# Alignment: Princeton Lifesaver Study Guides • The Calculus Lifesaver • All the Tools You Need to Excel at Calculus

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• Welcome
• 1 Functions, Graphs, and Lines
• 1.1 Functions
• 1.1.1 Interval Notation
• 1.1.2 Finding the Domain
• 1.1.3 Finding the Range Using the Graph
• 1.1.4 The Vertical Line Test
• 1.2 Inverse Functions
• 1.2.1 The Horizontal Line Test
• 1.2.2 Finding the Inverse
• 1.2.3 Restricting the Domain
• 1.2.4 Inverses of Inverse Functions
• 1.3 Composition of Functions
• 1.4 Odd and Even Functions
• 1.5 Graphs of Linear Functions
• 1.6 Common Functions and Graphs
• 2 Review of Trigonometry
• 2.1 The Basics
• 2.2 Extending the Domain of Trig Functions
• 2.2.1 The ASTC Method
• 2.2.2 Trig Functions outside [0, 2π]
• 2.3 The Graphs of Trig Functions
• 2.4 Trig Identities
• 3 Introduction to Limits
• 3.1 Limits: The Basic Idea
• 3.2 Left-Hand and Right-Hand Limits
• 3.3 When the Limit Does Not Exist
• 3.4 Limits at ∞ and −∞
• 3.4.1 Large Numbers and Small Numbers
• 3.5 Two Common Misconceptions about Asymptotes
• 3.6 The Sandwich Principle
• 3.7 Summary of Basic Types of Limits
• 4 How to Solve Limit Problems Involving Polynomials
• 4.1 Limits Involving Rational Functions as 𝑥 → 𝑎
• 4.2 Limits Involving Square Roots as 𝑥 → 𝑎
• 4.3 Limits Involving Rational Functions as 𝑥 → ∞
• 4.3.1 Method and Examples
• 4.4 Limits Involving Poly-Type Functions as 𝑥 → ∞
• 4.5 Limits Involving Rational Functions as 𝑥 → −∞
• 4.6 Limits Involving Absolute Values
• 5 Continuity and Differentiability
• 5.1 Continuity
• 5.1.1 Continuity at a Point
• 5.1.2 Continuity on an Interval
• 5.1.3 Examples of Continuous Functions
• 5.1.4 The Intermediate Value Theorem
• 5.1.5 A Harder IVT Example
• 5.1.6 Maxima and Minima of Continuous Functions
• 5.2 Differentiability
• 5.2.1 Average Speed
• 5.2.2 Displacement and Velocity
• 5.2.3 Instantaneous Velocity
• 5.2.4 The Graphical Interpretation of Velocity
• 5.2.5 Tangent Lines
• 5.2.6 The Derivative Function
• 5.2.7 The Derivative as a Limiting Ratio
• 5.2.8 The Derivative of Linear Functions
• 5.2.9 Second and Higher-Order Derivatives
• 5.2.10 When the Derivative Does Not Exist
• 5.2.11 Differentiability and Continuity
• 6 How to Solve Differentiation Problems
• 6.1 Finding Derivatives Using the Definition
• 6.2 Finding Derivatives (The Nice Way)
• 6.2.1 Constant Multiples of Functions
• 6.2.2 Sums and Differences of Functions
• 6.2.3 Products of Functions via the Product Rule
• 6.2.4 Quotients of Functions via the Quotient Rule
• 6.2.5 Composition of Functions via the Chain Rule
• 6.2.6 A Nasty Example
• 6.2.7 Justification of the Product Rule and the Chain Rule
• 6.3 Finding the Equation of a Tangent Line
• 6.4 Velocity and Acceleration
• 6.4.1 Constant Negative Acceleration
• 6.5 Limits Which Are Derivatives in Disguise
• 6.6 Derivatives of Piecewise-Defined Functions
• 6.7 Sketching Derivative Graphs Directly
• 7 Trig Limits and Derivatives
• 7.1 Limits Involving Trig Functions
• 7.1.1 The Small Case
• 7.1.2 Solving Problems—the Small Case
• 7.1.3 The Large Case
• 7.1.4 The “Other” Case
• 7.1.5 Proof of an Important Limit
• 7.2 Derivatives Involving Trig Functions
• 7.2.1 Examples of Differentiating Trig Functions
• 7.2.2 Simple Harmonic Motion
• 7.2.3 A Curious Function
• 8 Implicit Differentiation and Related Rates
• 8.1 Implicit Differentiation
• 8.1.1 Techniques and Examples
• 8.1.2 Finding the Second Derivative Implicitly
• 8.2 Related Rates
• 8.2.1 A Simple Example
• 8.2.2 A Slightly Harder Example
• 8.2.3 A Much Harder Example
• 8.2.4 A Really Hard Example
• 9 Exponentials and Logarithms
• 9.1 The Basics
• 9.1.1 Review of Exponentials
• 9.1.2 Review of Logarithms
• 9.1.3 Logarithms, Exponentials, and Inverses
• 9.1.4 Log Rules
• 9.2 Definition of 𝑒
• 9.2.1 A Question about Compound Interest
• 9.2.2 The Answer to Our Question
• 9.2.3 More about 𝑒 and Logs
• 9.3 Differentiation of Logs and Exponentials
• 9.3.1 Examples of Differentiating Exponentials and Logs
• 9.4 How to Solve Limit Problems Involving Exponentials or Logs
• 9.4.1 Limits Involving the Definition of 𝑒
• 9.4.2 Behavior of Exponentials near 0
• 9.4.3 Behavior of Logarithms near 1
• 9.4.4 Behavior of Exponentials near ∞ or −∞
• 9.4.5 Behavior of Logs near ∞
• 9.4.6 Behavior of Logs near 0
• 9.5 Logarithmic Differentiation
• 9.5.1 The Derivative of 𝑥ᵃ
• 9.6 Exponential Growth and Decay
• 9.6.1 Exponential Growth
• 9.6.2 Exponential Decay
• 9.7 Hyperbolic Functions
• 10 Inverse Functions and Inverse Trig Functions
• 10.1 The Derivative and Inverse Functions
• 10.1.1 Using the Derivative to Show That an Inverse Exists
• 10.1.2 Derivatives and Inverse Functions: What Can Go Wrong
• 10.1.3 Finding the Derivative of an Inverse Function
• 10.1.4 A Big Example
• 10.2 Inverse Trig Functions
• 10.2.1 Inverse Sine
• 10.2.2 Inverse Cosine
• 10.2.3 Inverse Tangent
• 10.2.4 Inverse Secant
• 10.2.5 Inverse Cosecant and Inverse Cotangent
• 10.2.6 Computing Inverse Trig Functions
• 10.3 Inverse Hyperbolic Functions
• 10.3.1 The Rest of the Inverse Hyperbolic Functions
• 11 The Derivative and Graphs
• 11.1 Extrema of Functions
• 11.1.1 Global and Local Extrema
• 11.1.2 The Extreme Value Theorem
• 11.1.3 How to Find Global Maxima and Minima
• 11.2 Rolle’s Theorem
• 11.3 The Mean Value Theorem
• 11.3.1 Consequences of the Mean Value Theorem
• 11.4 The Second Derivative and Graphs
• 11.4.1 More about Points of Inflection
• 11.5 Classifying Points Where the Derivative Vanishes
• 11.5.1 Using the First Derivative
• 11.5.2 Using the Second Derivative
• 12 Sketching Graphs
• 12.1 How to Construct a Table of Signs
• 12.1.1 Making a Table of Signs for the Derivative
• 12.1.2 Making a Table of Signs for the Second Derivative
• 12.2 The Big Method
• 12.3 Examples
• 12.3.1 An Example without Using Derivatives
• 12.3.2 The Full Method: Example 1
• 12.3.3 The Full Method: Example 2
• 12.3.4 The Full Method: Example 3
• 12.3.5 The Full Method: Example 4
• 13 Optimization and Linearization
• 13.1 Optimization
• 13.1.1 An Easy Optimization Example
• 13.1.2 Optimization Problems: The General Method
• 13.1.3 An Optimization Example
• 13.1.4 Another Optimization Example
• 13.1.5 Using Implicit Differentiation in Optimization
• 13.1.6 A Difficult Optimization Example
• 13.2 Linearization
• 13.2.1 Linearization in General
• 13.2.2 The Differential
• 13.2.3 Linearization Summary and Examples
• 13.2.4 The Error in Our Approximation
• 13.3 Newton’s Method
• 14 L’Hôpital’s Rule and Overview of Limits
• 14.1 L’Hôpital’s Rule
• 14.1.1 Type A: 0/0 Case
• 14.1.2 Type A: ±∞/±∞ Case
• 14.1.3 Type B1 (∞ − ∞)
• 14.1.4 Type B2 (0 × ±∞)
• 14.1.5 Type C (1^±∞, 0⁰, or ∞⁰)
• 14.1.6 Summary of L’Hôpital’s Rule Types
• 14.2 Overview of Limits
• 15 Introduction to Integration
• 15.1 Sigma Notation
• 15.1.1 A Nice Sum
• 15.1.2 Telescoping Series
• 15.2 Displacement and Area
• 15.2.1 Three Simple Cases
• 15.2.2 A More General Journey
• 15.2.3 Signed Area
• 15.2.4 Continuous Velocity
• 15.2.5 Two Special Approximations
• 16 Definite Integrals
• 16.1 The Basic Idea
• 16.1.1 Some Easy Examples
• 16.2 Definition of the Definite Integral
• 16.2.1 An Example of Using the Definition
• 16.3 Properties of Definite Integrals
• 16.4 Finding Areas
• 16.4.1 Finding the Unsigned Area
• 16.4.2 Finding the Area between Two Curves
• 16.4.3 Finding the Area between a Curve and the 𝑦-Axis
• 16.5 Estimating Integrals
• 16.5.1 A Simple Type of Estimation
• 16.6 Averages and the Mean Value Theorem for Integrals
• 16.6.1 The Mean Value Theorem for Integrals
• 16.7 A Nonintegrable Function
• 17 The Fundamental Theorems of Calculus
• 17.1 Functions Based on Integrals of Other Functions
• 17.2 The First Fundamental Theorem
• 17.2.1 Introduction to Antiderivatives
• 17.3 The Second Fundamental Theorem
• 17.4 Indefinite Integrals
• 17.5 How to Solve Problems: The First Fundamental Theorem
• 17.5.1 Variation 1: Variable Left-Hand Limit of Integration
• 17.5.2 Variation 2: One Tricky Limit of Integration
• 17.5.3 Variation 3: Two Tricky Limits of Integration
• 17.5.4 Variation 4: Limit Is a Derivative in Disguise
• 17.6 How to Solve Problems: The Second Fundamental Theorem
• 17.6.1 Finding Indefinite Integrals
• 17.6.2 Finding Definite Integrals
• 17.6.3 Unsigned Areas and Absolute Values
• 17.7 A Technical Point
• 17.8 Proof of the First Fundamental Theorem
• 18 Techniques of Integration, Part One
• 18.1 Substitution
• 18.1.1 Substitution and Definite Integrals
• 18.1.2 How to Decide What to Substitute
• 18.1.3 Theoretical Justification of the Substitution Method
• 18.2 Integration by Parts
• 18.2.1 Some Variations
• 18.3 Partial Fractions
• 18.3.1 The Algebra of Partial Fractions
• 18.3.2 Integrating the Pieces
• 18.3.3 The Method and a Big Example
• 19 Techniques of Integration, Part Two
• 19.1 Integrals Involving Trig Identities
• 19.2 Integrals Involving Powers of Trig Functions
• 19.2.1 Powers of sin and/or cos
• 19.2.2 Powers of tan
• 19.2.3 Powers of sec
• 19.2.4 Powers of cot
• 19.2.5 Powers of csc
• 19.2.6 Reduction Formulas
• 19.3 Integrals Involving Trig Substitutions
• 19.3.1 Type 1: √𝑎² − 𝑥²
• 19.3.2 Type 2: √𝑥² + 𝑎²
• 19.3.3 Type 3: √𝑥² − 𝑎²
• 19.3.4 Completing the Square and Trig Substitutions
• 19.3.5 Summary of Trig Substitutions
• 19.3.6 Technicalities of Square Roots and Trig Substitutions
• 19.4 Overview of Techniques of Integration
• 20 Improper Integrals: Basic Concepts
• 20.1 Convergence and Divergence
• 20.1.1 Some Examples of Improper Integrals
• 20.1.2 Other Blow-Up Points
• 20.2 Integrals over Unbounded Regions
• 20.3 The Comparison Test (Theory)
• 20.4 The Limit Comparison Test (Theory)
• 20.4.1 Functions Asymptotic to Each Other
• 20.4.2 The Statement of the Test
• 20.5 The 𝑝-Test (Theory)
• 20.6 The Absolute Convergence Test
• 21 Improper Integrals: How to Solve Problems
• 21.1 How to Get Started
• 21.1.1 Splitting up the Integral
• 21.1.2 How to Deal with Negative Function Values
• 21.2 Summary of Integral Tests
• 21.3 Behavior of Common Functions near ∞ and −∞
• 21.3.1 Polynomials and Poly-Type Functions near ∞ and −∞
• 21.3.2 Trig Functions near ∞ and −∞
• 21.3.3 Exponentials near ∞ and −∞
• 21.3.4 Logarithms near ∞
• 21.4 Behavior of Common Functions near 0
• 21.4.1 Polynomials and Poly-Type Functions near 0
• 21.4.2 Trig Functions near 0
• 21.4.3 Exponentials near 0
• 21.4.4 Logarithms near 0
• 21.4.5 The Behavior of More General Functions near 0
• 21.5 How to Deal with Problem Spots Not at 0 or ∞
• 22 Sequences and Series: Basic Concepts
• 22.1 Convergence and Divergence of Sequences
• 22.1.1 The Connection between Sequences and Functions
• 22.1.2 Two Important Sequences
• 22.2 Convergence and Divergence of Series
• 22.2.1 Geometric Series (Theory)
• 22.3 the 𝑛th Term Test (Theory)
• 22.4 Properties of Both Infinite Series and Improper Integrals
• 22.4.1 The Comparison Test (Theory)
• 22.4.2 The Limit Comparison Test (Theory)
• 22.4.3 The 𝑝-Test (Theory)
• 22.4.4 The Absolute Convergence Test
• 22.5 New Tests for Series
• 22.5.1 The Ratio Test (Theory)
• 22.5.2 The Root Test (Theory)
• 22.5.3 The Integral Test (Theory)
• 22.5.4 The Alternating Series Test (Theory)
• 23 How to Solve Series Problems
• 23.1 How to Evaluate Geometric Series
• 23.2 How to Use the 𝑛th Term Test
• 23.3 How to Use the Ratio Test
• 23.4 How to Use the Root Test
• 23.5 How to Use the Integral Test
• 23.6 Comparison Test, Limit Comparison Test, and 𝑝-Test
• 23.7 How to Deal with Series with Negative Terms
• 24 Taylor Polynomials, Taylor Series, and Power Series
• 24.1 Approximations and Taylor Polynomials
• 24.1.1 Linearization Revisited
• 24.1.3 Higher-Degree Approximations
• 24.1.4 Taylor’s Theorem
• 24.2 Power Series and Taylor Series
• 24.2.1 Power Series in General
• 24.2.2 Taylor Series and Maclaurin Series
• 24.2.3 Convergence of Taylor Series
• 24.3 A Useful Limit
• 25 How to Solve Estimation Problems
• 25.1 Summary of Taylor Polynomials and Series
• 25.2 Finding Taylor Polynomials and Series
• 25.3 Estimation Problems Using the Error Term
• 25.3.1 First Example
• 25.3.2 Second Example
• 25.3.3 Third Example
• 25.3.4 Fourth Example
• 25.3.5 Fifth Example
• 25.3.6 General Techniques for Estimating the Error Term
• 25.4 Another Technique for Estimating the Error
• 26 Taylor and Power Series: How to Solve Problems
• 26.1 Convergence of Power Series
• 26.1.2 How to Find the Radius and Region of Convergence
• 26.2 Getting New Taylor Series from Old Ones
• 26.2.1 Substitution and Taylor Series
• 26.2.2 Differentiating Taylor Series
• 26.2.3 Integrating Taylor Series
• 26.2.4 Adding and Subtracting Taylor Series
• 26.2.5 Multiplying Taylor Series
• 26.2.6 Dividing Taylor Series
• 26.3 Using Power and Taylor Series to Find Derivatives
• 26.4 Using Maclaurin Series to Find Limits
• 27 Parametric Equations and Polar Coordinates
• 27.1 Parametric Equations
• 27.1.1 Derivatives of Parametric Equations
• 27.2 Polar Coordinates
• 27.2.1 Converting to and from Polar Coordinates
• 27.2.2 Sketching Curves in Polar Coordinates
• 27.2.3 Finding Tangents to Polar Curves
• 27.2.4 Finding Areas Enclosed by Polar Curves
• 28 Complex Numbers
• 28.1 The Basics
• 28.1.1 Complex Exponentials
• 28.2 The Complex Plane
• 28.2.1 Converting to and from Polar Form
• 28.3 Taking Large Powers of Complex Numbers
• 28.4 Solving 𝑧ⁿ = 𝑤
• 28.4.1 Some Variations
• 28.5 Solving 𝑒ᶻ = 𝑤
• 28.6 Some Trigonometric Series
• 28.7 Euler’s Identity and Power Series
• 29 Volumes, Arc Lengths, and Surface Areas
• 29.1 Volumes of Solids of Revolution
• 29.1.1 The Disc Method
• 29.1.2 The Shell Method
• 29.1.3 Summary ... and Variations
• 29.1.4 Variation 1: Regions between a Curve and the 𝑦-Axis
• 29.1.5 Variation 2: Regions between Two Curves
• 29.1.6 Variation 3: Axes Parallel to the Coordinate Axes
• 29.2 Volumes of General Solids
• 29.3 Arc Lengths
• 29.3.1 Parametrization and Speed
• 29.4 Surface Areas of Solids of Revolution
• 30 Differential Equations
• 30.1 Introduction to Differential Equations
• 30.2 Separable First-Order Differential Equations
• 30.3 First-Order Linear Equations
• 30.3.1 Why the Integrating Factor Works
• 30.4 Constant-Coefficient Differential Equations
• 30.4.1 Solving First-Order Homogeneous Equations
• 30.4.2 Solving Second-Order Homogeneous Equations
• 30.4.3 Why the Characteristic Quadratic Method Works
• 30.4.4 Nonhomogeneous Equations and Particular Solutions
• 30.4.5 Finding a Particular Solution
• 30.4.6 Examples of Finding Particular Solutions
• 30.4.7 Resolving Conflicts between 𝑦_𝑃 and 𝑦_𝐻
• 30.4.8 Initial Value Problems (Constant-Coefficient Linear)
• 30.5 Modeling Using Differential Equations
• Appendix A Limits and Proofs
• A.1 Formal Definition of a Limit
• A.1.1 A Little Game
• A.1.2 The Actual Definition
• A.1.3 Examples of Using the Definition
• A.2 Making New Limits from Old Ones
• A.2.1 Sums and Differences of Limits—Proofs
• A.2.2 Products of Limits—Proof
• A.2.3 Quotients of Limits—Proof
• A.2.4 The Sandwich Principle—Proof
• A.3 Other Varieties of Limits
• A.3.1 Infinite Limits
• A.3.2 Left-Hand and Right-Hand Limits
• A.3.3 Limits at ∞ and −∞
• A.3.4 Two Examples Involving Trig
• A.4 Continuity and Limits
• A.4.1 Composition of Continuous Functions
• A.4.2 Proof of the Intermediate Value Theorem
• A.4.3 Proof of the Max-Min Theorem
• A.5 Exponentials and Logarithms Revisited
• A.6 Differentiation and Limits
• A.6.1 Constant Multiples of Functions
• A.6.2 Sums and Differences of Functions
• A.6.3 Proof of the Product Rule
• A.6.4 Proof of the Quotient Rule
• A.6.5 Proof of the Chain Rule
• A.6.6 Proof of the Extreme Value Theorem
• A.6.7 Proof of Rolle’s Theorem
• A.6.8 Proof of the Mean Value Theorem
• A.6.9 The Error in Linearization
• A.6.10 Derivatives of Piecewise-Defined Functions
• A.6.11 Proof of L’Hôpital’s Rule
• A.7 Proof of the Taylor Approximation Theorem
• Appendix B Estimating Integrals
• B.1 Estimating Integrals Using Strips
• B.1.1 Evenly Spaced Partitions
• B.2 The Trapezoidal Rule
• B.3 Simpson’s Rule
• B.3.1 Proof of Simpson’s Rule
• B.4 The Error in Our Approximations
• B.4.1 Examples of Estimating the Error
• B.4.2 Proof of an Error Term Inequality