Calculus
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1.1 Overview
1.1.1 An Introduction to Thinkwell's Calculus
1.1.2 The Two Questions of Calculus
1.1.3 Average Rates of Change
1.1.4 How to Do Math
1.2 Precalculus Review
1.2.1 Functions
1.2.2 Graphing Lines
1.2.3 Parabolas
1.2.4 Some Non-Euclidean Geometry
2.1 The Concept of the Limit
2.1.1 Finding Rate of Change over an Interval
2.1.2 Finding Limits Graphically
2.1.3 The Formal Definition of a Limit
2.1.4 The Limit Laws, Part I
2.1.5 The Limit Laws, Part II
2.1.6 One-Sided Limits
2.1.7 The Squeeze Theorem
2.1.8 Continuity and Discontinuity
2.2 Evaluating Limits
2.2.1 Evaluating Limits
2.2.2 Limits and Indeterminate Forms
2.2.3 Two Techniques for Evaluating Limits
2.2.4 An Overview of Limits
3.1 Understanding the Derivative
3.1.1 Rates of Change, Secants, and Tangents
3.1.2 Finding Instantaneous Velocity
3.1.3 The Derivative
3.1.4 Differentiability
3.2 Using the Derivative
3.2.1 The Slope of a Tangent Line
3.2.2 Instantaneous Rate
3.2.3 The Equation of a Tangent Line
3.2.4 More on Instantaneous Rate
3.3 Some Special Derivatives
3.3.1 The Derivative of the Reciprocal Function
3.3.2 The Derivative of the Square Root Function
4.1 The Power Rule
4.1.1 A Shortcut for Finding Derivatives
4.1.2 A Quick Proof of the Power Rule
4.1.3 Uses of the Power Rule
4.2 The Product and Quotient Rules
4.2.1 The Product Rule
4.2.2 The Quotient Rule
4.3 The Chain Rule
4.3.1 An Introduction to the Chain Rule
4.3.2 Using the Chain Rule
4.3.3 Combining Computational Techniques
5.1 Trigonometric Functions
5.1.1 A Review of Trigonometry
5.1.2 Graphing Trigonometric Functions
5.1.3 The Derivatives of Trigonometric Functions
5.1.4 The Number Pi
5.2 Exponential Functions
5.2.1 Graphing Exponential Functions
5.2.2 Derivatives of Exponential Functions
5.2.3 The Music of Math
5.3 Logarithmic Functions
5.3.1 Evaluating Logarithmic Functions
5.3.2 The Derivative of the Natural Log Function
5.3.3 Using the Derivative Rules with Transcendental Functions
6.1 Implicit Differentiation Basics
6.1.1 An Introduction to Implicit Differentiation
6.1.2 Finding the Derivative Implicitly
6.2 Applying Implicit Differentiation
6.2.1 Using Implicit Differentiation
6.2.2 Applying Implicit Differentiation
7.1 Position and Velocity
7.1.1 Acceleration and the Derivative
7.1.2 Solving Word Problems Involving Distance and Velocity
7.2 Linear Approximation
7.2.1 Higher-Order Derivatives and Linear Approximation
7.2.2 Using the Tangent Line Approximation Formula
7.2.3 Newton's Method
7.3 Related Rates
7.3.1 The Pebble Problem
7.3.2 The Ladder Problem
7.3.3 The Baseball Problem
7.3.4 The Blimp Problem
7.3.5 Math Anxiety
7.4 Optimization
7.4.1 The Connection Between Slope and Optimization
7.4.2 The Fence Problem
7.4.3 The Box Problem
7.4.4 The Can Problem
7.4.5 The Wire-Cutting Problem
8.1 Introduction
8.1.1 An Introduction to Curve Sketching
8.1.2 Three Big Theorems
8.1.3 Morale Moment
8.2 Critical Points
8.2.1 Critical Points
8.2.2 Maximum and Minimum
8.2.3 Regions Where a Function Increases or Decreases
8.2.4 The First Derivative Test
8.2.5 Math Magic
8.3 Concavity
8.3.1 Concavity and Inflection Points
8.3.2 Using the Second Derivative to Examine Concavity
8.3.3 The Möbius Band
8.4 Graphing Using the Derivative
8.4.1 Graphs of Polynomial Functions
8.4.2 Cusp Points and the Derivative
8.4.3 Domain-Restricted Functions and the Derivative
8.4.4 The Second Derivative Test
8.5 Asymptotes
8.5.1 Vertical Asymptotes
8.5.2 Horizontal Asymptotes and Infinite Limits
8.5.3 Graphing Functions with Asymptotes
8.5.4 Functions with Asymptotes and Holes
8.5.5 Functions with Asymptotes and Critical Points
9.1 Antiderivatives
9.1.1 Antidifferentiation
9.1.2 Antiderivatives of Powers of x
9.1.3 Antiderivatives of Trigonometric and Exponential Functions
9.2 Integration by Substitution
9.2.1 Undoing the Chain Rule
9.2.2 Integrating Polynomials by Substitution
9.3 Illustrating Integration by Substitution
9.3.1 Integrating Composite Trigonometric Functions by Substitution
9.3.2 Integrating Composite Exponential and Rational Functions by Substitution
9.3.3 More Integrating Trigonometric Functions by Substitution
9.3.4 Choosing Effective Function Decompositions
9.4 The Fundamental Theorem of Calculus
9.4.1 Approximating Areas of Plane Regions
9.4.2 Areas, Riemann Sums, and Definite Integrals
9.4.3 The Fundamental Theorem of Calculus, Part I
9.4.4 The Fundamental Theorem of Calculus, Part II
9.4.5 Illustrating the Fundamental Theorem of Calculus
9.4.6 Evaluating Definite Integrals
10.1 Motion
10.1.1 Antiderivatives and Motion
10.1.2 Gravity and Vertical Motion
10.1.3 Solving Vertical Motion Problems
10.2 Finding the Area between Two Curves
10.2.1 The Area between Two Curves
10.2.2 Limits of Integration and Area
10.2.3 Common Mistakes to Avoid When Finding Areas
10.2.4 Regions Bound by Several Curves
10.3 Integrating with Respect to y
10.3.1 Finding Areas by Integrating with Respect to y: Part One
10.3.2 Finding Areas by Integrating with Respect to y: Part Two
10.3.3 Area, Integration by Substitution, and Trigonometry
11.1 The Close of Calculus I
11.1.1 A Glimpse Into Calculus II
12.1 Paradoxes
12.1.1 An Introduction to Paradoxes
12.1.2 Paradoxes and Air Safety
12.1.3 Newcomb's Paradox
12.1.4 Zeno's Paradox
12.2 Sequences
12.2.1 Fibonacci Numbers
12.2.2 The Golden Ratio
13.1 Introduction
13.1.1 Welcome to Calculus II
13.1.2 Calculus I in 20 Minutes
14.1 Indeterminate Quotients
14.1.1 Indeterminate Forms
14.1.2 An Introduction to L'Hôpital's Rule
14.1.3 Basic Uses of L'Hôpital's Rule
14.1.4 More Exotic Examples of Indeterminate Forms
14.2 Other Indeterminate Forms
14.2.1 L'Hôpital's Rule and Indeterminate Products
14.2.2 L'Hôpital's Rule and Indeterminate Differences
14.2.3 L'Hôpital's Rule and One to the Infinite Power
14.2.4 Another Example of One to the Infinite Power
15.1 Inverse Functions
15.1.1 The Exponential and Natural Log Functions
15.1.2 Differentiating Logarithmic Functions
15.1.3 Logarithmic Differentiation
15.1.4 The Basics of Inverse Functions
15.1.5 Finding the Inverse of a Function
15.2 The Calculus of Inverse Functions
15.2.1 Derivatives of Inverse Functions
15.3 Inverse Trigonometric Functions
15.3.1 The Inverse Sine, Cosine, and Tangent Functions
15.3.2 The Inverse Secant, Cosecant, and Cotangent Functions
15.3.3 Evaluating Inverse Trigonometric Functions
15.4 The Calculus of Inverse Trigonometric Functions
15.4.1 Derivatives of Inverse Trigonometric Functions
15.4.2 More Calculus of Inverse Trigonometric Functions
15.5 The Hyperbolic Functions
15.5.1 Defining the Hyperbolic Functions
15.5.2 Hyperbolic Identities
15.5.3 Derivatives of Hyperbolic Functions
16.1 Integration Using Tables
16.1.1 An Introduction to the Integral Table
16.1.2 Making u-Substitutions
16.2 Integrals Involving Powers of Sine and Cosine
16.2.1 An Introduction to Integrals with Powers of Sine and Cosine
16.2.2 Integrals with Powers of Sine and Cosine
16.2.3 Integrals with Even and Odd Powers of Sine and Cosine
16.3 Integrals Involving Powers of Other Trigonometric Functions
16.3.1 Integrals of Other Trigonometric Functions
16.3.2 Integrals with Odd Powers of Tangent and Any Power of Secant
16.3.3 Integrals with Even Powers of Secant and Any Power of Tangent
16.4 An Introduction to Integration by Partial Fractions
16.4.1 Finding Partial Fraction Decompositions
16.4.2 Partial Fractions
16.4.3 Long Division
16.5 Integration by Partial Fractions with Repeated Factors
16.5.1 Repeated Linear Factors: Part One
16.5.2 Repeated Linear Factors: Part Two
16.5.3 Distinct and Repeated Quadratic Factors
16.5.4 Partial Fractions of Transcendental Functions
16.6 Integration by Parts
16.6.1 An Introduction to Integration by Parts
16.6.2 Applying Integration by Parts to the Natural Log Function
16.6.3 Inspirational Examples of Integration by Parts
16.6.4 Repeated Application of Integration by Parts
16.6.5 Algebraic Manipulation and Integration by Parts
16.7 An Introduction to Trigonometric Substitution
16.7.1 Converting Radicals into Trigonometric Expressions
16.7.2 Using Trigonometric Substitution to Integrate Radicals
16.7.3 Trigonometric Substitutions on Rational Powers
16.8 Trigonometric Substitution Strategy
16.8.1 An Overview of Trigonometric Substitution Strategy
16.8.2 Trigonometric Substitution Involving a Definite Integral: Part One
16.8.3 Trigonometric Substitution Involving a Definite Integral: Part Two
16.9 Numerical Integration
16.9.1 Deriving the Trapezoidal Rule
16.9.2 An Example of the Trapezoidal Rule
17.1 Improper Integrals
17.1.1 The First Type of Improper Integral
17.1.2 The Second Type of Improper Integral
17.1.3 Infinite Limits of Integration, Convergence, and Divergence
18.1 The Average Value of a Function
18.1.1 Finding the Average Value of a Function
18.2 Finding Volumes Using Cross-Sections
18.2.1 Finding Volumes Using Cross-Sectional Slices
18.2.2 An Example of Finding Cross-Sectional Volumes
18.3 Disks and Washers
18.3.1 Solids of Revolution
18.3.2 The Disk Method along the y-Axis
18.3.3 A Transcendental Example of the Disk Method
18.3.4 The Washer Method across the x-Axis
18.3.5 The Washer Method across the y-Axis
18.4 Shells
18.4.1 Introducing the Shell Method
18.4.2 Why Shells Can Be Better Than Washers
18.4.3 The Shell Method: Integrating with Respect to y
18.5 Arc Lengths and Functions
18.5.1 An Introduction to Arc Length
18.5.2 Finding Arc Lengths of Curves Given by Functions
18.6 Work
18.6.1 An Introduction to Work
18.6.2 Calculating Work
18.6.3 Hooke's Law
18.7 Moments and Centers of Mass
18.7.1 Center of Mass
18.7.2 The Center of Mass of a Thin Plate
19.1 Sequences
19.1.1 The Limit of a Sequence
19.1.2 Determining the Limit of a Sequence
19.1.3 The Squeeze and Absolute Value Theorems
19.2 Monotonic and Bounded Sequences
19.2.1 Monotonic and Bounded Sequences
19.3 Infinite Series
19.3.1 An Introduction to Infinite Series
19.3.2 The Summation of Infinite Series
19.3.3 Geometric Series
19.3.4 Telescoping Series
19.4 Convergence and Divergence
19.4.1 Properties of Convergent Series
19.4.2 The nth-Term Test for Divergence
19.5 The Integral Test
19.5.1 An Introduction to the Integral Test
19.5.2 Examples of the Integral Test
19.5.3 Using the Integral Test
19.5.4 Defining p-Series
19.6 The Direct Comparison Test
19.6.1 An Introduction to the Direct Comparison Test
19.6.2 Using the Comparison Test
19.7 The Limit Comparison Test
19.7.1 An Introduction to the Limit Comparison Test
19.7.2 Using the Limit Comparison Test
19.7.3 Inverting the Series in the Limit Comparison Test
19.8 The Alternating Series
19.8.1 Alternating Series
19.8.2 The Alternating Series Test
19.8.3 Estimating the Sum of an Alternating Series
19.9 Absolute and Conditional Convergences
19.9.1 Absolute and Conditional Convergence
19.10 The Ratio and Root Tests
19.10.1 The Ratio Test
19.10.2 Examples of the Ratio Test
19.10.3 The Root Test
19.11 Polynomial Approximations of Elementary Functions
19.11.1 Polynomial Approximation of Elementary Functions
19.11.2 Higher-Degree Approximations
19.12 Taylor and Maclaurin Polynomials
19.12.1 Taylor Polynomials
19.12.2 Maclaurin Polynomials
19.12.3 The Remainder of a Taylor Polynomial
19.12.4 Approximating the Value of a Function
19.13 Taylor and Maclaurin Series
19.13.1 Taylor Series
19.13.2 Examples of the Taylor and Maclaurin Series
19.13.3 New Taylor Series
19.13.4 The Convergence of Taylor Series
19.14 Power Series
19.14.1 The Definition of Power Series
19.14.2 The Interval and Radius of Convergence
19.14.3 Finding the Interval and Radius of Convergence: Part One
19.14.4 Finding the Interval and Radius of Convergence: Part Two
19.14.5 Finding the Interval and Radius of Convergence: Part Three
19.15 Power Series Representations of Functions
19.15.1 Differentiation and Integration of Power Series
19.15.2 Finding Power Series Representations by Differentiation
19.15.3 Finding Power Series Representations by Integration
19.15.4 Integrating Functions Using Power Series
20.1 Separable Differential Equations
20.1.1 An Introduction to Differential Equations
20.1.2 Solving Separable Differential Equations
20.1.3 Finding a Particular Solution
20.1.4 Direction Fields
20.2 Solving a Homogeneous Differential Equation
20.2.1 Separating Homogeneous Differential Equations
20.2.2 Change of Variables
20.3 Growth and Decay Problems
20.3.1 Exponential Growth
20.3.2 Radioactive Decay
20.4 Solving First-Order Linear Differential Equations
20.4.1 First-Order Linear Differential Equations
20.4.2 Using Integrating Factors
21.1 Understanding Parametric Equations
21.1.1 An Introduction to Parametric Equations
21.1.2 The Cycloid
21.1.3 Eliminating Parameters
21.2 Calculus and Parametric Equations
21.2.1 Derivatives of Parametric Equations
21.2.2 Graphing the Elliptic Curve
21.2.3 The Arc Length of a Parameterized Curve
21.2.4 Finding Arc Lengths of Curves Given by Parametric Equations
21.3 Understanding Polar Coordinates
21.3.1 The Polar Coordinate System
21.3.2 Converting between Polar and Cartesian Forms
21.3.3 Spirals and Circles
21.3.4 Graphing Some Special Polar Functions
21.4 Polar Functions and Slope
21.4.1 Calculus and the Rose Curve
21.4.2 Finding the Slopes of Tangent Lines in Polar Form
21.5 Polar Functions and Area
21.5.1 Heading toward the Area of a Polar Region
21.5.2 Finding the Area of a Polar Region: Part One
21.5.3 Finding the Area of a Polar Region: Part Two
21.5.4 The Area of a Region Bounded by Two Polar Curves: Part One
21.5.5 The Area of a Region Bounded by Two Polar Curves: Part Two
22.1 Vectors and the Geometry of R2 and R3
22.1.1 Coordinate Geometry in Three Dimensional Space
22.1.2 Introduction to Vectors
22.1.3 Vectors in R2 and R3
22.1.4 An Introduction to the Dot Product
22.1.5 Orthogonal Projections
22.1.6 An Introduction to the Cross Product
22.1.7 Geometry of the Cross Product
22.1.8 Equations of Lines and Planes in R3
22.2 Vector Functions
22.2.1 Introduction to Vector Functions
22.2.2 Derivatives of Vector Functions
22.2.3 Vector Functions: Smooth Curves
22.2.4 Vector Functions: Velocity and Acceleration
Edward Burger
Williams College
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
TestPrep
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A fast way to prepare for a final exam!
Thinkwell has identified the topics that are most commonly found on a final exam. Take a quick test to find out which topics you need to work on and watch only those videos. If you have a limited time to prepare for a final, this is for you!
Online subscription is not required. Workbooks or CDs are not available.
Thinkwell's Calculus TestPrep with Edward Burger helps you get ready for your finals with no wasted time! Thinkwell's Calculus TestPrep helps you get right to work on the exact topics you need help with. Focus only on what you need and be ready for your exam in less than 5 hours!
Diagnostic Tests
Time is tight during exam season, so we'll help you get going by offering a Pre-study Prep Test that tells you exactly which topics you need to review. Once you've studied the recommended topics, there's a Practice Final that checks to make sure you're up to speed and ready to ace your final. There's no wasted time with Thinkwell's Calculus TestPrep!
Comprehensive Video Tutorials
We've built Thinkwell's Calculus TestPrep around carefully selected multimedia tutorials that provide a quick and concise review of Calculus. Thinkwell's tutorials offer a more engaging, more effective way for you to learn. Watch one Thinkwell video lecture and you'll understand why Thinkwell works better.
Interactive Exercises with Feedback
Each of Thinkwell's Calculus TestPrep topics includes exercises with fully worked-out solutions and explanations. Practice and review as often as you like and be ready for your exam!
Review Notes and More
We include illustrated review notes, a glossary, transcripts of the video lectures, and links to relevant websites. These optional materials can be viewed online, and the notes and transcripts can be printed and kept for reference.
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Finals season is a breeze with Thinkwell's Calculus TestPrep? It's a sure-fire way to make the grade!
1.1 TestPrep Topics
1.1.1 Finding Limits Graphically
1.1.2 The Limit Laws, Part I
1.1.3 The Limit Laws, Part II
1.1.4 Continuity and Discontinuity
1.1.5 Evaluating Limits
1.1.6 Limits and Indeterminate Forms
1.1.7 Two Techniques for Evaluating Limits
1.1.8 Finding Instantaneous Velocity
1.1.9 The Derivative
1.1.10 Differentiability
1.1.11 The Slope of a Tangent Line
1.1.12 The Equation of a Tangent Line
1.1.13 Uses of the Power Rule
1.1.14 The Product Rule
1.1.15 The Quotient Rule
1.1.16 Using the Chain Rule
1.1.17 Combining Computational Techniques
1.1.18 The Derivatives of Trigonometric Functions
1.1.19 Derivatives of Exponential Functions
1.1.20 The Derivative of the Natural Log Function
1.1.21 Using the Derivative Rules with Transcendental Functions
1.1.22 Finding the Derivative Implicitly
1.1.23 Using Implicit Differentiation
1.1.24 Solving Word Problems Involving Distance and Velocity
1.1.25 The Fence Problem
1.1.26 The Box Problem
1.1.27 The Ladder Problem
1.1.28 Critical Points
1.1.29 Maximum and Minimum
1.1.30 The First Derivative Test
1.1.31 Concavity and Inflection Points
1.1.32 Using the Second Derivative to Examine Concavity
1.1.33 Graphs of Polynomial Functions
1.1.34 Antiderivatives of Powers of x
1.1.35 Antiderivatives of Trigonometric and Exponential Functions
1.1.36 Integrating Polynomials by Substitution
1.1.37 Integrating Composite Trigonometric Functions by Substitution
1.1.38 Integrating Composite Exponential and Rational Functions by Substitution
1.1.39 More Integrating Trigonometric Functions by Substitution
1.1.40 The Fundamental Theorem of Calculus, Part I
1.1.41 The Fundamental Theorem of Calculus, Part II
1.1.42 Illustrating the Fundamental Theorem of Calculus
1.1.43 Evaluating Definite Integrals
1.1.44 Antiderivatives and Motion
1.1.45 The Area between Two Curves
Edward Burger
Williams College
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
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