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