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Introduction to Calculus (Volume-1)

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English
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J.H.Heinbockel
calculus
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Introduction to Calculus (Volume-1)
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Entries
221
1. Sets, Functions, Graphs and Limits
Theory
Page 9
1.1 Elementary Set Theory
Theory
Page 9
1.2 Subsets
Theory
Page 11
1.3 Set Operations
Theory
Page 13
1.4 Coordinate Systems
Theory
Page 13
1.5 Distance Between Two Points in the Plane
Theory
Page 15
1.6 Graphs and Functions
Theory
Page 16
1.7 Increasing and Decreasing Functions
Theory
Page 20
1.8 Linear Dependence and Independence
Theory
Page 21
1.9 Single-Valued Functions
Theory
Page 22
1.10 Parametric Representation of Curve
Theory
Page 25
1.11 Equation of Circle
Theory
Page 26
1.12 Types of Functions
Theory
Page 28
1.13 The Exponential and Logarithmic Functions
Theory
Page 29
1.14 The Trigonometric Functions
Theory
Page 32
1.15 Graphs of Trigonometric Functions
Theory
Page 32
1.16 The Hyperbolic Functions
Theory
Page 33
1.17 Symmetry of Functions
Theory
Page 34
1.18 Translation and Scaling of Axes
Theory
Page 36
1.19 Inverse Functions
Theory
Page 39
1.20 Equations of Lines
Theory
Page 44
1.21 Perpendicular Lines
Theory
Page 47
1.22 Limits
Theory
Page 48
1.23 Infinitesimals
Theory
Page 49
1.24 Limiting Value of a Function
Theory
Page 51
1.25 Formal Definition of a Limit
Theory
Page 54
1.26 Special Considerations
Theory
Page 57
1.27 Properties of Limits
Theory
Page 58
1.28 The Squeeze Theorem
Theory
Page 61
1.29 Continuous Functions and Discontinuous Functions
Theory
Page 62
1.30 Asymptotic Lines
Theory
Page 63
1.31 Finding Asymptotic Lines
Theory
Page 64
1.32 Conic Sections
Theory
Page 65
1.33 Circle
Theory
Page 67
1.34 Parabola
Theory
Page 68
1.35 Ellipse
Theory
Page 71
1.36 Hyperbola
Theory
Page 74
1.37 Conic Sections in Polar Coordinates
Theory
Page 78
1.38 Rotation of Axes
Theory
Page 78
1.39 General Equation of the Second Degree
Theory
Page 79
1.40 Computer Languages
Theory
Page 81
2. Differential Calculus
Theory
Page 93
2.1 Slope of Tangent Line to Curve
Theory
Page 93
2.2 The Derivative of $y=f(x)$
Theory
Page 95
2.3 Right and Left-Hand Derivatives
Theory
Page 97
2.4 Alternative Notations for the Derivative
Theory
Page 98
2.5 Higher Derivatives
Theory
Page 98
2.6 Rules and Properties
Theory
Page 99
2.7 Differentiation of a Composite Function
Theory
Page 106
2.8 Differentials
Theory
Page 109
2.9 Differentiation of Implicit Functions
Theory
Page 110
2.10 Importance of Tangent Line and Derivative Function $f'(x)$
Theory
Page 114
2.11 Rolle's Theorem
Theory
Page 116
2.12 The Mean-Value Theorem
Theory
Page 116
2.13 Cauchy's Generalized Mean-Value Theorem
Theory
Page 119
2.14 Derivative of the Logarithm Function
Theory
Page 119
2.15 Derivative of the Exponential Function
Theory
Page 121
2.16 Derivative and Continuity
Theory
Page 124
2.17 Maxima and Minima
Theory
Page 124
2.18 Concavity of Curve
Theory
Page 126
2.19 Comments on Local Maxima and Minima
Theory
Page 127
2.20 First Derivative Test
Theory
Page 128
2.21 Second Derivative Test
Theory
Page 128
2.22 Logarithmic Differentiation
Theory
Page 135
2.23 Differentiation of Inverse Functions
Theory
Page 136
2.24 Differentiation of Parametric Equations
Theory
Page 137
2.25 Differentiation of the Trigonometric Functions
Theory
Page 138
2.26 Simple Harmonic Motion
Theory
Page 144
2.27 L'Hopital's Rule
Theory
Page 146
2.28 Differentiation of Inverse Trigonometric Functions
Theory
Page 148
2.29 Hyperbolic Functions and Their Derivatives
Theory
Page 150
2.30 Approximations
Theory
Page 152
2.31 Hyperbolic Identities
Theory
Page 153
2.32 Euler's Formula
Theory
Page 155
2.33 Derivatives of the Hyperbolic Functions
Theory
Page 157
2.34 Inverse Hyperbolic Functions and Their Derivatives
Theory
Page 157
2.35 Relations Between Inverse Hyperbolic Functions
Theory
Page 161
2.36 Derivatives of the Inverse Hyperbolic Functions
Theory
Page 162
2.37 Table of Derivatives
Theory
Page 164
2.38 Table of Differentials
Theory
Page 165
2.39 Partial Derivatives
Theory
Page 166
2.40 Total Differential
Theory
Page 167
2.41 Notation
Theory
Page 168
2.42 Differential Operator
Theory
Page 168
2.43 Maxima and Minima for Functions of Two Variables
Theory
Page 169
2.44 Implicit Differentiation
Theory
Page 170
3. Integral Calculus
Theory
Page 183
3.1 Summations
Theory
Page 183
3.2 Special Sums
Theory
Page 184
3.3 Integration
Theory
Page 187
3.4 Properties of the Integral Operator
Theory
Page 189
3.5 Notation
Theory
Page 190
3.6 Integration of Derivatives
Theory
Page 191
3.7 Polynomials
Theory
Page 191
3.8 General Considerations
Theory
Page 192
3.9 Table of Integrals
Theory
Page 194
3.10 Trigonometric Substitutions
Theory
Page 197
3.11 Products of Sines and Cosines
Theory
Page 201
3.12 Special Trigonometric Integrals
Theory
Page 202
3.13 Method of Partial Fractions
Theory
Page 203
3.14 Sums and Differences of Squares
Theory
Page 210
3.15 Summary of Integrals
Theory
Page 216
3.16 Reduction Formula
Theory
Page 219
3.17 The Definite Integral
Theory
Page 221
3.18 Fundamental Theorem of Integral Calculus
Theory
Page 225
3.19 Properties of the Definite Integral
Theory
Page 226
3.20 Solids of Revolution
Theory
Page 233
3.21 Slicing Method
Theory
Page 233
3.22 Integration by Parts
Theory
Page 240
3.23 Physical Interpretation
Theory
Page 241
3.24 Improper Integrals
Theory
Page 242
3.25 Integrals Used to Define Functions
Theory
Page 244
3.26 Arc Length
Theory
Page 246
3.27 Area in Polar Coordinates
Theory
Page 248
3.28 Arc Length in Polar Coordinates
Theory
Page 249
3.29 Surface of Revolution
Theory
Page 250
3.30 Mean Value Theorems for Integrals
Theory
Page 253
3.31 Proof of Mean Value Theorems
Theory
Page 254
3.32 Differentiation of Integrals
Theory
Page 255
3.33 Double Integrals
Theory
Page 257
3.34 Summations over Nonrectangular Regions
Theory
Page 260
3.35 Polar Coordinates
Theory
Page 263
3.36 Cylindrical Coordinates
Theory
Page 264
3.37 Spherical Coordinates
Theory
Page 265
3.38 Using Table of Integrals
Theory
Page 266
3.39 The Bliss Theorem
Theory
Page 266
4. Sequences, Summations and Products
Theory
Page 279
4.1 Sequences
Theory
Page 279
4.2 Limit of a Sequence
Theory
Page 280
4.3 Convergence of a Sequence
Theory
Page 280
4.4 Divergence of a Sequence
Theory
Page 281
4.5 Relation Between Sequences and Functions
Theory
Page 282
4.6 Establish Bounds for Sequences
Theory
Page 283
4.7 Additional Terminology Associated with Sequences
Theory
Page 285
4.8 Stolz–Cesàro Theorem
Theory
Page 287
4.9 Examples of Sequences
Theory
Page 289
4.10 Infinite Series
Theory
Page 289
4.11 Sequence of Partial Sums
Theory
Page 290
4.12 Convergence and Divergence of a Series
Theory
Page 291
4.13 Comparison of Two Series
Theory
Page 294
4.14 Test for Divergence
Theory
Page 294
4.15 Cauchy Convergence
Theory
Page 295
4.16 The Integral Test for Convergence
Theory
Page 296
4.17 Alternating Series Test
Theory
Page 301
4.18 Bracketing Terms of a Convergent Series
Theory
Page 303
4.19 Comparison Tests
Theory
Page 304
4.20 Ratio Comparison Test
Theory
Page 305
4.21 Absolute Convergence
Theory
Page 307
4.22 Slowly Converging or Slowly Diverging Series
Theory
Page 309
4.23 Certain Limits
Theory
Page 312
4.24 Power Series
Theory
Page 313
4.25 Operations with Power Series
Theory
Page 314
4.26 Maclaurin Series
Theory
Page 315
4.27 Taylor and Maclaurin Series
Theory
Page 319
4.28 Taylor Series for Functions of Two Variables
Theory
Page 323
4.29 Alternative Derivation of the Taylor Series
Theory
Page 326
4.30 Remainder Term for Taylor Series
Theory
Page 327
4.31 Schlömilch and Roche Remainder Term
Theory
Page 328
4.32 Indeterminate Forms
Theory
Page 330
4.33 Modification of a Series
Theory
Page 332
4.34 Conditional Convergence
Theory
Page 333
4.35 Algebraic Operations with Series
Theory
Page 333
4.36 Bernoulli Numbers
Theory
Page 336
4.37 Euler Numbers
Theory
Page 336
4.38 Functions Defined by Series
Theory
Page 338
4.39 Generating Functions
Theory
Page 339
4.40 Functions Defined by Products
Theory
Page 339
4.41 Continued Fractions
Theory
Page 340
4.42 Terminology
Theory
Page 341
4.43 Evaluation of Continued Fractions
Theory
Page 342
4.44 Convergent Continued Fraction
Theory
Page 344
4.45 Regular Continued Fractions
Theory
Page 344
4.46 Euler's Theorem for Continued Fractions
Theory
Page 345
4.47 Gauss Representation for the Hypergeometric Function
Theory
Page 345
4.48 Representation of Functions
Theory
Page 345
4.49 Fourier Series
Theory
Page 347
4.50 Properties of the Fourier Trigonometric Series
Theory
Page 352
4.51 Fourier Series of Odd Functions
Theory
Page 353
4.52 Fourier Series of Even Functions
Theory
Page 354
4.53 Options
Theory
Page 354
5. Applications of Calculus
Theory
Page 371
5.1 Related Rates
Theory
Page 371
5.2 Newton's Laws
Theory
Page 374
5.3 Newton's Law of Gravitation
Theory
Page 376
5.4 Work
Theory
Page 379
5.5 Energy
Theory
Page 380
5.6 First Moments and Center of Gravity
Theory
Page 382
5.7 Centroid and Center of Mass
Theory
Page 383
5.8 Centroid of an Area
Theory
Page 385
5.9 Symmetry
Theory
Page 388
5.10 Centroids of Composite Shapes
Theory
Page 391
5.11 Centroid for Curve
Theory
Page 392
5.12 Higher Order Moments
Theory
Page 393
5.13 Moment of Inertia of an Area
Theory
Page 398
5.14 Moment of Inertia of a Solid
Theory
Page 400
5.15 Moment of Inertia of Composite Shapes
Theory
Page 401
5.16 Pressure
Theory
Page 401
5.17 Chemical Kinetics
Theory
Page 403
5.18 Rates of Reactions
Theory
Page 403
5.19 The Law of Mass Action
Theory
Page 404
5.20 Differential Equations
Theory
Page 407
5.21 Spring-Mass System
Theory
Page 409
5.22 Simple Harmonic Motion
Theory
Page 411
5.23 Damping Forces
Theory
Page 413
5.24 Mechanical Resonance
Theory
Page 420
5.25 Particular Solution
Theory
Page 422
5.26 Torsional Vibrations
Theory
Page 425
5.27 The Simple Pendulum
Theory
Page 426
5.28 Electrical Circuits
Theory
Page 426
5.29 Thermodynamics
Theory
Page 432
5.30 Radioactive Decay
Theory
Page 434
5.31 Economics
Theory
Page 437
5.32 Population Models
Theory
Page 439
5.33 Approximations
Theory
Page 441
5.34 Partial Differential Equations
Theory
Page 442
5.35 Easy to Solve Partial Differential Equations
Theory
Page 443
A. Units of Measurement
Theory
Page 460
B. Background Material
Theory
Page 462
C. Table of Integrals
Theory
Page 474
D. Solutions to Selected Problems
Theory
Page 528
6. Index
Theory
Page 560
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