5.7 volumes of revolution . . . . . . . . . . . . . . . . . . . . . . 250
5.8 arc length and surface area . . . . . . . . . . . . . . . . . . 260
6 techniques of ıntegration 267
6.1 ıntegration by formulae . . . . . . . . . . . . . . . . . . . . . . 267
6.2 ıntegration by substitution . . . . . . . . . . . . . . . . . . . . 273
6.3 ıntegration by parts . . . . . . . . . . . . . . . . . . . . . . . 276
6.4 trigonometric ıntegrals . . . . . . . . . . . . . . . . . . . . . . 280
6.5 trigonometric substitutions . . . . . . . . . . . . . . . . . . . 282
6.6 ıntegration by partial fractions . . . . . . . . . . . . . . . . . 288
6.7 fractional power substitutions . . . . . . . . . . . . . . . . . . 289
6.8 tangent x/2 substitution . . . . . . . . . . . . . . . . . . . . 290
6.9 numerical ıntegration . . . . . . . . . . . . . . . . . . . . . . 291
7 ımproper ıntegrals and ındeterminate forms 294
7.1 ıntegrals over unbounded ıntervals . . . . . . . . . . . . . . . 294
7.2 discontinuities at end points . . . . . . . . . . . . . . . . . . 299
7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
7.4 ımproper ıntegrals . . . . . . . . . . . . . . . . . . . . . . . . 314
8 ınfinite series 315
8.1 sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.2 monotone sequences . . . . . . . . . . . . . . . . . . . . . . . 320
8.3 ınfinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
8.4 series with positive terms . . . . . . . . . . . . . . . . . . . . 327
8.5 alternating series . . . . . . . . . . . . . . . . . . . . . . . . . 341
8.6 power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
8.7 taylor polynomials and series . . . . . . . . . . . . . . . . . . 354
8.8 applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
9 analytic geometry and polar coordinates . . . . . . 361
9.1 parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
9.2 ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
9.3 hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.4 second-degree equations . . . . . . . . . . . . . . . . . . . . . 363
9.5 polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 364
9.6 graphs in polar coordinates . . . . . . . . . . . . . . . . . . . 365
9.7 areas in polar coordinates . . . . . . . . . . . . . . . . . . . . 366
9.8 parametric equations . . . . . . . . . . . . . . . . . . . . . . . 366
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