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| Welcome | |
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| How to Use This Book to Study for an Exam | |
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| Two all-purpose study tips | |
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| Key sections for exam review (by topic) | |
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| Acknowledgments | |
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| Functions, Graphs, and Lines | |
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| Functions | |
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| Interval notation | |
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| Finding the domain | |
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| Finding the range using the graph | |
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| The vertical line test | |
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| Inverse Functions | |
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| The horizontal line test | |
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| Finding the inverse | |
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| Restricting the domain | |
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| Inverses of inverse functions | |
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| Composition of Functions | |
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| Odd and Even Functions | |
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| Graphs of Linear Functions | |
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| Common Functions and Graphs | |
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| Review of Trigonometry | |
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| The Basics | |
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| Extending the Domain of Trig Functions | |
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| The ASTC method | |
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| Trig functions outside [0,2[pi]] | |
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| The Graphs of Trig Functions | |
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| Trig Identities | |
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| Introduction to Limits | |
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| Limits: The Basic Idea | |
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| Left-Hand and Right-Hand Limits | |
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| When the Limit Does Not Exist | |
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| Limits at [infinity] and [infinity] | |
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| Large numbers and small numbers | |
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| Two Common Misconceptions about Asymptotes | |
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| The Sandwich Principle | |
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| Summary of Basic Types of Limits | |
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| How to Solve Limit Problems Involving Polynomials | |
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| Limits Involving Rational Functions as x [RightArrow] a | |
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| Limits Involving Square Roots as x [RightArrow] a | |
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| Limits Involving Rational Functions as x [RightArrow infinity] a | |
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| Method and examples | |
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| Limits Involving Poly-type Functions as x [RightArrow infinity] | |
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| Limits Involving Rational Functions as x [RightArrow infinity] | |
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| Limits Involving Absolute Values | |
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| Continuity and Differentiability | |
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| Continuity | |
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| Continuity at a point | |
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| Continuity on an interval | |
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| Examples of continuous functions | |
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| The Intermediate Value Theorem | |
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| A harder IVT example | |
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| Maxima and minima of continuous functions | |
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| Differentiability | |
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| Average speed | |
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| Displacement and velocity | |
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| Instantaneous velocity | |
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| The graphical interpretation of velocity | |
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| Tangent lines | |
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| The derivative function | |
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| The derivative as a limiting ratio | |
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| The derivative of linear functions | |
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| Second and higher-order derivatives | |
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| When the derivative does not exist | |
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| Differentiability and continuity | |
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| How to Solve Differentiation Problems | |
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| Finding Derivatives Using the Definition | |
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| Finding Derivatives (the Nice Way) | |
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| Constant multiples of functions | |
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| Sums and differences of functions | |
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| Products of functions via the product rule | |
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| Quotients of functions via the quotient rule | |
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| Composition of functions via the chain rule | |
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| A nasty example | |
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| Justification of the product rule and the chain rule | |
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| Finding the Equation of a Tangent Line | |
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| Velocity and Acceleration | |
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| Constant negative acceleration | |
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| Limits Which Are Derivatives in Disguise | |
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| Derivatives of Piecewise-Defined Functions | |
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| Sketching Derivative Graphs Directly | |
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| Trig Limits and Derivatives | |
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| Limits Involving Trig Functions | |
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| The small case | |
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| Solving problems-the small case | |
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| The large case | |
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| The "other" case | |
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| Proof of an important limit | |
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| Derivatives Involving Trig Functions | |
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| Examples of differentiating trig functions | |
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| Simple harmonic motion | |
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| A curious function | |
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| Implicit Differentiation and Related Rates | |
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| Implicit Differentiation | |
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| Techniques and examples | |
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| Finding the second derivative implicitly | |
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| Related Rates | |
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| A simple example | |
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| A slightly harder example | |
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| A much harder example | |
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| A really hard example | |
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| Exponentials and Logarithms | |
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| The Basics | |
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| Review of exponentials | |
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| Review of logarithms | |
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| Logarithms, exponentials, and inverses | |
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| Log rules | |
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| Definition of e | |
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| A question about compound interest | |
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| The answer to our question | |
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| More about e and logs | |
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| Differentiation of Logs and Exponentials | |
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| Examples of differentiating exponentials and logs | |
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| How to Solve Limit Problems Involving Exponentials or Logs | |
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| Limits involving the definition of e | |
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| Behavior of exponentials near 0 | |
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| Behavior of logarithms near 1 | |
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| Behavior of exponentials near [infinity] or -[infinity] | |
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| Behavior of logs near [infinity] | |
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| Behavior of logs near 0 | |
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| Logarithmic Differentiation | |
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| The derivative of x[superscript a] | |
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| Exponential Growth and Decay | |
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| Exponential growth | |
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| Exponential decay | |
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| Hyperbolic Functions | |
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| Inverse Functions and Inverse Trig Functions | |
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| The Derivative and Inverse Functions | |
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| Using the derivative to show that an inverse exists | |
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| Derivatives and inverse functions: what can go wrong | |
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| Finding the derivative of an inverse function | |
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| A big example | |
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| Inverse Trig Functions | |
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| Inverse sine | |
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| Inverse cosine | |
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| Inverse tangent | |
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| Inverse secant | |
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| Inverse cosecant and inverse cotangent | |
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| Computing inverse trig functions | |
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| Inverse Hyperbolic Functions | |
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| The rest of the inverse hyperbolic functions | |
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| The Derivative and Graphs | |
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| Extrema of Functions | |
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| Global and local extrema | |
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| The Extreme Value Theorem | |
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| How to find global maxima and minima | |
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| Rolle's Theorem | |
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| The Mean Value Theorem | |
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| Consequences of the Mean Value Theorem | |
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| The Second Derivative and Graphs | |
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| More about points of inflection | |
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| Classifying Points Where the Derivative Vanishes | |
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| Using the first derivative | |
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| Using the second derivative | |
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| Sketching Graphs | |
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| How to Construct a Table of Signs | |
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| Making a table of signs for the derivative | |
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| Making a table of signs for the second derivative | |
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| The Big Method | |
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| Examples | |
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| An example without using derivatives | |
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| The full method: example 1 | |
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| The full method: example 2 | |
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| The full method: example 3 | |
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| The full method: example 4 | |
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| Optimization and Linearization | |
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| Optimization | |
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| An easy optimization example | |
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| Optimization problems: the general method | |
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| An optimization example | |
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| Another optimization example | |
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| Using implicit differentiation in optimization | |
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| A difficult optimization example | |
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| Linearization | |
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| Linearization in general | |
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| The differential | |
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| Linearization summary and examples | |
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| The error in our approximation | |
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| Newton's Method | |
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| L'Hopital's Rule and Overview of Limits | |
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| L'Hopital's Rule | |
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| Type A: 0/0 case | |
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| Type A: [PlusMinus infinity] / [PlusMinus infinity] case | |
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| Type B1 ([infinity] - [infinity]) | |
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| Type B2 (0 x [PlusMinus infinity]) | |
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| Type C (1[PlusMinus infinity], 0[superscript 0], or [infinity superscript 0]) | |
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| Summary of l'Hopital's Rule types | |
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| Overview of Limits | |
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| Introduction to Integration | |
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| Sigma Notation | |
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| A nice sum | |
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| Telescoping series | |
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| Displacement and Area | |
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| Three simple cases | |
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| A more general journey | |
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| Signed area | |
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| Continuous velocity | |
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| Two special approximations | |
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| Definite Integrals | |
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| The Basic Idea | |
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| Some easy examples | |
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| Definition of the Definite Integral | |
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| An example of using the definition | |
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| Properties of Definite Integrals | |
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| Finding Areas | |
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| Finding the unsigned area | |
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| Finding the area between two curves | |
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| Finding the area between a curve and the y-axis | |
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| Estimating Integrals | |
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| A simple type of estimation | |
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| Averages and the Mean Value Theorem for Integrals | |
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| The Mean Value Theorem for integrals | |
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| A Nonintegrable Function | |
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| The Fundamental Theorems of Calculus | |
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| Functions Based on Integrals of Other Functions | |
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| The First Fundamental Theorem | |
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| Introduction to antiderivatives | |
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| The Second Fundamental Theorem | |
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| Indefinite Integrals | |
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| How to Solve Problems: The First Fundamental Theorem | |
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| Variation 1: variable left-hand limit of integration | |
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| Variation 2: one tricky limit of integration | |
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| Variation 3: two tricky limits of integration | |
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| Variation 4: limit is a derivative in disguise | |
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| How to Solve Problems: The Second Fundamental Theorem | |
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| Finding indefinite integrals | |
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| Finding definite integrals | |
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| Unsigned areas and absolute values | |
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| A Technical Point | |
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| Proof of the First Fundamental Theorem | |
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| Techniques of Integration, Part One | |
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| Substitution | |
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| Substitution and definite integrals | |
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| How to decide what to substitute | |
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| Theoretical justification of the substitution method | |
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| Integration by Parts | |
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| Some variations | |
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| Partial Fractions | |
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| The algebra of partial fractions | |
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| Integrating the pieces | |
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| The method and a big example | |
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| Techniques of Integration, Part Two | |
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| Integrals Involving Trig Identities | |
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| Integrals Involving Powers of Trig Functions | |
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| Powers of sin and/or cos | |
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| Powers of tan | |
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| Powers of sec | |
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| Powers of cot | |
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| Powers of csc | |
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| Reduction formulas | |
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| Integrals Involving Trig Substitutions | |
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| Type 1: [Characters not reproducible] | |
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| Type 2: [Characters not reproducible] | |
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| Type 3: [Characters not reproducible] | |
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| Completing the square and trig substitutions | |
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| Summary of trig substitutions | |
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| Technicalities of square roots and trig substitutions | |
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| Overview of Techniques of Integration | |
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| Improper Integrals: Basic Concepts | |
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| Convergence and Divergence | |
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| Some examples of improper integrals | |
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| Other blow-up points | |
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| Integrals over Unbounded Regions | |
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| The Comparison Test (Theory) | |
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| The Limit Comparison Test (Theory) | |
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| Functions asymptotic to each other | |
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| The statement of the test | |
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| The p-test (Theory) | |
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| The Absolute Convergence Test | |
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| Improper Integrals: How to Solve Problems | |
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| How to Get Started | |
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| Splitting up the integral | |
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| How to deal with negative function values | |
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| Summary of Integral Tests | |
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| Behavior of Common Functions near [infinity] and -[infinity] | |
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| Polynomials and poly-type functions near [infinity] and -[infinity] | |
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| Trig functions near [infinity] and -[infinity] | |
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| Exponentials near [infinity] and -[infinity] | |
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| Logarithms near [infinity] | |
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| Behavior of Common Functions near 0 | |
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| Polynomials and poly-type functions near 0 | |
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| Trig functions near 0 | |
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| Exponentials near 0 | |
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| Logarithms near 0 | |
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| The behavior of more general functions near 0 | |
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| How to Deal with Problem Spots Not at 0 or [infinity] | |
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| Sequences and Series: Basic Concepts | |
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| Convergence and Divergence of Sequences | |
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| The connection between sequences and functions | |
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| Two important sequences | |
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| Convergence and Divergence of Series | |
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| Geometric series (theory) | |
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| The nth Terra Test (Theory) | |
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| Properties of Both Infinite Series and Improper Integrals | |
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| The comparison test (theory) | |
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| The limit comparison test (theory) | |
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| The p-test (theory) | |
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| The absolute convergence test | |
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| New Tests for Series | |
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| The ratio test (theory) | |
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| The root test (theory) | |
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| The integral test (theory) | |
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| The alternating series test (theory) | |
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| How to Solve Series Problems | |
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| How to Evaluate Geometric Series | |
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| How to Use the nth Term Test | |
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| How to Use the Ratio Test | |
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| How to Use the Root Test | |
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| How to Use the Integral Test | |
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| Comparison Test, Limit Comparison Test, and p-test | |
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| How to Deal with Series with Negative Terms | |
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| Taylor Polynomials, Taylor Series, and Power Series | |
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| Approximations and Taylor Polynomials | |
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| Linearization revisited | |
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| Quadratic approximations | |
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| Higher-degree approximations | |
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| Taylor's Theorem | |
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| Power Series and Taylor Series | |
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| Power series in general | |
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| Taylor series and Maclaurin series | |
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| Convergence of Taylor series | |
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| A Useful Limit | |
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| How to Solve Estimation Problems | |
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| Summary of Taylor Polynomials and Series | |
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| Finding Taylor Polynomials and Series | |
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| Estimation Problems Using the Error Term | |
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| First example | |
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| Second example | |
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| Third example | |
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| Fourth example | |
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| Fifth example | |
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| General techniques for estimating the error term | |
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| Another Technique for Estimating the Error | |
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| Taylor and Power Series: How to Solve Problems | |
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| Convergence of Power Series | |
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| Radius of convergence | |
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| How to find the radius and region of convergence | |
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| Getting New Taylor Series from Old Ones | |
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| Substitution and Taylor series | |
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| Differentiating Taylor series | |
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| Integrating Taylor series | |
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| Adding and subtracting Taylor series | |
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| Multiplying Taylor series | |
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| Dividing Taylor series | |
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| Using Power and Taylor Series to Find Derivatives | |
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| Using Maclaurin Series to Find Limits | |
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| Parametric Equations and Polar Coordinates | |
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| Parametric Equations | |
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| Derivatives of parametric equations | |
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| Polar Coordinates | |
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| Converting to and from polar coordinates | |
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| Sketching curves in polar coordinates | |
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| Finding tangents to polar curves | |
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| Finding areas enclosed by polar curves | |
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| Complex Numbers | |
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| The Basics | |
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| Complex exponentials | |
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| The Complex Plane | |
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| Converting to and from polar form | |
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| Taking Large Powers of Complex Numbers | |
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| Solving z[superscript n] = w | |
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| Some variations | |
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| Solving e[superscript z] = w | |
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| Some Trigonometric Series | |
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| Euler's Identity and Power Series | |
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| Volumes, Arc Lengths, and Surface Areas | |
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| Volumes of Solids of Revolution | |
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| The disc method | |
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| The shell method | |
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| Summary...and variations | |
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| Variation 1: regions between a curve and the y-axis | |
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| Variation 2: regions between two curves | |
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| Variation 3: axes parallel to the coordinate axes | |
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| Volumes of General Solids | |
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| Arc Lengths | |
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| Parametrization and speed | |
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| Surface Areas of Solids of Revolution | |
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| Differential Equations | |
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| Introduction to Differential Equations | |
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| Separable First-order Differential Equations | |
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| First-order Linear Equations | |
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| Why the integrating factor works | |
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| Constant-coefficient Differential Equations | |
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| Solving first-order homogeneous equations | |
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| Solving second-order homogeneous equations | |
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| Why the characteristic quadratic method works | |
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| Nonhomogeneous equations and particular solutions | |
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| Finding a particular solution | |
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| Examples of finding particular solutions | |
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| Resolving conflicts between y[subscript P] and y[subscript H] | |
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| Initial value problems (constant-coefficient linear) | |
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| Modeling Using Differential Equations | |
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| Limits and Proofs | |
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| Formal Definition of a Limit | |
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| A little game | |
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| The actual definition | |
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| Examples of using the definition | |
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| Making New Limits from Old Ones | |
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| Sum and differences of limits-proofs | |
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| Products of limits-proof | |
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| Quotients of limits-proof | |
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| The sandwich principle-proof | |
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| Other Varieties of Limits | |
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| Infinite limits | |
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| Left-hand and right-hand limits | |
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| Limits at [infinity] and -[infinity] | |
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| Two examples involving trig | |
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| Continuity and Limits | |
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| Composition of continuous functions | |
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| Proof of the Intermediate Value Theorem | |
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| Proof of the Max-Min Theorem | |
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| Exponentials and Logarithms Revisited | |
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| Differentiation and Limits | |
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| Constant multiples of functions | |
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| Sums and differences of functions | |
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| Proof of the product rule | |
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| Proof of the quotient rule | |
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| Proof of the chain rule | |
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| Proof of the Extreme Value Theorem | |
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| Proof of Rolle's Theorem | |
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| Proof of the Mean Value Theorem | |
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| The error in linearization | |
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| Derivatives of piecewise-defined functions | |
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| Proof of l'Hopital's Rule | |
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| Proof of the Taylor Approximation Theorem | |
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| Estimating Integrals | |
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| Estimating Integrals Using Strips | |
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| Evenly spaced partitions | |
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| The Trapezoidal Rule | |
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| Simpson's Rule | |
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| Proof of Simpson's rule | |
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| The Error in Our Approximations | |
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| Examples of estimating the error | |
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| Proof of an error term inequality | |
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| List of Symbols | |
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| Index | |