Calculus II
Book - 2023
Offers an introduction to the principles of calculus II, covering such topics as approximate integration, substitution, improper integrals, and numerical analysis.
Saved in:
- Subjects
- Published
-
Hoboken, NJ :
For Dummies, a Wiley brand
[2023]
- Language
- English
- Main 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