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

Table of Contents

  • 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 𝑥^𝑎
    • 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
      • 24.1.2 Quadratic Approximations
      • 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
      • 26.1.1 Radius of Convergence
      • 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 𝑧^𝑛 = 𝑤
      • 28.4.1 Some Variations
    • 28.5 Solving 𝑒^𝑧 = 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

Use Nagwa in conjunction with your preferred textbook. The recommended lessons from Nagwa for each section of this textbook are provided above. This alignment is not affiliated with, sponsored by, or endorsed by the publisher of the referenced textbook. Nagwa is a registered trademark of Nagwa Limited. All other trademarks and registered trademarks are the property of their respective owners.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.