Calculus II

Mark Zegarelli

Book - 2023

Offers an introduction to the principles of calculus II, covering such topics as approximate integration, substitution, improper integrals, and numerical analysis.

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2nd Floor 515/Zegarelli Due Apr 11, 2024
Subjects
Published
Hoboken, NJ : For Dummies, a Wiley brand [2023]
Language
English
Main Author
Mark Zegarelli (author)
Edition
3rd edition
Item Description
Includes index.
Physical Description
xiii, 375 pages : illustrations ; 24 cm
ISBN
9781119986614
  • Introduction
  • About This Book
  • Conventions Used in This Book
  • What You're Not to Read
  • Foolish Assumptions
  • Icons Used in This Book
  • Beyond the Book
  • Where to Go from Here
  • Part 1. Introduction to Integration
  • Chapter 1. An Aerial View of the Area Problem
  • Checking Out the Area
  • Comparing classical and analytic geometry
  • Finding definite answers with the definite integral
  • Slicing Things Up
  • Untangling a hairy problem using rectangles
  • Moving left, right, or center
  • Defining the Indefinite
  • Solving Problems with Integration
  • We can work it out: Finding the area between curves
  • Walking the long and winding road
  • You say you want a revolution
  • Differential Equations
  • Understanding Infinite Series
  • Distinguishing sequences and series
  • Evaluating series
  • Identifying convergent and divergent series
  • Chapter 2. Forgotten but Not Gone: Review of Algebra and Pre-Calculus
  • Quick Review of Pre-Algebra and Algebra
  • Working with fractions
  • Knowing the facts on factorials
  • Polishing off polynomials
  • Powering through powers (exponents)
  • Review of Pre-Calculus
  • Trigonometry
  • Asymptotes
  • Graphing common parent functions
  • Transforming continuous functions
  • Polar coordinates
  • Summing up sigma notation
  • chapter 3. Recent Memories: Review of Calculus I
  • Knowing Your Limits
  • Telling functions and limits apart
  • Evaluating limits
  • Hitting the Slopes with Derivatives
  • Referring to the limit formula for derivatives
  • Knowing two notations for derivatives
  • Understanding Differentiation
  • Memorizing key derivatives
  • Derivatives of the trig functions
  • Derivatives of the inverse trig functions
  • The Power rule
  • The Sum rule
  • The Constant Multiple rule
  • The Product rule
  • The Quotient rule
  • The Chain rule
  • Finding Limits Using L'Hôpital's Rule
  • Introducing L'Hôpital's rule
  • Alternative indeterminate forms
  • Part 2. From Definite to Indefinite Integrals
  • Chapter 4. Approximating Area with Riemann Sums
  • Three Ways to Approximate Area with Rectangles
  • Using left rectangles
  • Using right rectangles
  • Finding a middle ground: The Midpoint rule
  • Two More Ways to Approximate Area
  • Feeling trapped? The Trapezoid rule
  • Don't have a cow! Simpson's rule
  • Building the Riemann Sum Formula
  • Approximating the definite integral with the area formula for a rectangle
  • Widening your understanding of width
  • Limiting the margin of error
  • Summing things up with sigma notation
  • Heightening the functionality of height
  • Finishing with the slack factor
  • Chapter 5. There Must Be a Better Way - Introducing the indefinite Integral
  • FTC2: The Saga Begins
  • Introducing FTC2
  • Evaluating definite integrals using FTC2
  • Your New Best Friend: The Indefinite Integral
  • Introducing anti-differentiation
  • Solving area problems without the Riemann sum formula
  • Understanding signed area
  • Distinguishing definite and indefinite integrals
  • FTC1: The Journey Continues
  • Understanding area functions
  • Making sense of FTC1
  • Part 3. Evaluating Indefinite Integrals
  • Chapter 6. Instant Integration: Just Add Water (And C)
  • Evaluating Basic Integrals
  • Using the 17 basic antiderivatives for integrating
  • Three important integration rules
  • What happened to the other rules?
  • Evaluating More Difficult Integrals
  • Integrating polynomials
  • Integrating more complicated-looking functions
  • Understanding Integrability
  • Taking a look at two red herrings of integrability
  • Getting an idea of what integrable really means
  • Chapter 7. Sharpening Your Integration Moves
  • Integrating Rational and Radical Functions
  • Integrating simple rational functions
  • Integrating radical functions
  • Using Algebra to Integrate Using the Power Rule
  • Integrating by using inverse trig functions
  • Integrating Trig Functions
  • Recalling how to anti-differentiate the six basic trig functions
  • Using the Basic Five trig identities
  • Applying the Pythagorean trig identities
  • Integrating Compositions of Functions with Linear Inputs
  • Understanding how to integrate familiar functions that have linear inputs
  • Understanding why integrating compositions of functions with linear inputs actually works
  • Chapter 8. Here's Looking at U-Substitution
  • Knowing How to Use U-Substitution
  • Recognizing When to Use U-Substitution
  • The simpler case: f(x) c f′(x)
  • The more complex case: g(f(x)) c f′(x) when you know how to integrate g(x)
  • Using Substitution to Evaluate Definite Integrals
  • Part 4. Advanced Integration Techniques
  • Chapter 9. Parting Ways: Integration by Parts
  • Introducing Integration by Parts
  • Reversing the Product rule
  • Knowing how to integrate by parts
  • Knowing when to integrate by parts
  • Integrating by Parts with the DI-agonal Method
  • Looking at the DI-agonal chart
  • Using the DI-agonal method
  • Chapter 10. Trig Substitution: Knowing All the (Tri)Angles
  • Integrating the Six Trig Functions
  • Integrating Powers of Sines and Cosines
  • Odd powers of sines and cosines
  • Even powers of sines and cosines
  • Integrating Powers of Tangents and Secants
  • Even powers of secants
  • Odd powers of tangents
  • Other tangent and secant cases
  • Integrating Powers of Cotangents and Cosecants
  • Integrating Weird Combinations of Trig Functions
  • Using Trig Substitution
  • Distinguishing three cases for trig substitution
  • Integrating the three cases
  • Knowing when to avoid trig substitution
  • Chapter 11. Rational Solutions: Integration with Partial Fractions
  • Strange but True: Understanding Partial Fractions
  • Looking at partial fractions
  • Using partial fractions with rational expressions
  • Solving Integrals by Using Partial Fractions
  • Case 1. Distinct linear factors
  • Case 2. Repeated linear factors
  • Case 3. Distinct quadratic factors
  • Case 4. Repeated quadratic factors
  • Beyond the Four Cases: Knowing How to Set Up Any Partial Fraction
  • Integrating Improper Rationals
  • Distinguishing proper and improper rational expressions
  • Trying out an example
  • Part 5. Applications of Integrals
  • Chapter 12. Forging into New Areas: Solving Area Problems
  • Breaking Us in Two
  • Improper Integrals
  • Getting horizontal
  • Going vertical
  • Finding the Unsigned Area of Shaded Regions on the xy-Graph
  • Finding unsigned area when a region is separated horizontally
  • Measuring a single shaded region between two functions
  • Finding the area of two or more shaded regions between two functions
  • The Mean Value Theorem for Integrals
  • Calculating Arc Length
  • Chapter 13. Pump Up the Volume: Using Calculus to Solve 3-D Problems
  • Slicing Your Way to Success
  • Finding the volume of a solid with congruent cross sections
  • Finding the volume of a solid with similar cross sections
  • Measuring the volume of a pyramid
  • Measuring the volume of a weird solid
  • Turning a Problem on Its Side
  • Two Revolutionary Problems
  • Solidifying your understanding of solids of revolution
  • Skimming the surface of revolution
  • Finding the Space Between
  • Playing the Shell Game
  • Peeling and measuring a can of soup
  • Using the shell method without inverses
  • Knowing When and How to Solve 3-D Problems
  • Chapter 14. What's So Different about Differential Equations?
  • Basics of Differential Equations
  • Classifying DEs
  • Looking more closely at DEs
  • Solving Differential Equations
  • Solving separable equations
  • Solving initial-value problems
  • Part 6. Infinite Series
  • Chapter 15. Following a Sequence, Winning the Series
  • Introducing Infinite Sequences
  • Understanding notations for sequences
  • Looking at converging and diverging sequences
  • Introducing Infinite Series
  • Getting Comfy with Sigma Notation
  • Writing sigma notation in expanded form
  • Seeing more than one way to use sigma notation
  • Discovering the Constant Multiple rule for series
  • Examining the Sum rule for series
  • Connecting a Series with Its Two Related Sequences
  • A series and its defining sequence
  • A series and its sequences of partial sums
  • Recognizing Geometric Series and p-Series
  • Getting geometric series
  • Pinpointing p-series
  • Chapter 16. Where Is This Going? Testing for Convergence and Divergence
  • Starting at the Beginning
  • Using the nth-Term Test for Divergence
  • Let Me Count the Ways
  • One-way tests
  • Two-way tests
  • Choosing Comparison Tests
  • Getting direct answers with the direct comparison test
  • Testing your limits with the limit comparison test
  • Two-Way Tests for Convergence and Divergence
  • Integrating a solution with the integral test
  • Rationally solving problems with the ratio test
  • Rooting out answers with the root test
  • Looking at Alternating Series
  • Eyeballing two forms of the basic alternating series
  • Making new series from old ones
  • Alternating series based on convergent positive series
  • Checking out the alternating series test
  • Understanding absolute and conditional convergence
  • Testing alternating series
  • Chapter 17. Dressing Up Functions with the Taylor Series
  • Elementary Functions
  • Identifying two drawbacks of elementary functions
  • Appreciating why polynomials are so friendly
  • Representing elementary functions as series
  • Power Series: Polynomials on Steroids
  • Integrating power series
  • Understanding the interval of convergence
  • Expressing Functions as Series
  • Expressing sin x as a series
  • Expressing cos x as a series
  • Introducing the Maclaurin Series
  • Introducing the Taylor Series
  • Computing with the Taylor series
  • Examining convergent and divergent Taylor series
  • Expressing functions versus approximating functions
  • Understanding Why the Taylor Series Works
  • Part 7. The Part of Tens
  • Chapter 18. Ten "Aha!" Insights in Calculus II
  • Integrating Means Finding the Area
  • When You Integrate, Area Means Signed Area
  • Integrating Is Just Fancy Addition
  • Integration Uses Infinitely Many Infinitely Thin Slices
  • Integration Contains a Slack Factor
  • A Definite Integral Evaluates to a Number
  • An Indefinite Integral Evaluates to a Function
  • Integration Is Inverse Differentiation
  • Every Infinite Series Has Two Related Sequences
  • Every Infinite Series Either Converges or Diverges
  • Chapter 19. Ten Tips to Take to the Test
  • Breathe
  • Start by Doing a Memory Dump as You Read through the Exam
  • Solve the Easiest Problem First
  • Don't Forget to Write dx and + C
  • Take the Easy Way Out Whenever Possible
  • If You Get Stuck, Scribble
  • If You Really Get Stuck, Move On
  • Check Your Answers
  • If an Answer Doesn't Make Sense, Acknowledge It
  • Repeat the Mantra, "I'm Doing My Best," and Then Do Your Best
  • Index