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PAGE
CHAPTER I. CO-ORDINATES 1
CHAPTEE II. THE STRAIGHT LINE 12
Examples on Chapter II 43
CHAPTER III. CHANGE OF AXES. ANHAKMONIC RATIOS OR
CROSS RATIOS. INVOLUTION 47
CHAPTER IV. THE CIRCLE 60
Examples on Chapter IV 83
CHAPTER V. THE PARABOLA , 88
Examples on Chapter V 105
CHAPTER VI. THE ELLIPSE 112
Examples on Chapter VI 138
CHAPTER VII. THE HYPERBOLA . ..... 146
Examples on Chapter VII 162
CHAPTER VIII. POLAR EQUATION OF A CONIC, THE FOCUS BEING
THE POLE 166
Examples on Chapter VIII 176
CHAPTER IX. GENERAL EQUATION OF THE SECOND DEGREE.
Every curve whose equation is of the second degree is a conic 179
Co-ordinates of the centre of a conic 181
The Discriminant 182
Position and magnitude of the axes of a central conic . . 183
Vlll CONTENTS.
PAGE
Axis and latus rectum of a parabola . . . . 184
Tracing conies 185
Equation of the asymptotes of a conic . . . . 188
Condition for a rectangular hyperbola 189
Examples on Chapter IX 190
CHAPTEE X. MISCELLANEOUS PROPOSITIONS.
Equation of the tangent at any point of a conic . . . 192
Condition that a given straight line may touch a conic . 193
Equation of the polar of any point with respect to a conic . 194
Conjugate points and conjugate lines .... 195
A chord of a conic is cut harmonically by a point and its
polar 195
Diameters of a conic 196
Condition that two given lines may be parallel to conjugate
diameters 196
Equi-con jugate diameters of a conic 197
Common conjugate diameters of two conies . . . 197
Segments of chords of a conic 198
Pairs of common chords of a circle and a conic are equally
inclined to the axes of the conic 199
Meaning of S - S = 0, S - uv = and S - u- = . . 200
Equation of a pair of tangents 202
Equation of the director-circle 203
Four foci of a conic 204
Eccentricities of a conic 205
Equations giving the foci 207
Equation of the axes 209
Equation of a conic referred to tangent and normal . . 211
Normals 211
Similar conies 215
Examples on Chapter X 219
CHAPTEE XI. SYSTEMS OF CONICS.
One conic through five points 231
Conies through four points. [See also 295] .... 233
Two parabolas through four points 234
Centre-locus of conies through four points. [See also 296] 234
Diagonal-points of a quadrangle are angular points of a
triangle self-polar with respect to any circumscribing
conic. [See also 297.] 237
CONTENTS. IX
PAGE
Diagonals of a quadrilateral are sides of a triangle self -polar
with respect to any inscribed conic .... 238
Centre-locus of conies touching four fixed lines. [See also
137, 270, and 297] 240
Parabola touching the axes of co-ordinates .... 241
Confocal conies 244
Osculating conies 251
Pairs of common chords of two conies 253
Invariants 254
Examples on Chapter XI 257
CHAPTER XII. ENVELOPES.
Envelopes, how found . . 264
Tangential co-ordinates and equations .... 267
Director-circle of envelope (cf. 203) 268
Foci of envelope 269
Meaning of the tangential equation S-S . . . 269
Director-circles of conies which touch four given straight lines
have a common radical axis. [See also 308 and 319.] . 270
Examples 271
CHAPTER XIII. THILIXEAB CO-ORDINATES.
Definition of Trilinear Co-ordinates 276
Equations and intersections of straight lines . . . 278
The line at infinity 280
Parallel lines 280
Condition of perpendicularity 281
Length of perpendicular from a point on a line . . 282
Co-ordinates of four points in the form i /, =b g, =k h . 283
Equation of four lines in the form lam(iny=0 . . 284
Examples 285
Tangent 286
Condition of tangency 287
/ Co-ordinates of the centre of a conic 288
Condition for a parabola 288
T The asymptotes 289
Condition for a rectangular hyperbola .... 290
The circumscribing circle 290
The circular points at infinity 291
Conditions for a circle . 291
X CONTENTS.
I AGE
Condition for ellipse, parabola, or hyperbola . . . 292
Foci 293
Equation of a circumscribing conic 293
Equation of an inscribed conic 294
Equation of the inscribed circle 294
Conies through four fixed points ...... 295
Conies touching four fixed lines 296
Conies with respect to which the triangle of reference is
self-polar 297
Conies referred to two tangents and the chord of contact . 298
The circle with respect to which the triangle of reference is
self-polar 298
The nine-point circle 299
Pascal s theorem. [See also 328.] 299
Brianchon s theorem 301
ABEAL CO-OEDINATES 302
TANGENTIAL CO-OBDINATES 303
Identical relation between the tangential co-ordinates of any
straight line 304
Tangential equation of a conic 305
The equation of the pole of a line 305
Examples on Chapter XIII. 309
CHAPTEK XIV. KECIPBOCAL POLABS. PBOJECTIONS.
Definition of polar reciprocal 315
The degree of a curve is the same as the class of its reciprocal 315
Examples of reciprocal theorems 317
Eeciprocation with respect to a circle 318
Eeciprocal of a circle with respect to a point . . . 320
Co-axial circles reciprocated into confocal conies . . 322
Projection. Definition of projection 323
The projection of any curve is a curve of the same degree . 323
Projections of tangents, poles and polars .... 324
Straight lines meeting in a point projected into parallel
straight lines . . 324
Any line can be projected to infinity, and at the same time
any two angles into given angles 324
A quadrilateral projected into a square .... 326
Any conic projected into a circle 327
CONTENTS. XI
PAQR
A system of conies inscribed in a quadrilateral projected into
confocal conies ........ 327
Cross ratios of pencils and ranges unaltered by projection . 329
Conies through four fixed points are cut by any straight line
in pairs of points in involution ..... 329
The cross ratio of a pencil of four lines equal to that of the
range formed by their poles ...... 329
Anharmonic properties of points on a conic, and of tangents
to a conic ......... 330
Homographic ranges and pencils ..... 331
Two lines at right angles together with the lines to the
circular points at infinity form a harmonic pencil . 334
Examples .......... 336
Appendix .......... 339
CHAPTER I.
CO-ORDINATES.
1. IF in a plane two fixed straight lines XOX , YOY
be taken, and through any point P in the plane the two
straight lines PM, PL be drawn parallel to XOX , YOY
respectively; the position of the point P can be found
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Link : http://rapidshare.com/files/115287749/CalcI_Complete.pdf
Preface ........................................................................................................................................... iii
Outline ........................................................................................................................................... iv
Review............................................................................................................................................. 2
Introduction .............................................................................................................................................. 2
Review : Functions ................................................................................................................................... 4
Review : Inverse Functions .................................................................................................................... 10
Review : Trig Functions ......................................................................................................................... 17
Review : Solving Trig Equations ............................................................................................................ 24
Review : Solving Trig Equations with Calculators, Part I .................................................................... 33
Review : Solving Trig Equations with Calculators, Part II ................................................................... 44
Review : Exponential Functions ............................................................................................................ 49
Review : Logarithm Functions ............................................................................................................... 52
Review : Exponential and Logarithm Equations .................................................................................. 58
Review : Common Graphs ...................................................................................................................... 64
Limits ............................................................................................................................................ 76
Introduction ............................................................................................................................................ 76
Rates of Change and Tangent Lines ...................................................................................................... 78
The Limit ................................................................................................................................................. 87
One?Sided Limits .................................................................................................................................... 97
Limit Properties .....................................................................................................................................103
Computing Limits ..................................................................................................................................109
Infinite Limits ........................................................................................................................................117
Limits At Infinity, Part I .........................................................................................................................126
Limits At Infinity, Part II .......................................................................................................................135
Continuity ...............................................................................................................................................144
The Definition of the Limit ....................................................................................................................151
Derivatives .................................................................................................................................. 166
Introduction ...........................................................................................................................................166
The Definition of the Derivative ...........................................................................................................168
Interpretations of the Derivative .........................................................................................................174
Differentiation Formulas ......................................................................................................................179
Product and Quotient Rule ...................................................................................................................187
Derivatives of Trig Functions ...............................................................................................................193
Derivatives of Exponential and Logarithm Functions ........................................................................204
Derivatives of Inverse Trig Functions ..................................................................................................209
Derivatives of Hyperbolic Functions ....................................................................................................215
Chain Rule ..............................................................................................................................................217
Implicit Differentiation .........................................................................................................................227
Related Rates .........................................................................................................................................236
Higher Order Derivatives ......................................................................................................................250
Logarithmic Differentiation ..................................................................................................................255
Applications of Derivatives ....................................................................................................... 258
Introduction ...........................................................................................................................................258
Rates of Change......................................................................................................................................260
Critical Points .........................................................................................................................................263
Minimum and Maximum Values ...........................................................................................................269
Finding Absolute Extrema ....................................................................................................................277
The Shape of a Graph, Part I ..................................................................................................................283
The Shape of a Graph, Part II ................................................................................................................292
The Mean Value Theorem .....................................................................................................................301
Optimization ..........................................................................................................................................308
More Optimization Problems ...............................................................................................................322 Calculus I
© 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx
Indeterminate Forms and L’Hospital’s Rule ........................................................................................336
Linear Approximations .........................................................................................................................342
Differentials ...........................................................................................................................................345
Newton’s Method ...................................................................................................................................348
Business Applications ...........................................................................................................................353
Integrals ...................................................................................................................................... 359
Introduction ...........................................................................................................................................359
Indefinite Integrals ................................................................................................................................360
Computing Indefinite Integrals ............................................................................................................366
Substitution Rule for Indefinite Integrals ............................................................................................376
More Substitution Rule .........................................................................................................................389
Area Problem .........................................................................................................................................402
The Definition of the Definite Integral .................................................................................................412
Computing Definite Integrals ...............................................................................................................422
Substitution Rule for Definite Integrals ...............................................................................................434
Applications of Integrals ........................................................................................................... 445
Introduction ...........................................................................................................................................445
Average Function Value ........................................................................................................................446
Area Between Curves ............................................................................................................................449
Volumes of Solids of Revolution / Method of Rings ............................................................................460
Volumes of Solids of Revolution / Method of Cylinders .....................................................................470
Work .......................................................................................................................................................478
Extras .......................................................................................................................................... 482
Introduction ...........................................................................................................................................482
Proof of Various Limit Properties ........................................................................................................483
Proof of Various Derivative Facts/Formulas/Properties ...................................................................494
Proof of Trig Limits ...............................................................................................................................507
Proofs of Derivative Applications Facts/Formulas .............................................................................512
Proof of Various Integral Facts/Formulas/Properties .......................................................................523
Area and Volume Formulas ..................................................................................................................535
Types of Infinity .....................................................................................................................................539
Summation Notation .............................................................................................................................543
Constants of Integration .......................................................................................................................545
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http://rapidshare.com/files/115286157/Alg_Complete.rar
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Link: http://rapidshare.com/files/115286157/Alg_Complete.rar
Contents
Preface ........................................................................................................................................... iii
Outline ........................................................................................................................................... iv
Preliminaries .................................................................................................................................. 1
Introduction ................................................................................................................................................ 1
Integer Exponents ...................................................................................................................................... 2
Rational Exponents .................................................................................................................................... 9
Real Exponents ........................................................................................................................................ 15
Radicals .................................................................................................................................................... 16
Polynomials .............................................................................................................................................. 25
Factoring Polynomials ............................................................................................................................. 31
Rational Expressions ................................................................................................................................ 41
Complex Numbers ................................................................................................................................... 52
Solving Equations and Inequalities ............................................................................................ 58
Introduction .............................................................................................................................................. 58
Solutions and Solution Sets ...................................................................................................................... 59
Linear Equations ...................................................................................................................................... 63
Application of Linear Equations .............................................................................................................. 71
Equations With More Than One Variable ................................................................................................ 81
Quadratic Equations – Part I .................................................................................................................... 85
Quadratic Equations – Part II ................................................................................................................... 93
Solving Quadratic Equations : A Summary ............................................................................................104
Application of Quadratic Equations ........................................................................................................107
Equations Reducible to Quadratic Form .................................................................................................111
Equations with Radicals ..........................................................................................................................116
Linear Inequalities ...................................................................................................................................122
Polynomial Inequalities ...........................................................................................................................129
Rational Inequalities ...............................................................................................................................135
Absolute Value Equations .......................................................................................................................140
Absolute Value Inequalities ....................................................................................................................147
Graphing and Functions ........................................................................................................... 152
Introduction .............................................................................................................................................152
Graphing .................................................................................................................................................153
Lines ........................................................................................................................................................159
Circles .....................................................................................................................................................169
The Definition of a Function ...................................................................................................................175
Graphing Functions .................................................................................................................................186
Combining Functions ..............................................................................................................................190
Inverse Functions ....................................................................................................................................197
Common Graphs ....................................................................................................................... 204
Introduction .............................................................................................................................................204
Lines, Circles and Piecewise Functions ..................................................................................................205
Parabolas .................................................................................................................................................206
Ellipses ....................................................................................................................................................216
Hyperbolas ..............................................................................................................................................220
Miscellaneous Functions .........................................................................................................................224
Transformations ......................................................................................................................................227
Symmetry ................................................................................................................................................233
Rational Functions ..................................................................................................................................238
Polynomial Functions ................................................................................................................ 244
Introduction .............................................................................................................................................244
Dividing Polynomials..............................................................................................................................245
Zeroes/Roots of Polynomials ..................................................................................................................250 College Algebra
© 2007 Paul Dawkins ii
Graphing Polynomials .............................................................................................................................255
Finding Zeroes of Polynomials ...............................................................................................................263
Partial Fractions ......................................................................................................................................271
Exponential and Logarithm Functions .................................................................................... 279
Introduction .............................................................................................................................................279
Exponential Functions .............................................................................................................................280
Logarithm Functions ...............................................................................................................................285
Solving Exponential Equations ...............................................................................................................295
Solving Logarithm Equations .................................................................................................................302
Applications ............................................................................................................................................308
Systems of Equations ................................................................................................................. 315
Introduction .............................................................................................................................................315
Linear Systems with Two Variables .......................................................................................................316
Linear Systems with Three Variables .....................................................................................................324
Augmented Matrices ...............................................................................................................................326
More on the Augmented Matrix ..............................................................................................................335
Non-Linear Systems ................................................................................................................................341
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Link:
Contents
Preface ........................................................................................................................................... iii
Outline ........................................................................................................................................... iv
Preliminaries .................................................................................................................................. 1
Introduction ................................................................................................................................................ 1
Integer Exponents ...................................................................................................................................... 2
Rational Exponents .................................................................................................................................... 9
Real Exponents ........................................................................................................................................ 15
Radicals .................................................................................................................................................... 16
Polynomials .............................................................................................................................................. 25
Factoring Polynomials ............................................................................................................................. 31
Rational Expressions ................................................................................................................................ 41
Complex Numbers ................................................................................................................................... 52
Solving Equations and Inequalities ............................................................................................ 58
Introduction .............................................................................................................................................. 58
Solutions and Solution Sets ...................................................................................................................... 59
Linear Equations ...................................................................................................................................... 63
Application of Linear Equations .............................................................................................................. 71
Equations With More Than One Variable ................................................................................................ 81
Quadratic Equations – Part I .................................................................................................................... 85
Quadratic Equations – Part II ................................................................................................................... 93
Solving Quadratic Equations : A Summary ............................................................................................104
Application of Quadratic Equations ........................................................................................................107
Equations Reducible to Quadratic Form .................................................................................................111
Equations with Radicals ..........................................................................................................................116
Linear Inequalities ...................................................................................................................................122
Polynomial Inequalities ...........................................................................................................................129
Rational Inequalities ...............................................................................................................................135
Absolute Value Equations .......................................................................................................................140
Absolute Value Inequalities ....................................................................................................................147
Graphing and Functions ........................................................................................................... 152
Introduction .............................................................................................................................................152
Graphing .................................................................................................................................................153
Lines ........................................................................................................................................................159
Circles .....................................................................................................................................................169
The Definition of a Function ...................................................................................................................175
Graphing Functions .................................................................................................................................186
Combining Functions ..............................................................................................................................190
Inverse Functions ....................................................................................................................................197
Common Graphs ....................................................................................................................... 204
Introduction .............................................................................................................................................204
Lines, Circles and Piecewise Functions ..................................................................................................205
Parabolas .................................................................................................................................................206
Ellipses ....................................................................................................................................................216
Hyperbolas ..............................................................................................................................................220
Miscellaneous Functions .........................................................................................................................224
Transformations ......................................................................................................................................227
Symmetry ................................................................................................................................................233
Rational Functions ..................................................................................................................................238
Polynomial Functions ................................................................................................................ 244
Introduction .............................................................................................................................................244
Dividing Polynomials..............................................................................................................................245
Zeroes/Roots of Polynomials ..................................................................................................................250 College Algebra
© 2007 Paul Dawkins ii
Graphing Polynomials .............................................................................................................................255
Finding Zeroes of Polynomials ...............................................................................................................263
Partial Fractions ......................................................................................................................................271
Exponential and Logarithm Functions .................................................................................... 279
Introduction .............................................................................................................................................279
Exponential Functions .............................................................................................................................280
Logarithm Functions ...............................................................................................................................285
Solving Exponential Equations ...............................................................................................................295
Solving Logarithm Equations .................................................................................................................302
Applications ............................................................................................................................................308
Systems of Equations ................................................................................................................. 315
Introduction .............................................................................................................................................315
Linear Systems with Two Variables .......................................................................................................316
Linear Systems with Three Variables .....................................................................................................324
Augmented Matrices ...............................................................................................................................326
More on the Augmented Matrix ..............................................................................................................335
Non-Linear Systems ................................................................................................................................341
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