First Year Calculus (For Students of Mathematics and Related Disciplines)
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Michael M.Dougherty
calculus
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First Year Calculus (For Students of Mathematics and Related Disciplines)
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1. Mathematical Logic and Sets
Theory
Page 21
1.1 Logic Symbols and Truth Tables
Theory
Page 22
1.1.1 Lexicographical Listings of Possible Truth Values
Theory
Page 23
1.1.2 The Logic Operations
Theory
Page 24
1.1.3 Constructing Further Truth Tables
Theory
Page 29
1.1.4 Tautologies and Contradictions, A First Look
Theory
Page 31
1.2 Valid Logical Equivalences as Tautologies
Theory
Page 35
1.2.1 The Idea, and Definition, of Logical Equivalence
Theory
Page 35
1.2.2 Equivalences for Negations
Theory
Page 37
1.2.3 Equivalent Forms of the Implication
Theory
Page 38
1.2.4 Other Valid Equivalences
Theory
Page 39
1.2.5 Circuits and Logic
Theory
Page 43
1.2.6 The Statements $T$ and $F$
Theory
Page 46
1.3 Valid Implications and Arguments
Theory
Page 49
1.3.1 Valid Implications Defined
Theory
Page 49
1.3.2 Partial List of Valid Implications
Theory
Page 51
1.3.3 Fallacies and Valid Arguments
Theory
Page 52
1.3.4 Analyzing Arguments Without Truth Tables
Theory
Page 59
1.4 Quantifiers and Sets
Theory
Page 65
1.4.1 Sets
Theory
Page 65
1.4.2 Quantifiers
Theory
Page 67
1.4.3 Statements with Multiple Quantifiers
Theory
Page 67
1.4.4 Detour: Uniqueness as an Independent Concept
Theory
Page 69
1.4.5 Negating Universally and Existentially Quantified Statements
Theory
Page 70
1.4.6 Negating Statements Containing Mixed Quantifiers
Theory
Page 71
1.5 Sets Proper
Theory
Page 74
1.5.1 Subsets and Set Equality
Theory
Page 74
1.5.2 Intervals and Inequalities in $\mathbb{R}$
Theory
Page 77
1.5.3 Most General Venn Diagrams
Theory
Page 79
1.5.4 Set Operations
Theory
Page 80
1.5.5 More on Subsets
Theory
Page 84
1.6 Epilogue: Proofs Without Truth Tables
Theory
Page 89
1.7 Logic Epilogue
Theory
Page 93
2. Real Numbers, Algebra and Functions (Optional)
Theory
Page 94
2.1 Real Numbers Defined
Theory
Page 96
2.1.1 Axioms
Theory
Page 98
2.2 Arithmetic with Real Numbers
Theory
Page 106
2.2.1 Subtraction is Addition of the Additive Inverse
Theory
Page 106
2.2.2 Division is Multiplication by the Reciprocal
Theory
Page 107
2.2.3 Addition & Subtraction with Multiplication & Division
Theory
Page 108
2.2.4 Integer Exponents
Theory
Page 110
2.2.5 Miscellaneous Manipulations
Theory
Page 112
2.2.6 Order of Operations
Theory
Page 113
2.2.7 Absolute Value, Size, and Distance on the Real Line
Theory
Page 115
2.2.8 Binomial Expansion
Theory
Page 119
2.3 Arithmetic of Units and Significant Digits
Theory
Page 121
2.3.1 Adding Units
Theory
Page 121
2.3.2 Multiplying Units: Simplest Cases
Theory
Page 122
2.3.3 Division of Units: Rates
Theory
Page 123
2.3.4 Conversions
Theory
Page 124
2.3.5 Tolerances
Theory
Page 126
2.3.6 Significant Digits
Theory
Page 127
2.4 Algebra
Theory
Page 131
2.4.1 Polynomial Equations: Using Theorem 2.1.8
Theory
Page 133
2.4.2 More with Radicals
Theory
Page 136
2.4.3 Completing the Square
Theory
Page 139
2.4.4 The Quadratic Formula
Theory
Page 140
2.5 Functions-I
Theory
Page 143
2.5.1 Various Definitions of Functions and Notation
Theory
Page 144
2.5.2 Functions as Processors of Inputs
Theory
Page 146
2.5.3 Graphs, Domain and Range
Theory
Page 146
2.5.4 Functions By Formulas
Theory
Page 147
2.6 Basic Functions of One Variable
Theory
Page 149
2.6.1 “Linear” Functions
Theory
Page 149
2.6.2 Simple Powers
Theory
Page 151
2.6.3 Roots
Theory
Page 153
2.6.4 Reciprocal Function
Theory
Page 155
2.6.5 Absolute Value and Piecewise Defined Functions
Theory
Page 156
2.6.6 Exponential Functions
Theory
Page 157
2.6.7 Logarithmic Functions
Theory
Page 160
2.6.8 Sine and Cosine Functions
Theory
Page 161
2.6.9 Arcsine and Arccosine Functions
Theory
Page 164
2.7 Functions-III: Simple Transformations of Functions
Theory
Page 166
2.7.1 Diagramming Functions
Theory
Page 166
2.7.2 Shift Transformations
Theory
Page 167
2.7.3 Reflection Transformations
Theory
Page 169
2.7.4 Rescaling: Contractions and Dilations
Theory
Page 171
2.7.5 Multiple Simple Transformations of Basic Functions
Theory
Page 172
2.7.6 Reflection Around Other Vertical Lines
Theory
Page 174
2.8 Functions-IV: Combinations of Functions
Theory
Page 178
2.8.1 Adding Functions
Theory
Page 179
2.8.2 Multiplying Functions
Theory
Page 182
2.8.3 Compositions of Functions
Theory
Page 187
3. Continuity and Limits of Functions
Theory
Page 189
3.1 Definition of Continuity at a Point
Theory
Page 190
3.2 Continuity Theorems
Theory
Page 201
3.2.1 Basic Theorems
Theory
Page 202
3.2.2 One-Sided Continuity
Theory
Page 207
3.2.3 Essential versus Removable Discontinuities
Theory
Page 211
3.3 Continuity on Intervals
Theory
Page 215
3.3.1 Continuity on Intervals: Main Definitions and Theorems
Theory
Page 215
3.3.2 Simple Applications of the Intermediate Value Theorem
Theory
Page 218
3.3.3 Polynomial Inequalities
Theory
Page 219
3.3.4 Rational Inequalities
Theory
Page 224
3.4 Finite Limits at Points
Theory
Page 229
3.4.1 Definition, Theorems and Examples
Theory
Page 230
3.4.2 Further Limit Notation
Theory
Page 238
3.4.3 Proofs of Limit Theorems
Theory
Page 239
3.5 One-Sided Finite Limits
Theory
Page 243
3.6 Infinite Limits at Points
Theory
Page 250
3.7 Sandwich, Composition, Trigonometric Continuity
Theory
Page 260
3.7.1 Sandwich Theorem
Theory
Page 260
3.7.2 “Approaches” for Independent Versus Dependent Variables
Theory
Page 263
3.7.3 “Sandwiching” From One Side
Theory
Page 263
3.7.4 Limits with Compositions of Functions
Theory
Page 265
3.7.5 Continuity Considerations for Trigonometric Functions
Theory
Page 267
3.7.6 Proofs
Theory
Page 269
3.8 Limits “At Infinity”
Theory
Page 270
3.9 Further Limit Theorems and Trigonometric Limits
Theory
Page 287
3.9.1 Simple Limit Theorems
Theory
Page 287
3.9.2 A Trigonometric Limit
Theory
Page 291
3.9.3 Limits by Substitution
Theory
Page 295
3.9.4 Epilogue
Theory
Page 300
3.10 Limit of Sequences: A First Look
Theory
Page 303
3.10.1 Definitions and First Examples
Theory
Page 303
3.10.2 Subsequences
Theory
Page 308
3.11 Sequences II
Theory
Page 312
4. The Derivative
Theory
Page 314
4.1 Derivative: Rate of Change, Velocity, Slope
Theory
Page 313
4.1.1 Slope of a Curve
Theory
Page 318
4.1.2 Marginal Cost
Theory
Page 322
4.1.3 Some Further Applications
Theory
Page 325
4.2 First Differentiation Rules and Leibniz Notation
Theory
Page 329
4.2.1 Positive Integer Power Rule
Theory
Page 329
4.2.2 Leibniz Notation
Theory
Page 330
4.2.3 Sum and Constant Derivative Rules
Theory
Page 331
4.2.4 Increasing and Decreasing Functions, Graphing Polynomials
Theory
Page 333
4.2.5 Derivatives of Sine and Cosine
Theory
Page 338
4.2.6 Absolute Values and Piece-wise Defined Functions
Theory
Page 342
4.3 First Applications of Derivatives
Theory
Page 346
4.3.1 Velocity: Vertical and Horizontal
Theory
Page 346
4.3.2 Fluid Flow
Theory
Page 348
4.3.3 Energy, Work and Power
Theory
Page 349
4.3.4 Electrical Current
Theory
Page 350
4.3.5 Ohm’s Law
Theory
Page 355
4.3.6 Electromagnetic Inductance
Theory
Page 357
4.4 Chain Rule I
Theory
Page 363
4.4.1 Chain Rule in Prime Notation
Theory
Page 363
4.4.2 Partial Proof of Chain Rule
Theory
Page 365
4.4.3 Leibniz Notation and the Chain Rule
Theory
Page 365
4.4.4 Chain Rule Derivatives of Sine and Cosine
Theory
Page 368
4.4.5 Power Rule for Rational Powers
Theory
Page 371
4.4.6 Proof Of Power Rule for Rational Powers
Theory
Page 373
4.4.7 Further Examples
Theory
Page 374
4.5 Product, Quotient and Other Trigonometric Rules
Theory
Page 380
4.5.1 Product Rule Stated and First Applied
Theory
Page 380
4.5.2 Product Rule Proof
Theory
Page 384
4.5.3 Quotient Rule
Theory
Page 385
4.5.4 Tangent, Cotangent, Secant and Cosecant Rules
Theory
Page 388
4.5.5 Putting Rules Together—Carefully
Theory
Page 390
4.6 Chain Rule II: Implicit Differentiation
Theory
Page 395
4.6.1 Review of Differentiation Rules
Theory
Page 395
4.6.2 Implicit Functions and Their Derivatives
Theory
Page 397
4.6.3 A Mistake to Avoid
Theory
Page 407
4.6.4 An Application
Theory
Page 407
4.7 Arctrigonometric Functions and their Derivatives
Theory
Page 410
4.7.1 One-to-one Functions and Inverses, Briefly
Theory
Page 410
4.7.2 The Arctrigonometric Functions and Their Derivatives
Theory
Page 411
4.7.3 The Arcsine Function and Its Derivative
Theory
Page 417
4.7.4 The Arccosine and Its Derivative
Theory
Page 419
4.7.5 The Arctangent Function and Its Derivative
Theory
Page 421
4.7.6 The Arcsecant Function and Its Derivative
Theory
Page 424
4.7.7 Geometric and Algebraic Arguments
Theory
Page 426
4.8 Exponential Functions
Theory
Page 432
4.8.1 Exponential Functions
Theory
Page 432
4.8.2 Derivative of a Special Exponential Function
Theory
Page 435
4.8.3 A Note on Differences Between Polynomials, Exponentials
Theory
Page 439
4.9 The Natural Logarithm I
Theory
Page 441
4.9.1 Algebra of Logarithms
Theory
Page 441
4.9.2 Graph of the Natural Logarithm Function
Theory
Page 444
4.9.3 Derivative of the Natural Logarithm
Theory
Page 445
4.10 The Natural Logarithm II: Further Results
Theory
Page 452
4.10.1 Logarithmic Differentiation
Theory
Page 452
4.10.2 Bases Other Than $e$: Logarithms
Theory
Page 456
4.10.3 Bases Other than $e$: Exponential Functions
Theory
Page 457
4.10.4 Derivative of $f(x)^{g(x)}$
Theory
Page 459
4.11 Further Interpretations of the Derivative
Theory
Page 464
4.11.1 Velocity and Acceleration
Theory
Page 464
4.11.2 Power
Theory
Page 464
4.11.3 Fluid Flow and Volume
Theory
Page 464
4.11.4 Population Growth and Decay
Theory
Page 464
4.11.5 Consumption
Theory
Page 466
4.11.6 Marginal Cost and Profit
Theory
Page 466
5. Using Derivatives to Analyze Functions and Applications
Theory
Page 467
5.1 Higher-Order Derivatives and Graphing
Theory
Page 468
5.1.1 Higher-Order Derivatives (Derivatives of Derivatives)
Theory
Page 468
5.1.2 Graphical Significance of the Second Derivative
Theory
Page 469
5.2 Extrema on Closed Intervals
Theory
Page 489
5.2.1 Argument for Main Theorem
Theory
Page 489
5.2.2 Main Theorem and Examples
Theory
Page 492
5.2.3 Applications (Max/Min Problems)
Theory
Page 496
5.3 The Mean Value Theorem
Theory
Page 499
5.3.1 Linear Interpolation and Average Rate of Change
Theory
Page 499
5.3.2 Mean Value Theorem
Theory
Page 500
5.3.3 Applications to Graphing Theorems
Theory
Page 501
5.3.4 Numerical Applications
Theory
Page 503
5.4 Differentials and the Linear Approximation Method
Theory
Page 505
5.4.1 Some Approximation Problems
Theory
Page 505
5.4.2 Linear Approximation
Theory
Page 507
5.4.3 Linear Approximations and Implicit Functions
Theory
Page 511
5.4.4 Differentials
Theory
Page 513
5.5 Newton’s Approximation Method
Theory
Page 516
5.5.1 Approximating Methods Applied to an Example
Theory
Page 516
5.6 Chain Rule III: Related Rates
Theory
Page 524
5.6.1 The Basic Idea and an Example
Theory
Page 524
5.6.2 Choosing the Form of the Relationship
Theory
Page 525
5.6.3 Examples: Finding a Relationship Among Variables
Theory
Page 529
5.6.4 Further Examples
Theory
Page 531
5.6.5 A Modeling Example
Theory
Page 532
6. Basic Integration
Theory
Page 535
6.1 First Indefinite Integrals (Antiderivatives)
Theory
Page 536
6.1.1 Indefinite Integrals and Constants of Integration
Theory
Page 536
6.1.2 Power Rule for Integrals
Theory
Page 538
6.1.3 Finding $C$ (Where Possible)
Theory
Page 541
6.1.4 First Trigonometric Rules
Theory
Page 544
6.1.5 Integrals Yielding Inverse Trigonometric Functions
Theory
Page 546
6.1.6 Integrals Yielding Exponential Functions
Theory
Page 548
6.2 Summation (Sigma) Notation
Theory
Page 550
6.3 Riemann Sums and the Fundamental Theorem of Calculus
Theory
Page 556
6.3.1 Absolute, Relative and Percent Errors
Theory
Page 557
6.3.2 A Physics Example
Theory
Page 558
6.3.3 General Riemann Sums, FTC
Theory
Page 559
6.3.4 Computing Areas: Geometric Interpretation of FTC
Theory
Page 562
6.3.5 Physics Application: Net Displacement versus Distance Traveled
Theory
Page 566
6.3.6 Infinitesimals
Theory
Page 566
6.4 Substitution With Power Rule
Theory
Page 569
6.4.1 The Technique
Theory
Page 570
6.4.2 A Slight Twist on the Method
Theory
Page 575
6.4.3 Other Miscellaneous Power Rule Substitutions
Theory
Page 577
6.5 Second Trigonometric Rules
Theory
Page 580
6.5.1 Antiderivatives of the Six Trigonometric Functions
Theory
Page 580
6.5.2 Substitution and the Basic Trigonometric Functions, Part I
Theory
Page 581
6.6 Substitution with All Basic Forms
Theory
Page 588
6.6.1 List of Basic Forms
Theory
Page 589
6.7 Further Arctrigonometric Forms
Theory
Page 595
6.7.1 Factoring to Achieve “1”
Theory
Page 595
6.7.2 Completing the Square
Theory
Page 598
6.8 Hyperbolic Functions
Theory
Page 600
6.8.1 Hyperbolic Functions and Their Basic Intrinsic Structures
Theory
Page 600
7. Advanced Integration Techniques
Theory
Page 602
7.1 Substitution-Type Integration by Inspection
Theory
Page 602
7.1.1 The Method
Theory
Page 603
7.1.2 Limitations of the Method
Theory
Page 606
7.2 Integration By Parts
Theory
Page 610
7.2.1 The Idea by Examples
Theory
Page 610
7.2.2 The Technique in Its Simpler Applications
Theory
Page 611
7.2.3 Repeated Use of Integration by Parts
Theory
Page 615
7.2.4 Integrals of Certain Other Functions
Theory
Page 616
7.2.5 An Indirect Method
Theory
Page 617
7.2.6 Miscellaneous Considerations
Theory
Page 620
7.3 Trigonometric Integrals
Theory
Page 627
7.3.1 Sample Problems
Theory
Page 627
7.3.2 Odd Powers of Sine or Cosine
Theory
Page 628
7.3.3 Even Powers of Secant or Odd Powers of Tangent
Theory
Page 633
7.3.4 Even Powers of Secant or Odd Powers of Cotangent
Theory
Page 635
7.3.5 Even Powers of Sine and Cosine
Theory
Page 637
7.3.6 Miscellaneous Problems and Methods-I
Theory
Page 640
7.3.7 Miscellaneous Problems and Methods-II
Theory
Page 644
7.4 Trigonometric Substitution
Theory
Page 650
7.4.1 Introduction
Theory
Page 650
7.4.2 The General Approach
Theory
Page 655
7.5 Partial Fractions and Integration
Theory
Page 661
7.5.1 Theory Behind the Forms of PFD’s (Optional)
Theory
Page 661
7.5.2 Partial Fraction Decompositions: The Rules
Theory
Page 665
7.5.3 Finding the Coefficients of PFD’s
Theory
Page 666
7.6 Miscellaneous Methods
Theory
Page 677
8. Applications of Definite Integrals I: General Arguments
Theory
Page 678
8.1 Riemann Sums and Approximations of Cumulative Quantities
Theory
Page 678
8.2 Other Complications
Theory
Page 679
8.3 Approximation Methods for Definite Integrals
Theory
Page 679
8.4 Physical Applications of Definite Integrals
Theory
Page 679
8.4.1 Geometric Applications
Theory
Page 679
9. Improper Integrals and Advanced Limit Techniques
Theory
Page 681
9.1 Some Asymptotics of Functions
Theory
Page 681
9.2 L’Hôpital’s Rule
Theory
Page 686
9.3 Other Indeterminate Forms
Theory
Page 695
9.3.1 $\infty-\infty$, $0\cdot\infty$
Theory
Page 695
9.3.2 $1^\infty$, $0^0$, $\infty^0$ and Similar Forms
Theory
Page 697
9.4 Improper Integrals
Theory
Page 700
9.4.1 First Examples
Theory
Page 700
9.4.2 The Background and Method
Theory
Page 701
9.4.3 Infinite Areas do not Cancel
Theory
Page 704
9.4.4 More Complicated Integrals
Theory
Page 706
9.4.5 A Complication With Oscillatory and Other Integrals
Theory
Page 709
10. Series of Constants
Theory
Page 711
10.1 Series and Partial Sums
Theory
Page 713
10.1.1 Telescoping Series
Theory
Page 714
10.1.2 Geometric Series
Theory
Page 717
10.1.3 Convergence/Divergence in Geometric Series
Theory
Page 720
10.2 NTTFD and Integral Test
Theory
Page 723
10.2.1 First Divergence Test: NTTFD
Theory
Page 725
10.2.2 Integral Test Proper (IT)
Theory
Page 727
10.3 Comparison Tests
Theory
Page 732
10.3.1 Direct Comparison Test (DCT)
Theory
Page 733
10.3.2 A Hierarchy of Functions and DCT
Theory
Page 735
10.3.3 Limit Comparison Test (LCT)
Theory
Page 737
10.3.4 Summary
Theory
Page 741
10.3.5 Elementary Series Theorems
Theory
Page 742
10.4 Alternating Series and Absolute Convergence
Theory
Page 744
10.4.1 Alternating Series Test (AST)
Theory
Page 744
10.4.2 Absolutely and Conditionally Convergent Series
Theory
Page 747
10.4.3 One Last Remark Concerning Absolute Convergence
Theory
Page 749
10.5 Ratio and Root Tests
Theory
Page 752
11. Taylor Series
Theory
Page 759
11.1 Taylor Polynomials: Examples and Derivation
Theory
Page 760
11.2 Accuracy of $P_N(x)$
Theory
Page 781
11.3 Taylor/Maclaurin Series
Theory
Page 788
11.3.1 Checking Validity of Taylor Series
Theory
Page 788
11.3.2 Techniques for Writing Series using $\Sigma$-Notation
Theory
Page 791
11.4 Power Series, Interval of Convergence
Theory
Page 794
11.4.1 Definition of General Power Series
Theory
Page 794
11.4.2 Abel’s Theorem
Theory
Page 794
11.4.3 Finding the Interval and Radius of Convergence
Theory
Page 795
11.4.4 Taylor/Power Series Connection
Theory
Page 797
11.5 Valid Manipulations with Taylor/Power Series
Theory
Page 800
11.5.1 Algebraic Manipulations
Theory
Page 800
11.5.2 Derivatives and Integrals
Theory
Page 802
11.5.3 The Binomial Series and an Application
Theory
Page 805
11.6 Complications and the Role of Complex Numbers
Theory
Page 810
11.6.1 Troubles Stemming from Continuity Problems
Theory
Page 810
11.6.2 The Complex Plane
Theory
Page 813
11.6.3 The Complex Plane’s Role
Theory
Page 817
11.6.4 Summary
Theory
Page 819
12. Topics in Two-Dimensional Analytic Geometry
Theory
Page 821
12.1 Conic Sections
Theory
Page 822
12.1.1 Conic Sections Defined Geometrically
Theory
Page 822
12.1.2 Simplest Equations of Conic Sections
Theory
Page 823
12.1.3 Equations of Conic Sections from Definitions
Theory
Page 832
12.1.4 Eccentricity
Theory
Page 837
12.2 Parametric Curves
Theory
Page 838
12.3 Polar Coordinates
Theory
Page 838
12.4 Calculus in Polar Coordinates
Theory
Page 838
12.5 Vectors in $\mathbb{R}^2$ and Scalar (Dot) Products
Theory
Page 839
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