Calculus Set Free: Infinitesimals to the Rescue

Cover
Titlepage
Copyright
Contents
Preface for the Student
	Preparation for success
	Features of this textbook
Preface for the Instructor
	Features of this textbook
	Teaching infinitesimals
	More about section dependencies
Acknowledgments
Chapter
0 Review
	0.1 Algebra Review, Part I
		Real number line
		Inequalities
		Intervals
		Absolute value
		Absolute value equationsand inequalities
		Distance on the number line
		Exercises 0.1
	0.2 Algebra Review, Part II
		Coordinate plane
		Graphs of equations
		Distance formula
		Slopes of lines
		Point-slope form of the equation of a line
		Slope-intercept form of the equationof a line
		Parallel and perpendicular lines
		Vertical lines
		Exercises 0.2
	0.3 Trigonometry Review
		Angles
		Radians
		Trigonometric functions and their values
		Finding additional trig values
		Useful trig identities
		Basic trig graphs
		Exercises 0.3
	0.4 Functions Review, Part I
		Machine description of a function
		Representations of functions
		Vertical line test
		Function values
		Finding domains and ranges
		Piecewise-defined functions
		Catalog of essential functions
		Exercises 0.4
	0.5 Functions Review, Part II
		Vertical shifts
		Horizontal shifts
		Reflections
		Stretching
		Combining functions
		Composite functions
		Exercises 0.5
	0.6 Avoiding Common Errors
		Squaring a binomial
		Exercises for squaring a binomial
		Square root of a sum
		Exercises for square root of a sum
		Cancellation
		Exercises for cancellation
		Polynomial equations
		Exercises for polynomial equations
		Inside vs. outside: being careful with function arguments
		Exercises for inside vs. outside
		Trig notation
		Exercises for trig notation
		Parentheses
		Exercises for parentheses
		When not to distribute
		Exercises for when not to distribute
Chapter
I Hyperreals, Limits, and Continuity
	1.0 Motivation
		Tangent line to a curve
	1.1 Infinitesimals
		It can't be done …or can it?
		Arithmetic with infinitesimals
		More levels of infinitesimals
		Infinite numbers
		Transfer principle
		Exercises 1.1
	1.2 Approximation
		Hyperreal approximations: a few details
		Additional examples
		Approximation principle
		Approximations with fractions
		Absolute value approximations
		Exercises 1.2
	1.3 Hyperreals and Functions
		Evaluating functions at hyperreals
		The 3 Rs principle
		Another twist: arbitrary infinitesimals
		Almost dividing by zero
		Evaluating a function at an infinite number
		Exercises 1.3
	1.4 Limits, Part I
		Limits: holes in a graph
		Limit example with a square root
		Limit example with trig
		Limit example with compound fraction
		Limit example with polynomial
		One-sided limits
		Vertical asymptotes
		Exercises 1.4
	1.5 Limits, Part II
		Limit examples with piecewise-defined functions
		Finding limits graphically
		Sketching functions from limit information
		Numerical estimation of limits
		Limitations of graphical and numerical methods for estimating limits
		Squeeze theorem
		Exercises 1.5
	1.6 Continuity, Part I
		Verifying continuity
		Alternate definition of continuity
		Identifying discontinuities graphically
		Identifying discontinuities algebraically
		Oscillatory discontinuities
		Exercises 1.6
	1.7 Continuity, Part II
		Linear functions are continuous
		Continuity of polynomial and rational functions
		Where is f continuous?
		Using continuity to evaluate limits
		Intermediate value theorem
		Application of the IVT to root-finding
		Proof of continuity of polynomial and rational functions
		Proof of continuity of trigonometric functions
		Approximating inside a continuous function
		Exercises 1.7
	1.8 Slope, Velocity, and Rates of Change
		Tangent line to a curve
		Slope of the tangent line example with a parabola
		Slope of the tangent line example with a rational function
		Slope of the tangent line example with a square root
		Average velocity
		Average velocity example
		Instantaneous velocity
		Instantaneous rate of change
		Exercises 1.8
Chapter
II Derivatives
	2.1 The Derivative
		The definition of derivative
		Differentiability implies continuity
		Differentiability, corners, and smoothness
		Vertical tangents
		Recognizing nondifferentiability visually
		Notation for derivatives
		Proofs of theorems
		Exercises 2.1
	2.2 Derivative Rules
		Derivative of a constant function
		Derivatives of linear functions
		Power rule (version 1)
		Constant multiple rule
		Sum and difference rules
		Derivatives of polynomials
		Higher-order derivatives
		Product rule
		Quotient rule
		Power rule for negative integer exponents
		General power rule
		Exercises 2.2
	2.3 Tangent Lines Revisited
		Equations of tangent lines revisited
		Detecting horizontal tangent lines
		Comparing the graphs of f and f'
		Local linearity
		Linearizations
		Differentials
		Exercises 2.3
	2.4 Derivatives of Trigonometric Functions
		Two trig limits
		Derivatives of sine and cosine
		Derivatives of the remaining trig functions
		Derivative examples with trig functions
		Why study derivatives of trig functions?
		Exercises 2.4
	2.5 Chain Rule
		The chain rule: derivatives of compositions
		Applying the chain rule: trig functions
		Applying the chain rule: algebraic functions
		Chain rule with product rule
		Multiple-link chains
		Why radians instead of degrees?
		Differentials and the chain rule
		Exercises 2.5
	2.6 Implicit Differentiation
		Differentiating implicitly defined functions
		Implicit differentiation examples
		Second derivatives implicitly
		Making implicit explicit
		Proof of the power rule for rational exponents
		Exercises 2.6
	2.7 Rates of Change: Motion and Marginals
		Motion
		Motion examples
		Rates of change in economics
		Exercises 2.7
	2.8 Related Rates: Pythagorean Relationships
		Sliding ladder
		Submarine passing under radar station
		Passing ships
		Exercises 2.8
	2.9 Related Rates: Non-Pythagorean Relationships
		Expanding sphere
		Melting ice
		Sliding ladder, revisited
		Streetlight shadows
		Rising water
		Exercises 2.9
Chapter
III Applications of the Derivative
	3.1 Absolute Extrema
		Local vs. absolute extrema
		Derivatives and local extrema
		The extreme value theorem
		Examples of finding absolute extrema
		Proof of Fermat's theorem
		Exercises 3.1
	3.2 Mean Value Theorem
		Rolle's theorem
		Mean value theorem
		Functions with the same derivative
		Proof of the mean value theorem
		Exercises 3.2
	3.3 Local Extrema
		Increasing and decreasing functions
		Finding intervals of increase or decrease
		First derivative test
		Examples of finding local extrema
		Exercises 3.3
	3.4 Concavity
		Concavity and derivatives
		Examples of determining concavity and inflection points
		Second derivative test
		Exercises 3.4
	3.5 Curve Sketching: Polynomials
		Turning information into a sketch
		Polynomial sketch: cubic
		Polynomial sketch: quintic
		Graphing polynomials using technology
		Exercises 3.5
	3.6 Limits at Infinity
		Horizontal asymptotes and limits at infinity
		Examples of limits at infinity
		Examples of finding asymptotes
		Limits at infinity: trig examples
		The approximation principle still applies!
		Exercises 3.6
	3.7 Curve Sketching: General Functions
		Turning information into a sketch
		Complete curve-sketching examples
		Exercises 3.7
	3.8 Optimization
		Optimization example: maximum enclosed area
		Optimization example: maximum volume
		Optimization example: best path
		Optimization example: maximum profit
		Optimization example: minimum material
		Exercises 3.8
	3.9 Newton's Method
		The idea of Newton's method
		Newton's method example
		Newton's method example: two solutions
		When does Newton's method work?
		Exercises 3.9
Chapter
IV Integration
	4.1 Antiderivatives
		Antiderivatives: reversing the differentiation process
		Indefinite integrals
		Antiderivative rules
		Manipulating integrands
		Trig antiderivatives
		Exercises 4.1
	4.2 Finite Sums
		Estimating areas using rectangles: left- and right-hand endpoints
		More rectangles
		Midpoints
		Upper and lower estimates
		Estimating distance traveled
		Exercises 4.2
	4.3 Areas and Sums
		Summation notation
		Writing summation notation
		Helpful summation formulas
		Omega sums
		Areas revisited
		Exercises 4.3
	4.4 Definite Integral
		Definition of the definite integral
		Notation
		Net area
		Properties of definite integrals
		Exercises 4.4
	4.5 Fundamental Theorem of Calculus
		Fundamental theorem of calculus, part I
		Fundamental theorem of calculus, part II
		Exercises 4.5
	4.6 Substitution for Indefinite Integrals
		Substitution: reversing the chain rule
		More substitution examples
		Substitutions sometimes fail
		Exercises 4.6
	4.7 Substitution for Definite Integrals
		Substitution with definite integrals: method 1
		Substitution with definite integrals: method 2
		Proof of validity of method 2
		Exercises 4.7
	4.8 Numerical Integration, Part I
		Exploring the options
		Trapezoid rule
		Trapezoid rule example
		Using a calculator
		Error bound for the trapezoid rule
		Exercises 4.8
	4.9 Numerical Integration, Part II
		Midpoint rule
		Error bound for the midpoint rule
		Simpson's rule
		Error bound for Simpson's rule
		Derivation of Simpson's rule
		Exercises 4.9
	4.10 Initial Value Problems and Net Change
		Initial value problems
		IVP: gravity
		Net change
		Displacement vs. total distance traveled
		Exercises 4.10
Chapter
V Transcendental Functions
	5.1 Logarithms, Part I
		Introducing the natural logarithmic function
		Graph of y= ln x
		Laws of logarithms
		Using the laws of logarithms
		Differentiating logs, simplified
		Asymptotes on y=ln x
		Limits with logs
		Exercises 5.1
	5.2 Logarithms, Part II
		Domains of logarithmic functions
		An antiderivative of y=1x, x "2260 0
		Antiderivatives of tangent and cotangent
		Logarithmic differentiation
		Exercises 5.2
	5.3 Inverse Functions
		Inverse functions: review
		Finding inverses
		Graphs of inverse functions
		Calculus of inverse functions
		Near-proof of the continuity of inverse functions theorem
		Exercises 5.3
	5.4 Exponentials
		The natural exponential function
		Calculus of the natural exponential function
		Graph of y=ex
		Limits with exponentials
		Laws of exponents
		Algebra with logs and exponentials
		A caution about notation
		Exercises 5.4
	5.5 General Exponentials
		General exponential functions
		Calculus of general exponentials
		Graph of y=ax
		Limits with general exponentials
		Laws of exponents for general exponentials
		Comparing derivative rules
		Power rule, general case
		Exercises 5.5
	5.6 General Logarithms
		General logarithmic functions
		Laws of logarithms for general logarithmic functions
		Change of base property
		Calculus with general logarithmic functions
		Alternate definition of e
		Calculating exponentials by hand
		Exercises 5.6
	5.7 Exponential Growth and Decay
		Exponential change
		Exponential change example: growth
		Exponential change example: decay
		Continuously compounded interest example
		Newton's law of cooling
		Newton's law of cooling example
		Alternate derivation of the continuously compounded interest formula
		Exercises 5.7
	5.8 Inverse Trigonometric Functions
		Inverse sine
		Derivative of y=sin-1x
		Inverse cosine
		Inverse tangent
		Limits with inverse tangent
		Inverse cotangent, secant, and cosecant
		Integral formulas: inverse sine and cosine
		Integral formulas: inverse tangent
		Review: trig composed with inverse trig
		Exercises 5.8
	5.9 Hyperbolic and Inverse Hyperbolic Functions
		Hyperbolic functions
		Derivatives of hyperbolic functions
		Integrals of hyperbolic functions
		Inverse hyperbolic functions
		Derivatives of inverse hyperbolic functions
		Integrals with inverse hyperbolic functions
		Exercises 5.9
	5.10 Comparing Rates of Growth
		Faster growth means higher-level numbers
		Faster growth means eventually higher values
		Logarithmic growth rates
		Computer science: big-oh notation
		Exponential growth rates
		The fun never ends: even more levels …
		Exercises 5.10
	5.11 Limits with Transcendental Functions: L'Hospital's Rule, Part I
		Limits and indeterminate forms
		L'Hospital's Rule
		L'Hospital's rule is for indeterminate forms only
		L'Hospital's rule might not be easier
		Exercises 5.11
	5.12 L'Hospital's Rule, Part II: More Indeterminate Forms
		Indeterminate form 0·∞
		Indeterminate form ∞-∞
		Indeterminate forms 00, ∞0, and 1∞
		Another alternate definition of e
		Exercises 5.12
	5.13 Functions without End
		Sine integral: definition and derivative
		Sine integral: exploration using the derivative
		The graph of sine integral
		Sine integral: antiderivatives
		Exercises 5.13
Chapter
VI Applications of Integration
	6.1 Area between Curves
		Area between curves: definition
		Area enclosed by two curves
		Area between curves: more complicated regions
		Area between curves, sideways
		Omega sums and areas between curves
		Why units2?
		Exercises 6.1
	6.2 Volumes, Part I
		Volume of a solid of revolution
		Volume of a solid of revolution: examples
		Rotating about the y-axis
		Volumes by cross-sectional area
		Exercises 6.2
	6.3 Volumes, Part II
		Solids of revolution: rotations about y=k or x=k
		Washer-shaped cross sections
		Exercises 6.3
	6.4 Shell Method for Volumes
		Slicing parallel to the axis of rotation
		Shell method examples: rotating about the y-axis
		Shell method: rotating regions between curves
		Shell method: rotating about x=k
		Shell method: rotating about the x-axis
		Volume: summary of methods
		Exercises 6.4
	6.5 Work, Part I
		Force
		Work with constant force
		Work with variable force
		Work: springs
		Exercises 6.5
	6.6 Work, Part II
		Work: pumping fluids
		Exercises 6.6
	6.7 Average Value of a Function
		Computing the average value of a function
		Average value: geometric interpretation
		Mean value theorem for integrals
		Exercises 6.7
Chapter
VII Techniques of Integration
	7.1 Algebra for Integration
		Review: long division of polynomials
		Integrating using long division
		Substitution: u=x+k
		Evaluating 1ax+bdx
		Review: completing the square
		Integration using completing the square
		Exercises 7.1
	7.2 Integration by Parts
		Integration by parts: the formula
		Integration by parts: basic examples
		Tabular integration
		Substitution vs. parts
		Integration by parts: exponential times trig
		Integration by parts: logs, inverse trig, and inverse hyperbolic
		Combining substitution and parts
		Integration by parts: definite integrals
		Exercises 7.2
	7.3 Trigonometric Integrals
		sec xdx
		(cosn x)(sinm x)dx, n or m odd
		Review: sinkxdx and coskxdx
		(cosn x)(sinm x)dx, n and m even
		Integrating powers of tangent and secant
		sinmxcosnxdx
		Exercises 7.3
	7.4 Trigonometric Substitution
		A motivating example
		Table of trigonometric substitutions
		Multiple-technique example
		Exercises 7.4
	7.5 Partial Fractions, Part I
		Two examples
		The method of partial fractions
		Partial fractions: complete examples
		Partial fractions: improper fractions
		Partial fractions example: three linear factors
		Multiple-technique example
		Exercises 7.5
	7.6 Partial Fractions, Part II
		Partial fraction forms
		Partial fractions example: repeated linear factor
		Partial fractions examples: irreducible quadratic factor
		Exercises 7.6
	7.7 Other Techniques of Integration
		Rationalizing fractional powers
		Rational trigonometric integrands: the substitution tanu2=z
		The reciprocal substitution
		Exercises 7.7
	7.8 Strategy for Integration
		Order for trial and error
		Applying the strategy
		Exercises 7.8
	7.9 Tables of Integrals and Use of Technology
		Using technology
		Tables of integrals
		Exercises 7.9
	7.10 Type I Improper Integrals
		Improper integrals, type I: integrating to infinity
		Substitution with type I improper integrals
		Integrating from -∞ to ∞
		Divergent but not to infinity
		p-Test for integrals
		Comparison theorem
		Proof of the p-test for integrals
		Exercises 7.10
	7.11 Type II Improper Integrals
		Improper integrals, type II: handling discontinuities
		Type II improper integrals: examples
		Divergent but not to infinity
		Type II improper integrals: why?
		Exercises 7.11
Chapter
VIII Alternate Representations: Parametric and Polar Curves
	8.1 Parametric Equations
		Describing motion in two dimensions
		Eliminating the parameter
		Parametric equations: example
		Parameterization of a line segment
		Parameterization of a circle
		Traversing a curve exactly once
		Exercises 8.1
	8.2 Tangents to Parametric Curves
		Parametric curves: tangent lines
		Multiple tangent lines at a single point
		Second derivatives with parameterized curves
		Exercises 8.2
	8.3 Polar Coordinates
		Polar coordinates
		Polar equations and inequalities
		Polar graph paper
		Graphing polar curves
		Exercises 8.3
	8.4 Tangents to Polar Curves
		Polar–rectangular conversions
		Tangents to polar curves
		More polar circles
		Exercises 8.4
	8.5 Conic Sections
		Review: circles
		Parabolas
		Ellipses
		Hyperbolas
		Why the name conic sections?
		A more general example
		Reflective properties of conics
		Exercises 8.5
	8.6 Conic Sections in Polar Coordinates
		Eccentricity
		Directrices
		Polar equation of a conic section, e>0
		Alternate polar form for ellipses
		Exercises 8.6
Chapter
IX Additional Applications of Integration
	9.1 Arc Length
		Arc length: how long is a curved path?
		Arc length example: extracting the square root
		The arc length function
		Arc length: a strategy that doesn't work
		Exercises 9.1
	9.2 Areas and Lengths in Polar Coordinates
		Review: area of a circular sector
		Areas in polar coordinates
		Lengths in polar coordinates
		Intersections of polar curves
		Additional examples
		Exercises 9.2
	9.3 Surface Area
		Area of a surface of revolution, horizontal axis
		Area of a surface of revolution, vertical axis
		Improper surface area
		A few more hints
		A mnemonic device
		Exercises 9.3
	9.4 Lengths and Surface Areas with Parametric Curves
		Lengths of parametric curves
		Surface areas using parametric curves
		Exercises 9.4
	9.5 Hydrostatic Pressure and Force
		Fluid pressure
		Fluid force
		Fluid force on a vertical plate
		Exercises 9.5
	9.6 Centers of Mass
		Point-mass systems
		Center of mass: thin flat plates of constant density
		Exercises 9.6
	9.7 Applications to Economics
		Introduction: supply and demand
		Consumers' surplus
		Producers' surplus
		Gini coefficient of income distribution
		Numerical approximation of the Gini coefficient
		Exercises 9.7
	9.8 Logistic Growth
		Unrestrained growth
		Restrained growth
		Comparison of exponential and logistic models
		Logistic growth example: declining population
		Have you heard …?
		Exercises 9.8
Chapter
X Sequences and Series
	10.1 Sequences
		What is a sequence?
		Recursively defined sequences
		Plotting sequences
		Estimating sequence limits graphically
		Monotone convergence theorem
		Exercises 10.1
	10.2 Sequence Limits
		Sequence limit definition
		Sequence limits: handling (-1)n
		Sequence limits: approximating inside continuous functions
		The squeeze theorem for sequences
		Sequence limits: level analysis
		The similar function rule
		List of common sequence limits
		Exercises 10.2
	10.3 Infinite Series
		Infinite series: defining the sum
		Geometric series
		Telescoping series
		Test for divergence
		Harmonic series
		Series rules
		Starting points
		Exercises 10.3
	10.4 Integral Test
		The integral test: convergence or divergence of a series
		Integral test examples
		p-series
		Estimating sums using the integral test
		Proof of the sum of powers approximation formula
		Exercises 10.4
	10.5 Comparison Tests
		The comparison test
		Comparison test examples
		The level comparison test
		Level comparison test examples
		Locating a level in the correct zone
		Refining the boundary
		Exercises 10.5
	10.6 Alternating Series
		Alternating series: definition
		The alternating series test
		Estimating sums of alternating series
		Another alternating series test example
		Exercises 10.6
	10.7 Ratio and Root Tests
		Absolute and conditional convergence
		Absolute and conditional convergence: examples
		Ratio test
		Ratio test examples
		Ratio test vs. level comparison test
		Root test
		Rearrangements
		A final detail
		Exercises 10.7
	10.8 Strategy for Testing Series
		Strategy checklist
		Applying the strategy
		Exercises 10.8
	10.9 Power Series
		Power series: definition
		Examples: convergence of power series
		Power series as functions
		Radius and interval of convergence
		Examples: radius and interval of convergence
		Functions as power series
		Exercises 10.9
	10.10 Taylor and Maclaurin Series
		Finding coefficients for a power series
		Maclaurin series examples
		Taylor series example
		Taylor polynomials
		Limits using series
		Derivatives and integrals of power series
		Multiplication and division of power series
		Taylor's inequality
		Exercises 10.10
Index
Answers to Odd-numbered Exercises