Calculus

Cover Page......Page 1
ISBN 9789380168111......Page 2
Table of Contents......Page 3
What is calculus?......Page 4
What you should know before using this text......Page 5
Classical understanding of functions......Page 7
Modern understanding of functions......Page 8
The vertical line test......Page 9
Important functions......Page 10
Manipulating functions......Page 11
Domain and Range......Page 16
Graphing Functions......Page 20
Rules of arithmetic and algebra......Page 21
Exercises......Page 22
Intuitive Look......Page 24
Limit rules......Page 26
Examples......Page 27
The Squeeze Theorem......Page 31
Finding limits......Page 32
Using limit notation to describe asymptotes......Page 34
Continuity......Page 36
Removable Discontinuities......Page 37
Intermediate value theorem (IVT)......Page 38
On a Line......Page 40
On a Function......Page 41
The Slope at a Point......Page 44
An example......Page 45
Understanding the Derivative Notation......Page 47
Derivative of a Constant Function......Page 48
Constant multiple and addition rules......Page 49
The Addition and Subtraction Rules......Page 50
The Power Rule......Page 51
Derivatives of polynomials......Page 52
Extrema and Points of Inflexion......Page 53
The Extremum Test......Page 55
Chain Rule......Page 56
Product and Quotient Rules......Page 57
Proof......Page 58
Application, proof of the power rule......Page 59
Quotient rule......Page 60
Implicit Differentiation......Page 61
Examples......Page 62
Exponential......Page 63
Logarithms......Page 64
Trigonometric Functions......Page 66
Inverse Trigonometric Functions......Page 70
Exercises......Page 71
Applications of Derivatives Newton's Method......Page 72
Notes......Page 73
Filling Tank......Page 74
Examples......Page 75
Kinematics......Page 80
Instantaneous Velocity and Acceleration......Page 81
Optimization......Page 82
Examples......Page 83
Definition of the Integral......Page 86
Left and Right Handed Riemann Sums......Page 91
Example 1......Page 93
Example 2......Page 94
The Constant Rule......Page 95
The Comparison Rule......Page 97
Linearity with respect to endpoints......Page 99
Statement of the Fundamental Theorem......Page 100
Proofs......Page 101
Integration of Polynomials......Page 103
Indefinite Integrals......Page 105
Integral of the Inverse function......Page 106
Integral of the Exponential function......Page 107
Integration Rules......Page 108
The Substitution Rule......Page 109
Proof of the substitution rule......Page 111
Integration of even and odd functions......Page 112
Integration by Parts......Page 113
Example 1......Page 117
Example 2......Page 118
Recognizing Derivatives and Reversing Derivative Rules......Page 119
Integrating with the derivative present......Page 120
Integration by Parts......Page 122
Example......Page 123
Partial Fraction Decomposition......Page 126
Integration by Complexifying......Page 124
Trigonometric Substitution......Page 129
Sine substitution......Page 130
Tangent substitution......Page 131
Secant substitution......Page 132
Powers of Sine and Cosine......Page 133
Powers of Tan and Secant......Page 135
More trigonometric combinations......Page 137
Reduction Formula......Page 138
The "tan half angle" substitution......Page 139
Alternate Method......Page 141
Irrational Functions......Page 142
Riemann Sum......Page 144
Improper Integrals......Page 145
Area between two curves......Page 147
Revolution solids......Page 149
Arc length......Page 150
The Arclength Formula......Page 151
Arclength of a parametric curve......Page 152
The Surface Area Formula......Page 153
Work......Page 155
Infinite Series Series......Page 156
Convergence......Page 157
Absolute convergence......Page 158
Ratio test......Page 159
Limit Comparison......Page 160
Geometric series......Page 161
Taylor Series......Page 162
List of Taylor series......Page 165
Constructing a Taylor Series......Page 166
Motivations......Page 168
Radius of convergence......Page 169
Abstraction......Page 170
Differentiation and Integration......Page 171
Introduction......Page 173
Component Form......Page 175
Magnitude......Page 176
Direction......Page 177
Vector Operations......Page 178
Applications of Scalar Multiplication and Dot Product......Page 181
Polar coordinates......Page 185
Basic definition......Page 186
Cylindrical coordinates......Page 188
Spherical coordinates......Page 189
Triple Products......Page 190
Three-Dimensional Lines and Planes......Page 191
Vector-Valued Functions......Page 192
Limits, Derivatives, and Integrals......Page 193
Velocity, Acceleration, Curvature, and a brief mention of the Binormal......Page 195
Lengths and distances......Page 197
Boundary points......Page 198
Collision and intersection points......Page 199
Tangent vectors......Page 200
Surfaces......Page 202
Rules......Page 203
Special note about limits......Page 204
Differentiable functions......Page 205
Properties......Page 206
Alternate notations......Page 207
Directional derivatives......Page 208
Gradient vectors......Page 209
Divergence......Page 210
Curl......Page 211
Product and chain rules......Page 212
Second order differentials......Page 213
Riemann sums......Page 214
Order of integration......Page 215
Line integrals......Page 216
Surface and Volume Integrals......Page 220
Gauss's divergence theorem......Page 221
Stokes' curl theorem......Page 222
Some simple differential equations......Page 225
Basic first order Differential Equations......Page 226
Separable equations......Page 227
Homogeneous equations......Page 228
Linear equations......Page 230
Exact equations......Page 232
Reducible ODE's......Page 233
Linear ODEs......Page 236
Special cases......Page 241
Quasilinear......Page 244
Fully non-linear PDEs......Page 254
Second order......Page 256
Parabolic......Page 259
First order systems......Page 263
Formal limits......Page 265
Examples......Page 267
Fields......Page 269
Constructing the Real Numbers......Page 270
Properties of Real Numbers......Page 271
Theory of Sequences......Page 272
Types and properties of sequences......Page 273
Summation notation......Page 274
Common summations......Page 275
Basic Trigonometric Functions......Page 276
Inverse Trigonometric Functions......Page 277
General Rules......Page 278
Exponential and Logarithmic Functions......Page 279
Hyperbolic and Inverse Hyperbolic Functions......Page 280
Double Angle Identities......Page 281
Product-to-sum identities......Page 282