Table of Contents for Calculus, First Edition
Preface
Lesson A The real numbers * Fundamental concept review
Lesson B More concept review * Geometry review
Lesson 1 Deductive reasoning * The contrapositive * Converse and inverse
Lesson 2 Radian measure of angles * Trigonometric ratios * Four quadrant signs * Simplifying trigonometric expressions
Lesson 3 Word problem review
Lesson 4 Functions: Their equations and graphs * Functional notation * Domain and range
Lesson 5 The unit circle * Graphing sinusoids
Lesson 6 Similar triangles * Functions of theta
Lesson 7 Quadratic equations
Lesson 8 Pythagorean identities * Trigonometric identities Cofunctions
Lesson 9 Abstract word problems
Lesson 10 Important numbers * Exponential functions
Lesson 11 Polar coordinates (vectors) * Polar coordinates (complex numbers)
Lesson 12 Absolute value as a distance * The line as a locus * The circle as a locus
Lesson 13 Special functions
Lesson 14 The logarithmic form of the exponential * Base 10 and base e * Simple logarithm problems
Lesson 15 Evaluating polynomials
Lesson 16 Continuity * Left-hand and right-hand limits
Lesson 17 Sum and difference identities for trigonometric functions * Double-angle identities for sine and cosines
Lesson 18 Graphs of logarithmic functions * Period of a function
Lesson 19 Limit of a function
Lesson 20 The parabola as a locus * Translated parabolas
Lesson 21 Inverse trigonometric functions * Trigonometric equations
Lesson 22 Interval notation * Products of linear factors * Tangents * Increasing and decreasing functions
Lesson 23 Logarithms of products and quotients * Logarithms of powers * Exponential equations
Lesson 24 Infinity as a limit * Undefined limits
Lesson 25 Sums, products, and quotients of functions * Composition of functions
Lesson 26 Locus development * Equation of the ellipse * Foci
Lesson 27 The derivative
Lesson 28 Change of base * Logarithmic inequalities
Lesson 29 Translation of functions * Rational functions I
Lesson 30 The hyperbola
Lesson 31 Binomial expansion * Recognizing the equations of conic sections
Lesson 32 Roots of complex numbers * Trigonometric functions of theta
Lesson 33 The derivative of xn * Notations for the derivative
Lesson 34 Identities for the tangent function * Area and volume
Lesson 35 The constant-multiple rule * The derivatives of sums and differences
Lesson 36 Exponential growth and decay
Lesson 37 Derivative of ex and ln |x| * Derivative of sin x and cos x
Lesson 38 Equation of the tangent line * Higher-order derivatives
Lesson 39 Graphs of rational functions II * A special limit Lesson 40 Newton and Leibniz * The differential
Lesson 41 Graph of tan theta * Graphs of reciprocal functions
Lesson 42 Product rule for derivatives and differentials * Proof of the product rule
Lesson 43 An antiderivative * Integration
Lesson 44 Factors of polynomial functions * Graphs of polynomial functions
Lesson 45 Implicit differentiation
Lesson 46 The integral of a constant * Integral of Cf(x)* Integral of xn
Lesson 47 Critical numbers
Lesson 48 Differentiation by usubstitution
Lesson 49 Integral of a sum * Integral of 1/x
Lesson 50 Units for the derivative * Normal lines
Lesson 51 Graphs of rational functions III * Repeated factors
Lesson 52 The derivative of a quotient * Proof of the quotient rule
Lesson 53 Area under a curve
Lesson 54 The chain rule * Equivalent forms for the derivative
Lesson 55 Using f' to characterize f * Using f' to define max and min
Lesson 56 Related rate problems
Lesson 57 Fundamental theorem of integral calculus
Lesson 58 Derivatives of trigonometric functions * Summary of rules for derivatives and differentials
Lesson 59 Concavity and inflection points * Application of the second derivative
Lesson 60 Derivatives of composite functions * Derivatives of products and quotients
Lesson 61 Integration by guessing
Lesson 62 Maximization and minimization problems
Lesson 63 Riemann sum * The definite integral
Lesson 64 Velocity and acceleration (motion I) * Motion due to gravity
Lesson 65 More integration by guessing
Lesson 66 Properties of the definite integral
Lesson 67 Explicit and implicit equations * Inverse functions
Lesson 68 Computing areas
Lesson 69 Area between two curves
Lesson 70 Game playing with f, f', and f''
Lesson 71 Applications of the definite integral I
Lesson 72 Critical numbers (closed interval) theorem
Lesson 73 Derivatives of inverse trigonometric functions * What to memorize
Lesson 74 Falling body problems
Lesson 75 Usubstitution * Change of variables * Proof of the substitution theorem
Lesson 76 Functions of y
Lesson 77 Even and odd functions
Lesson 78 Integration by parts
Lesson 79 Properties of limits * Some special limits
Lesson 80 Solids of revolution
Lesson 81 Derivatives and integrals of a and logq x * Derivative of |x|
Lesson 82 Fluid force
Lesson 83 Continuity of functions
Lesson 84 Integration of odd powers of sin x and cos x
Lesson 85 Applications of the definite integral (work II)
Lesson 86 Particle motion III
Lesson 87 L'Hopital's rule * Proof of L'Hopital's rule
Lesson 88 Asymptotes of rational functions
Lesson 89 Balance points
Lesson 90 Volume by washers
Lesson 91 Limits and continuity * Differentiability
Lesson 92 Integration of even powers of sin xand cos x
Lesson 93 Centroids
Lesson 94 Logarithmic differentiation
Lesson 95 The mean value theorem * Application of the mean value theorem * Proof of Rolle's theorem
Lesson 96 Rules for even and odd functions
Lesson 97 Volume by shells
Lesson 98 Separable differential equations
Lesson 99 Average value of a function * Mean value theorem for integrals
Lesson 100 Particle motion IV
Lesson 101 Derivatives of inverse functions
Lesson 102 Solids of revolution IV
Lesson 103 Absolute value
Lesson 104 Integral of tannx* Integral of cotnx
Lesson 105 Second fundamental theorem of integral calculus * The natural logarithm function
Lesson 106 Approximation with differentials
Lesson 107 Limit of (sin x)/x* A note (optional)
Lesson 108 Integrals of sec u and csc u * Trig substitution
Lesson 109 Polar equations * Polar graphing
Lesson 110 Partial fractions I
Lesson 111 Polar graphing II
Lesson 112 Partial fractions II
Lesson 113 Integration by parts II
Lesson 114 Implicit differentiation II
Lesson 115 Partial fractions III
Lesson 116 Derivative of ex and of ln x * Derivative of sin x
Lesson 117 Proofs of the fundamental theorem * Epsilon delta proofs
Answers to odd-numbered problems
Appendix Important formulas, facts, and rules
Index