1. Limits and continuity 1.1 Introducing Calculus: Can Change Occur at an Instant?0/01.1.1 Average rate of change from data and graphs1.1.2 Instantaneous rate of change and the idea of a limit1.1.3 Why average rates fail at a single instant1.2 Defining Limits and Using Limit Notation0/01.2.1 Informal definition of a limit at a point1.2.2 Writing and reading limit notation correctly1.2.3 Interpreting limits given in analytic notation1.3 Estimating Limit Values from Graphs0/01.3.1 One-sided and two-sided limits from graphs1.3.2 Recognizing when graphical limits exist or do not exist1.3.3 Limit estimation pitfalls due to graph scale and detail1.4 Estimating Limit Values from Tables0/01.4.1 Using tables of values to estimate limits1.4.2 One-sided limits using tables of values1.4.3 Recognizing patterns in tables when a limit does not exist1.5 Determining Limits Using Algebraic Properties of Limits0/01.5.1 Using limit laws for sums, differences, and products1.5.2 Using limit laws for quotients and composite functions1.5.3 One-sided limits evaluated algebraically and graphically1.6 Determining Limits Using Algebraic Manipulation0/01.6.1 Factoring and cancelling common factors to find limits1.6.2 Rationalizing expressions and simplifying radicals in limits1.6.3 Using algebraic rearrangements and equivalent forms strategically1.7 Selecting Procedures for Determining Limits0/01.7.1 Classifying limit problems by structure1.7.2 Choosing between numerical, graphical, and algebraic methods1.7.3 Building a decision strategy for complicated limit problems1.8 Determining Limits Using the Squeeze Theorem0/01.8.1 Statement and intuition of the Squeeze Theorem1.8.2 Applying Squeeze Theorem to trigonometric limits1.8.3 Constructing bounds to squeeze other functions1.9 Connecting Multiple Representations of Limits0/01.9.1 Converting between graphs, tables, and formulas for limits1.9.2 Translating verbal descriptions into limit notation1.9.3 Multi-representation limit problems1.10 Exploring Types of Discontinuities0/01.10.1 What it means for a function to be discontinuous at a point1.10.2 Identifying removable and jump discontinuities1.10.3 Discontinuities caused by vertical asymptotes1.11 Defining Continuity at a Point0/01.11.1 Intuitive idea of continuity at a point1.11.2 The formal three-part definition of continuity1.11.3 Using the definition to justify continuity or discontinuity1.12 Confirming Continuity over an Interval0/01.12.1 Continuity on intervals: open, closed, and half-open1.12.2 Continuity of common function families1.12.3 Using algebraic rules to combine continuous functions1.13 Removing Discontinuities0/01.13.1 When a discontinuity can be removed using limits1.13.2 Redefining function values to remove holes1.13.3 Making piecewise functions continuous at boundaries1.14 Connecting Infinite Limits and Vertical Asymptotes0/01.14.1 Infinite limits and notation for unbounded behavior1.14.2 Identifying vertical asymptotes using infinite limits1.14.3 Describing unbounded behavior from graphs and formulas1.15 Connecting Limits at Infinity and Horizontal Asymptotes0/01.15.1 Limits at infinity and end behavior of functions1.15.2 Horizontal asymptotes from limits at infinity1.15.3 Comparing growth rates using limits at infinity1.16 Working with the Intermediate Value Theorem (IVT)0/01.16.1 Statement and intuition of the Intermediate Value Theorem1.16.2 Using IVT to prove the existence of a root1.16.3 Applying IVT to model real-world continuous change1. Limits and continuity 1.1 Introducing Calculus: Can Change Occur at an Instant?0/01.1.1 Average rate of change from data and graphs1.1.2 Instantaneous rate of change and the idea of a limit1.1.3 Why average rates fail at a single instant1.2 Defining Limits and Using Limit Notation0/01.2.1 Informal definition of a limit at a point1.2.2 Writing and reading limit notation correctly1.2.3 Interpreting limits given in analytic notation1.3 Estimating Limit Values from Graphs0/01.3.1 One-sided and two-sided limits from graphs1.3.2 Recognizing when graphical limits exist or do not exist1.3.3 Limit estimation pitfalls due to graph scale and detail1.4 Estimating Limit Values from Tables0/01.4.1 Using tables of values to estimate limits1.4.2 One-sided limits using tables of values1.4.3 Recognizing patterns in tables when a limit does not exist1.5 Determining Limits Using Algebraic Properties of Limits0/01.5.1 Using limit laws for sums, differences, and products1.5.2 Using limit laws for quotients and composite functions1.5.3 One-sided limits evaluated algebraically and graphically1.6 Determining Limits Using Algebraic Manipulation0/01.6.1 Factoring and cancelling common factors to find limits1.6.2 Rationalizing expressions and simplifying radicals in limits1.6.3 Using algebraic rearrangements and equivalent forms strategically1.7 Selecting Procedures for Determining Limits0/01.7.1 Classifying limit problems by structure1.7.2 Choosing between numerical, graphical, and algebraic methods1.7.3 Building a decision strategy for complicated limit problems1.8 Determining Limits Using the Squeeze Theorem0/01.8.1 Statement and intuition of the Squeeze Theorem1.8.2 Applying Squeeze Theorem to trigonometric limits1.8.3 Constructing bounds to squeeze other functions1.9 Connecting Multiple Representations of Limits0/01.9.1 Converting between graphs, tables, and formulas for limits1.9.2 Translating verbal descriptions into limit notation1.9.3 Multi-representation limit problems1.10 Exploring Types of Discontinuities0/01.10.1 What it means for a function to be discontinuous at a point1.10.2 Identifying removable and jump discontinuities1.10.3 Discontinuities caused by vertical asymptotes1.11 Defining Continuity at a Point0/01.11.1 Intuitive idea of continuity at a point1.11.2 The formal three-part definition of continuity1.11.3 Using the definition to justify continuity or discontinuity1.12 Confirming Continuity over an Interval0/01.12.1 Continuity on intervals: open, closed, and half-open1.12.2 Continuity of common function families1.12.3 Using algebraic rules to combine continuous functions1.13 Removing Discontinuities0/01.13.1 When a discontinuity can be removed using limits1.13.2 Redefining function values to remove holes1.13.3 Making piecewise functions continuous at boundaries1.14 Connecting Infinite Limits and Vertical Asymptotes0/01.14.1 Infinite limits and notation for unbounded behavior1.14.2 Identifying vertical asymptotes using infinite limits1.14.3 Describing unbounded behavior from graphs and formulas1.15 Connecting Limits at Infinity and Horizontal Asymptotes0/01.15.1 Limits at infinity and end behavior of functions1.15.2 Horizontal asymptotes from limits at infinity1.15.3 Comparing growth rates using limits at infinity1.16 Working with the Intermediate Value Theorem (IVT)0/01.16.1 Statement and intuition of the Intermediate Value Theorem1.16.2 Using IVT to prove the existence of a root1.16.3 Applying IVT to model real-world continuous change2. Differentiation: Definition and Fundamental Properties 2.1 Defining Average and Instantaneous Rates of Change at a Point0/02.1.1 Average Rate of Change from Graphs and Tables2.1.2 Average Rate of Change from Formulas and Contexts2.1.3 Instantaneous Rate of Change as a Limit2.2 Defining the Derivative of a Function and Using Derivative Notation0/02.2.1 Defining the Derivative Function with Limits2.2.2 Understanding Derivative Notation in Problems2.2.3 Representing Derivatives Numerically and Graphically2.2.4 Tangent Lines and Derivatives at a Point2.3 Estimating Derivatives of a Function at a Point0/02.3.1 Estimating Derivatives from Tables of Values2.3.2 Estimating Derivatives from Graphs2.3.3 Using Technology to Approximate Derivatives2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist0/02.4.1 Differentiability Implies Continuity2.4.2 Continuous but Not Differentiable2.4.3 Vertical Tangents and Non-Differentiability2.5 Applying the Power Rule0/02.5.1 Deriving the Power Rule from First Principles2.5.2 Applying the Power Rule to Integer Powers2.5.3 Extending the Power Rule to Rational Exponents2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple0/02.6.1 Constant Multiple Rule for Derivatives2.6.2 Sum and Difference Rules for Derivatives2.6.3 Differentiating Polynomial Functions2.6.4 Interpreting Polynomial Derivatives in Context2.7 Derivatives of cos x, sin x, e^x, and ln x0/02.7.1 Derivatives of Sine and Cosine Functions2.7.2 Derivatives of Exponential and Natural Logarithm Functions2.7.3 Combining Trigonometric, Exponential, and Logarithmic Derivatives2.7.4 Using Known Derivatives to Evaluate Limits2.8 The Product Rule0/02.8.1 Deriving the Product Rule Conceptually2.8.2 Applying the Product Rule to Algebraic Functions2.8.3 Product Rule with Trigonometric, Exponential, and Logarithmic Functions2.9 The Quotient Rule0/02.9.1 Deriving the Quotient Rule Conceptually2.9.2 Applying the Quotient Rule to Rational Functions2.9.3 Choosing Between Quotient Rule and Algebraic Simplification2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions0/02.10.1 Derivatives of Tangent and Cotangent Functions2.10.2 Derivatives of Secant and Cosecant Functions2.10.3 Using Identities to Prepare Trig Functions for Differentiation2. Differentiation: Definition and Fundamental Properties 2.1 Defining Average and Instantaneous Rates of Change at a Point0/02.1.1 Average Rate of Change from Graphs and Tables2.1.2 Average Rate of Change from Formulas and Contexts2.1.3 Instantaneous Rate of Change as a Limit2.2 Defining the Derivative of a Function and Using Derivative Notation0/02.2.1 Defining the Derivative Function with Limits2.2.2 Understanding Derivative Notation in Problems2.2.3 Representing Derivatives Numerically and Graphically2.2.4 Tangent Lines and Derivatives at a Point2.3 Estimating Derivatives of a Function at a Point0/02.3.1 Estimating Derivatives from Tables of Values2.3.2 Estimating Derivatives from Graphs2.3.3 Using Technology to Approximate Derivatives2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist0/02.4.1 Differentiability Implies Continuity2.4.2 Continuous but Not Differentiable2.4.3 Vertical Tangents and Non-Differentiability2.5 Applying the Power Rule0/02.5.1 Deriving the Power Rule from First Principles2.5.2 Applying the Power Rule to Integer Powers2.5.3 Extending the Power Rule to Rational Exponents2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple0/02.6.1 Constant Multiple Rule for Derivatives2.6.2 Sum and Difference Rules for Derivatives2.6.3 Differentiating Polynomial Functions2.6.4 Interpreting Polynomial Derivatives in Context2.7 Derivatives of cos x, sin x, e^x, and ln x0/02.7.1 Derivatives of Sine and Cosine Functions2.7.2 Derivatives of Exponential and Natural Logarithm Functions2.7.3 Combining Trigonometric, Exponential, and Logarithmic Derivatives2.7.4 Using Known Derivatives to Evaluate Limits2.8 The Product Rule0/02.8.1 Deriving the Product Rule Conceptually2.8.2 Applying the Product Rule to Algebraic Functions2.8.3 Product Rule with Trigonometric, Exponential, and Logarithmic Functions2.9 The Quotient Rule0/02.9.1 Deriving the Quotient Rule Conceptually2.9.2 Applying the Quotient Rule to Rational Functions2.9.3 Choosing Between Quotient Rule and Algebraic Simplification2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions0/02.10.1 Derivatives of Tangent and Cotangent Functions2.10.2 Derivatives of Secant and Cosecant Functions2.10.3 Using Identities to Prepare Trig Functions for Differentiation3. Differentiation: Composite, Implicit, and Inverse Functions 3.1 The Chain Rule0/03.1.1 Recognizing Composite Functions 3.1.2 The Chain Rule Formula and Notation3.1.3 Chain Rule with Powers and Polynomials3.1.4 Chain Rule with Exponential and Logarithmic Functions3.1.5 Chain Rule with Trigonometric Functions3.1.6 Multiple Applications of the Chain Rule3.2 Implicit Differentiation0/03.2.1 Understanding Implicitly Defined Functions3.2.2 Using the Chain Rule in Implicit Differentiation3.2.3 Slopes and Tangent Lines from Implicit Derivatives3.2.4 Second Derivatives with Implicit Differentiation3.2.5 When Implicit Differentiation Is Useful3.3 Differentiating Inverse Functions0/03.3.1 Understanding Inverse Functions and Graphs3.3.2 Deriving the Derivative Formula for an Inverse3.3.3 Derivatives of Inverses at Specific Points3.3.4 Interpreting Derivatives of Inverse Functions3.3.5 Inverses, One-to-One Functions, and Validity3.4 Differentiating Inverse Trigonometric Functions0/03.4.1 Inverse Trig Functions and Principal Values3.4.2 Derivatives of arcsin and arccos3.4.3 Derivatives of arctan and Other Inverse Trig Functions3.4.4 Chain Rule with Inverse Trigonometric Composites3.5 Selecting Procedures for Calculating Derivatives0/03.5.1 Choosing an Appropriate Differentiation Rule3.5.2 Combining Multiple Rules in One Problem3.5.3 Using Graphs, Tables, and Technology3.5.4 Explaining and Justifying Your Method3.6 Calculating Higher-Order Derivatives0/03.6.1 From First to Second Derivative3.6.2 Notation for Higher-Order Derivatives3.6.3 Higher-Order Derivatives of Polynomials3.6.4 Higher-Order Derivatives Using the Chain Rule3.6.5 Interpreting Higher-Order Derivatives3. Differentiation: Composite, Implicit, and Inverse Functions 3.1 The Chain Rule0/03.1.1 Recognizing Composite Functions 3.1.2 The Chain Rule Formula and Notation3.1.3 Chain Rule with Powers and Polynomials3.1.4 Chain Rule with Exponential and Logarithmic Functions3.1.5 Chain Rule with Trigonometric Functions3.1.6 Multiple Applications of the Chain Rule3.2 Implicit Differentiation0/03.2.1 Understanding Implicitly Defined Functions3.2.2 Using the Chain Rule in Implicit Differentiation3.2.3 Slopes and Tangent Lines from Implicit Derivatives3.2.4 Second Derivatives with Implicit Differentiation3.2.5 When Implicit Differentiation Is Useful3.3 Differentiating Inverse Functions0/03.3.1 Understanding Inverse Functions and Graphs3.3.2 Deriving the Derivative Formula for an Inverse3.3.3 Derivatives of Inverses at Specific Points3.3.4 Interpreting Derivatives of Inverse Functions3.3.5 Inverses, One-to-One Functions, and Validity3.4 Differentiating Inverse Trigonometric Functions0/03.4.1 Inverse Trig Functions and Principal Values3.4.2 Derivatives of arcsin and arccos3.4.3 Derivatives of arctan and Other Inverse Trig Functions3.4.4 Chain Rule with Inverse Trigonometric Composites3.5 Selecting Procedures for Calculating Derivatives0/03.5.1 Choosing an Appropriate Differentiation Rule3.5.2 Combining Multiple Rules in One Problem3.5.3 Using Graphs, Tables, and Technology3.5.4 Explaining and Justifying Your Method3.6 Calculating Higher-Order Derivatives0/03.6.1 From First to Second Derivative3.6.2 Notation for Higher-Order Derivatives3.6.3 Higher-Order Derivatives of Polynomials3.6.4 Higher-Order Derivatives Using the Chain Rule3.6.5 Interpreting Higher-Order Derivatives4. Contextual Applications of Differentiation 4.1 Interpreting the Meaning of the Derivative in Context0/04.1.1 Derivatives as Instantaneous Rates of Change4.1.2 Interpreting Derivative Values in Real Situations4.1.3 Units and Notation for f′(x)4.1.4 Comparing Average and Instantaneous Rates4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration0/04.2.1 Position Functions and Motion on a Line4.2.2 Velocity as the Derivative of Position4.2.3 Acceleration and Higher Derivatives4.2.4 Analyzing Motion Using Graphs and Tables4.2.5 Solving Straight-Line Motion Problems4.3 Rates of Change in Applied Contexts Other Than Motion0/04.3.1 Setting Up Non-Motion Rate Problems4.3.2 Rates of Change in Population and Biology4.3.3 Rates in Economics and Business Contexts4.3.4 Geometric and Physical Rate Problems4.3.5 Comparing Structures Across Different Contexts4.4 Introduction to Related Rates0/04.4.1 Understanding Related Rates Situations4.4.2 Drawing Diagrams and Identifying Given Rates4.4.3 Using the Chain Rule in Related Rates4.4.4 Product and Quotient Rules in Related Rates4.4.5 Organizing Work and Units in Related Rates4.5 Solving Related Rates Problems0/04.5.1 Step-by-Step Method for Solving Related Rates4.5.2 Geometric Related Rates: Area and Volume4.5.3 Related Rates with Similar Triangles and Trigonometry4.5.4 Interpreting and Explaining Related Rate Answers4.5.5 Checking Reasonableness of Related Rates Solutions4.6 Approximating Values of a Function Using Local Linearity and Linearization0/04.6.1 Local Linearity and Zooming In on Curves4.6.2 Finding the Equation of a Tangent Line4.6.3 Using Linearization to Approximate Function Values4.6.4 Overestimates and Underestimates with Tangent Lines4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms0/04.7.1 Understanding Indeterminate Forms4.7.2 Conditions for Applying L’Hospital’s Rule4.7.3 Evaluating Limits Using L’Hospital’s Rule4.7.4 Interpreting L’Hospital’s Rule in Context4. Contextual Applications of Differentiation 4.1 Interpreting the Meaning of the Derivative in Context0/04.1.1 Derivatives as Instantaneous Rates of Change4.1.2 Interpreting Derivative Values in Real Situations4.1.3 Units and Notation for f′(x)4.1.4 Comparing Average and Instantaneous Rates4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration0/04.2.1 Position Functions and Motion on a Line4.2.2 Velocity as the Derivative of Position4.2.3 Acceleration and Higher Derivatives4.2.4 Analyzing Motion Using Graphs and Tables4.2.5 Solving Straight-Line Motion Problems4.3 Rates of Change in Applied Contexts Other Than Motion0/04.3.1 Setting Up Non-Motion Rate Problems4.3.2 Rates of Change in Population and Biology4.3.3 Rates in Economics and Business Contexts4.3.4 Geometric and Physical Rate Problems4.3.5 Comparing Structures Across Different Contexts4.4 Introduction to Related Rates0/04.4.1 Understanding Related Rates Situations4.4.2 Drawing Diagrams and Identifying Given Rates4.4.3 Using the Chain Rule in Related Rates4.4.4 Product and Quotient Rules in Related Rates4.4.5 Organizing Work and Units in Related Rates4.5 Solving Related Rates Problems0/04.5.1 Step-by-Step Method for Solving Related Rates4.5.2 Geometric Related Rates: Area and Volume4.5.3 Related Rates with Similar Triangles and Trigonometry4.5.4 Interpreting and Explaining Related Rate Answers4.5.5 Checking Reasonableness of Related Rates Solutions4.6 Approximating Values of a Function Using Local Linearity and Linearization0/04.6.1 Local Linearity and Zooming In on Curves4.6.2 Finding the Equation of a Tangent Line4.6.3 Using Linearization to Approximate Function Values4.6.4 Overestimates and Underestimates with Tangent Lines4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms0/04.7.1 Understanding Indeterminate Forms4.7.2 Conditions for Applying L’Hospital’s Rule4.7.3 Evaluating Limits Using L’Hospital’s Rule4.7.4 Interpreting L’Hospital’s Rule in Context5. Analytical Applications of Differentiation 5.1 Using the Mean Value Theorem0/05.1.1 Conditions for the Mean Value Theorem5.1.2 Guaranteed point where rates of change match5.1.3 Geometric meaning of the Mean Value Theorem5.1.4 Using the Mean Value Theorem in applications5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points0/05.2.1 Understanding the Extreme Value Theorem5.2.2 Global versus local extrema on an interval5.2.3 Defining critical points of a function5.2.4 Finding critical points using derivatives5.2.5 Relating critical points to local extrema5.3 Determining Intervals on Which a Function Is Increasing or Decreasing0/05.3.1 Using the sign of f′ to determine behavior5.3.2 Sign charts for increasing and decreasing intervals5.3.3 Describing behavior from a graph of f′5.3.4 Writing justifications about increasing and decreasing5.4 Using the First Derivative Test to Determine Relative (Local) Extrema0/05.4.1 Statement of the First Derivative Test5.4.2 Classifying local maxima with the First Derivative Test5.4.3 Classifying local minima with the First Derivative Test5.4.4 When the First Derivative Test is inconclusive5.5 Using the Candidates Test to Determine Absolute (Global) Extrema0/05.5.1 Candidates for absolute extrema on a closed interval5.5.2 Applying the Candidates Test step by step5.5.3 Identifying absolute maximum and minimum values5.5.4 Checking domain and conditions for extrema5.6 Determining Concavity of Functions over Their Domains0/05.6.1 Concavity and the behavior of the first derivative5.6.2 Understanding concavity using the second derivative5.6.3 Using tables and graphs to describe concavity5.6.4 Defining points of inflection5.6.5 Finding points of inflection using derivatives5.7 Using the Second Derivative Test to Determine Extrema0/05.7.1 Statement of the Second Derivative Test5.7.2 When the Second Derivative Test is inconclusive5.7.3 Relating concavity and local extrema5.7.4 One critical point giving an absolute extremum5.8 Sketching Graphs of Functions and Their Derivatives0/05.8.1 Relating key features of f and f′5.8.2 Using f′ and f″ to understand the shape of f5.8.3 Sketching f′ from a graph of f5.8.4 Sketching f from a graph of f′5.8.5 Connecting graphical, numerical, and algebraic views5.9 Connecting a Function, Its First Derivative, and Its Second Derivative0/05.9.1 Matching graphs of f, f′, and f″5.9.2 Using f′ to connect to f and f″5.9.3 Describing behavior consistently across all three graphs5.9.4 Explaining relationships in words5.10 Introduction to Optimization Problems0/05.10.1 Identifying quantities to maximize or minimize5.10.2 Building an objective function from a context5.10.3 Using derivatives to locate optimal values5.10.4 Interpreting optimization results realistically5.11 Solving Optimization Problems0/05.11.1 Understanding the meaning of optimal values5.11.2 Communicating solutions with correct units and context5.11.3 Checking that solutions satisfy constraints5.12 Exploring Behaviors of Implicit Relations0/05.12.1 Understanding implicit relations and functions5.12.2 Finding dy/dx for implicit relations5.12.3 Critical points of implicit relations5.12.4 Applying derivative tests to implicit curves5.12.5 Second derivatives for implicit relations5.12.6 Interpreting behavior from implicit derivatives5. Analytical Applications of Differentiation 5.1 Using the Mean Value Theorem0/05.1.1 Conditions for the Mean Value Theorem5.1.2 Guaranteed point where rates of change match5.1.3 Geometric meaning of the Mean Value Theorem5.1.4 Using the Mean Value Theorem in applications5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points0/05.2.1 Understanding the Extreme Value Theorem5.2.2 Global versus local extrema on an interval5.2.3 Defining critical points of a function5.2.4 Finding critical points using derivatives5.2.5 Relating critical points to local extrema5.3 Determining Intervals on Which a Function Is Increasing or Decreasing0/05.3.1 Using the sign of f′ to determine behavior5.3.2 Sign charts for increasing and decreasing intervals5.3.3 Describing behavior from a graph of f′5.3.4 Writing justifications about increasing and decreasing5.4 Using the First Derivative Test to Determine Relative (Local) Extrema0/05.4.1 Statement of the First Derivative Test5.4.2 Classifying local maxima with the First Derivative Test5.4.3 Classifying local minima with the First Derivative Test5.4.4 When the First Derivative Test is inconclusive5.5 Using the Candidates Test to Determine Absolute (Global) Extrema0/05.5.1 Candidates for absolute extrema on a closed interval5.5.2 Applying the Candidates Test step by step5.5.3 Identifying absolute maximum and minimum values5.5.4 Checking domain and conditions for extrema5.6 Determining Concavity of Functions over Their Domains0/05.6.1 Concavity and the behavior of the first derivative5.6.2 Understanding concavity using the second derivative5.6.3 Using tables and graphs to describe concavity5.6.4 Defining points of inflection5.6.5 Finding points of inflection using derivatives5.7 Using the Second Derivative Test to Determine Extrema0/05.7.1 Statement of the Second Derivative Test5.7.2 When the Second Derivative Test is inconclusive5.7.3 Relating concavity and local extrema5.7.4 One critical point giving an absolute extremum5.8 Sketching Graphs of Functions and Their Derivatives0/05.8.1 Relating key features of f and f′5.8.2 Using f′ and f″ to understand the shape of f5.8.3 Sketching f′ from a graph of f5.8.4 Sketching f from a graph of f′5.8.5 Connecting graphical, numerical, and algebraic views5.9 Connecting a Function, Its First Derivative, and Its Second Derivative0/05.9.1 Matching graphs of f, f′, and f″5.9.2 Using f′ to connect to f and f″5.9.3 Describing behavior consistently across all three graphs5.9.4 Explaining relationships in words5.10 Introduction to Optimization Problems0/05.10.1 Identifying quantities to maximize or minimize5.10.2 Building an objective function from a context5.10.3 Using derivatives to locate optimal values5.10.4 Interpreting optimization results realistically5.11 Solving Optimization Problems0/05.11.1 Understanding the meaning of optimal values5.11.2 Communicating solutions with correct units and context5.11.3 Checking that solutions satisfy constraints5.12 Exploring Behaviors of Implicit Relations0/05.12.1 Understanding implicit relations and functions5.12.2 Finding dy/dx for implicit relations5.12.3 Critical points of implicit relations5.12.4 Applying derivative tests to implicit curves5.12.5 Second derivatives for implicit relations5.12.6 Interpreting behavior from implicit derivatives6. Integration and Accumulation of Change 6.1 Exploring Accumulations of Change0/06.1.1 Accumulation of Change as Area Under a Rate Graph6.1.2 Using Geometry to Find Accumulated Change6.1.3 Positive and Negative Accumulated Change6.1.4 Units for Area and Accumulation6.2 Approximating Areas with Riemann Sums0/06.2.1 Approximating Integrals from Graphs and Tables6.2.2 Left and Right Riemann Sums6.2.3 Midpoint and Trapezoidal Approximations6.2.4 Uniform and Nonuniform Partitions6.2.5 Overestimates and Underestimates6.2.6 Using Technology for Numerical Integration6.3 Riemann Sums, Summation Notation, and Definite Integral Notation0/06.3.1 Riemann Sums as Sums of Products6.3.2 Definite Integrals as Limits of Riemann Sums6.3.3 Writing a Riemann Sum for a Given Integral6.3.4 Converting a Riemann Sum to Integral Notation6.4 The Fundamental Theorem of Calculus and Accumulation Functions0/06.4.1 Defining Functions Using Definite Integrals6.4.2 Accumulation Functions and Their Meaning6.4.3 Differentiating Functions Defined by Integrals (FTC Part 1)6.5 Interpreting the Behavior of Accumulation Functions Involving Area0/06.5.1 Relating f and Its Accumulation Function g6.5.2 Using Graphs to Sketch Accumulation Functions6.5.3 Contextual Meaning of Accumulation Functions6.6 Applying Properties of Definite Integrals0/06.6.1 Evaluating Integrals Using Area and Geometry6.6.2 Linearity of Definite Integrals6.6.3 Integrals Over Adjacent Intervals6.6.4 Integrals with Removable or Jump Discontinuities6.7 The Fundamental Theorem of Calculus and Definite Integrals0/06.7.1 Understanding Antiderivatives6.7.2 Accumulation Functions as Antiderivatives6.7.3 Evaluating Definite Integrals Using FTC Part 26.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation0/06.8.1 Indefinite Integrals and Notation6.8.2 Reversing Basic Derivative Rules6.8.3 The Constant of Integration and Families of Curves6.8.4 Functions Without Elementary Antiderivatives6.9 Integrating Using Substitution0/06.9.1 Connecting u-Substitution to the Chain Rule6.9.2 Indefinite Integrals Using Substitution6.9.3 Definite Integrals and Changing Limits with Substitution6.10 Integrating Functions Using Long Division and Completing the Square0/06.10.1 Using Algebraic Long Division Before Integrating6.10.2 Completing the Square to Simplify Integrals6.10.3 Choosing Algebraic Techniques for Integration6.11 Integrating Using Integration by Parts0/06.11.1 Deriving the Integration by Parts Formula6.11.2 Choosing u and dv in Integration by Parts6.11.3 Definite Integrals with Integration by Parts6.12 Integrating Using Linear Partial Fractions0/06.12.1 Decomposing Rational Functions into Linear Factors6.12.2 Integrating Decomposed Rational Functions6.12.3 Applying Partial Fractions to Definite Integrals6.13 Evaluating Improper Integrals0/06.13.1 Recognizing Improper Integrals6.13.2 Using Limits to Evaluate Improper Integrals6.13.3 Convergent and Divergent Improper Integrals6.14 Selecting Techniques for Antidifferentiation0/06.14.1 Classifying Integrands by Type6.14.2 Choosing Between Basic Rules and Substitution6.14.3 Deciding When to Use Algebraic Rearrangements6.14.4 Strategy Review for Antiderivatives and Definite Integrals6. Integration and Accumulation of Change 6.1 Exploring Accumulations of Change0/06.1.1 Accumulation of Change as Area Under a Rate Graph6.1.2 Using Geometry to Find Accumulated Change6.1.3 Positive and Negative Accumulated Change6.1.4 Units for Area and Accumulation6.2 Approximating Areas with Riemann Sums0/06.2.1 Approximating Integrals from Graphs and Tables6.2.2 Left and Right Riemann Sums6.2.3 Midpoint and Trapezoidal Approximations6.2.4 Uniform and Nonuniform Partitions6.2.5 Overestimates and Underestimates6.2.6 Using Technology for Numerical Integration6.3 Riemann Sums, Summation Notation, and Definite Integral Notation0/06.3.1 Riemann Sums as Sums of Products6.3.2 Definite Integrals as Limits of Riemann Sums6.3.3 Writing a Riemann Sum for a Given Integral6.3.4 Converting a Riemann Sum to Integral Notation6.4 The Fundamental Theorem of Calculus and Accumulation Functions0/06.4.1 Defining Functions Using Definite Integrals6.4.2 Accumulation Functions and Their Meaning6.4.3 Differentiating Functions Defined by Integrals (FTC Part 1)6.5 Interpreting the Behavior of Accumulation Functions Involving Area0/06.5.1 Relating f and Its Accumulation Function g6.5.2 Using Graphs to Sketch Accumulation Functions6.5.3 Contextual Meaning of Accumulation Functions6.6 Applying Properties of Definite Integrals0/06.6.1 Evaluating Integrals Using Area and Geometry6.6.2 Linearity of Definite Integrals6.6.3 Integrals Over Adjacent Intervals6.6.4 Integrals with Removable or Jump Discontinuities6.7 The Fundamental Theorem of Calculus and Definite Integrals0/06.7.1 Understanding Antiderivatives6.7.2 Accumulation Functions as Antiderivatives6.7.3 Evaluating Definite Integrals Using FTC Part 26.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation0/06.8.1 Indefinite Integrals and Notation6.8.2 Reversing Basic Derivative Rules6.8.3 The Constant of Integration and Families of Curves6.8.4 Functions Without Elementary Antiderivatives6.9 Integrating Using Substitution0/06.9.1 Connecting u-Substitution to the Chain Rule6.9.2 Indefinite Integrals Using Substitution6.9.3 Definite Integrals and Changing Limits with Substitution6.10 Integrating Functions Using Long Division and Completing the Square0/06.10.1 Using Algebraic Long Division Before Integrating6.10.2 Completing the Square to Simplify Integrals6.10.3 Choosing Algebraic Techniques for Integration6.11 Integrating Using Integration by Parts0/06.11.1 Deriving the Integration by Parts Formula6.11.2 Choosing u and dv in Integration by Parts6.11.3 Definite Integrals with Integration by Parts6.12 Integrating Using Linear Partial Fractions0/06.12.1 Decomposing Rational Functions into Linear Factors6.12.2 Integrating Decomposed Rational Functions6.12.3 Applying Partial Fractions to Definite Integrals6.13 Evaluating Improper Integrals0/06.13.1 Recognizing Improper Integrals6.13.2 Using Limits to Evaluate Improper Integrals6.13.3 Convergent and Divergent Improper Integrals6.14 Selecting Techniques for Antidifferentiation0/06.14.1 Classifying Integrands by Type6.14.2 Choosing Between Basic Rules and Substitution6.14.3 Deciding When to Use Algebraic Rearrangements6.14.4 Strategy Review for Antiderivatives and Definite Integrals7. Differential Equations 7.1 Modeling Situations with Differential Equations 0/07.1.1 Understanding Differential Equations7.1.2 Identifying Dependent and Independent Variables in Context7.1.3 Writing Differential Equations from Verbal Descriptions7.1.4 Interpreting Units and Meaning of Derivative Expressions7.2 Verifying Solutions for Differential Equations0/07.2.1 Checking Solutions Using Derivatives7.2.2 General Solutions and Families of Functions7.2.3 Relating General and Particular Solutions7.3 Sketching Slope Fields0/07.3.1 Introduction to Slope Fields7.3.2 Constructing Slope Fields by Hand7.3.3 Interpreting Solution Curves from a Slope Field7.3.4 Using Technology to Explore Slope Fields7.4 Reasoning Using Slope Fields0/07.4.1 Matching Slope Fields to Differential Equations7.4.2 Describing Long‑Term Behavior from Slope Fields7.4.3 Initial Conditions and Particular Solution Curves7.5 Approximating Solutions Using Euler’s Method (BC only)0/07.5.1 Idea Behind Euler’s Method7.5.2 Constructing an Euler’s Method Table7.5.3 Step Size and Accuracy in Euler’s Method7.6 Finding General Solutions Using Separation of Variables0/07.6.1 Recognizing Separable Differential Equations7.6.2 Separating Variables and Integrating Both Sides7.6.3 Solving for y and Interpreting the Constant of Integration7.6.4 Verifying General Solutions by Differentiation7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables0/07.7.1 Using Initial Conditions to Determine Constants7.7.2 Accumulation Functions as Particular Solutions7.7.3 Domain Restrictions for Solutions7.7.4 Interpreting Particular Solutions in Context7.8 Exponential Models with Differential Equations0/07.8.1 From Verbal Description to Exponential Differential Equation7.8.2 Solving dy/dt = ky and the Form y = y₀e^{kt}7.8.3 Growth versus Decay and Graph Behavior7.8.4 Applying Exponential Models to Real‑World Contexts7.8.5 Using Data to Determine Parameters in Exponential Models7.9 Logistic Models with Differential Equations (BC only)0/07.9.1 Introduction to Logistic Growth and Carrying Capacity7.9.2 Writing Logistic Differential Equations from Context7.9.3 Interpreting Logistic Solutions without Solving the Equation7.9.4 Long‑Term Behavior and Carrying Capacity7.9.5 Maximum Growth Rate and Inflection Point7. Differential Equations 7.1 Modeling Situations with Differential Equations 0/07.1.1 Understanding Differential Equations7.1.2 Identifying Dependent and Independent Variables in Context7.1.3 Writing Differential Equations from Verbal Descriptions7.1.4 Interpreting Units and Meaning of Derivative Expressions7.2 Verifying Solutions for Differential Equations0/07.2.1 Checking Solutions Using Derivatives7.2.2 General Solutions and Families of Functions7.2.3 Relating General and Particular Solutions7.3 Sketching Slope Fields0/07.3.1 Introduction to Slope Fields7.3.2 Constructing Slope Fields by Hand7.3.3 Interpreting Solution Curves from a Slope Field7.3.4 Using Technology to Explore Slope Fields7.4 Reasoning Using Slope Fields0/07.4.1 Matching Slope Fields to Differential Equations7.4.2 Describing Long‑Term Behavior from Slope Fields7.4.3 Initial Conditions and Particular Solution Curves7.5 Approximating Solutions Using Euler’s Method (BC only)0/07.5.1 Idea Behind Euler’s Method7.5.2 Constructing an Euler’s Method Table7.5.3 Step Size and Accuracy in Euler’s Method7.6 Finding General Solutions Using Separation of Variables0/07.6.1 Recognizing Separable Differential Equations7.6.2 Separating Variables and Integrating Both Sides7.6.3 Solving for y and Interpreting the Constant of Integration7.6.4 Verifying General Solutions by Differentiation7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables0/07.7.1 Using Initial Conditions to Determine Constants7.7.2 Accumulation Functions as Particular Solutions7.7.3 Domain Restrictions for Solutions7.7.4 Interpreting Particular Solutions in Context7.8 Exponential Models with Differential Equations0/07.8.1 From Verbal Description to Exponential Differential Equation7.8.2 Solving dy/dt = ky and the Form y = y₀e^{kt}7.8.3 Growth versus Decay and Graph Behavior7.8.4 Applying Exponential Models to Real‑World Contexts7.8.5 Using Data to Determine Parameters in Exponential Models7.9 Logistic Models with Differential Equations (BC only)0/07.9.1 Introduction to Logistic Growth and Carrying Capacity7.9.2 Writing Logistic Differential Equations from Context7.9.3 Interpreting Logistic Solutions without Solving the Equation7.9.4 Long‑Term Behavior and Carrying Capacity7.9.5 Maximum Growth Rate and Inflection Point8. Applications of Integration 8.1 Finding the Average Value of a Function on an Interval0/08.1.1 Understanding the Average Value of a Function8.1.2 Formula for the Average Value Using Integrals8.1.3 Calculating Average Value from Formulas, Tables, and Graphs8.1.4 Interpreting Average Value in Real-World Contexts8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals0/08.2.1 Position Functions and Rectilinear Motion8.2.2 Displacement as the Integral of Velocity8.2.3 Total Distance Traveled as the Integral of Speed8.2.4 Motion Problems Interpreting Integrals of Velocity and Speed8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts0/08.3.1 Defining Accumulation Functions with Integrals8.3.2 Interpreting Graphs and Tables of Accumulation Functions8.3.3 Net Change as the Integral of a Rate of Change8.3.4 Net Change in Real-World Applied Contexts8.3.5 Setting Up Integrals for Accumulation and Net Change8.4 Finding the Area Between Curves Expressed as Functions of x0/08.4.1 Area Under a Single Curve as a Definite Integral8.4.2 Area Between Two Functions of x8.4.3 Using Graphs and Intersection Points to Set Up Area Integrals8.4.4 Applied Area Problems Using Functions of x8.5 Finding the Area Between Curves Expressed as Functions of y0/08.5.1 Describing Regions Using Functions of y8.5.2 Area Formula with Right Function Minus Left Function8.5.3 Choosing Whether to Integrate with Respect to x or y8.6 Finding the Area Between Curves That Intersect at More Than Two Points0/08.6.1 Splitting Regions with Multiple Intersections into Subregions8.6.2 Using the Integral of the Absolute Value of a Difference8.6.3 Applied Area Problems with Changing Top and Bottom Curves8.7 Volumes with Cross Sections: Squares and Rectangles0/08.7.1 Solids with Known Square or Rectangular Cross Sections8.7.2 Volume Integrals for Square Cross Sections8.7.3 Rectangular Cross Sections and Applications8.8 Volumes with Cross Sections: Triangles and Semicircles0/08.8.1 Volume of Solids with Triangular Cross Sections8.8.2 Volume of Solids with Semicircular Cross Sections8.8.3 Volumes with Other Geometric Cross Sections8.8.4 Viewing Cross-Sectional Volume as Accumulated Area8.9 Volume with Disc Method: Revolving Around the x- or y-Axis0/08.9.1 Solids of Revolution Around Coordinate Axes8.9.2 Disc Method Formulas for Principal Axes8.9.3 Setting Up and Evaluating Disc Method Integrals8.10 Volume with Disc Method: Revolving Around Other Axes0/08.10.1 Revolving Regions Around Horizontal or Vertical Lines8.10.2 Writing Disc Method Integrals for Shifted Axes8.10.3 Comparing Different Lines of Revolution Using the Disc Method8.11 Volume with Washer Method: Revolving Around the x- or y-Axis0/08.11.1 Ring-Shaped Cross Sections and the Washer Idea8.11.2 Washer Method Volume Formula for Principal Axes8.11.3 Applying the Washer Method Around Coordinate Axes8.12 Volume with Washer Method: Revolving Around Other Axes0/08.12.1 Washers for Revolution Around Shifted Lines8.12.2 Washer Integrals for Horizontal or Vertical Lines8.12.3 Interpreting Graphs and Diagrams in Washer Problems8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC only)0/08.13.1 Arc Length as Accumulated Change in Position8.13.2 Using a Definite Integral to Find Arc Length8.13.3 Distance Traveled Along a Planar Curve (BC Only)8. Applications of Integration 8.1 Finding the Average Value of a Function on an Interval0/08.1.1 Understanding the Average Value of a Function8.1.2 Formula for the Average Value Using Integrals8.1.3 Calculating Average Value from Formulas, Tables, and Graphs8.1.4 Interpreting Average Value in Real-World Contexts8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals0/08.2.1 Position Functions and Rectilinear Motion8.2.2 Displacement as the Integral of Velocity8.2.3 Total Distance Traveled as the Integral of Speed8.2.4 Motion Problems Interpreting Integrals of Velocity and Speed8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts0/08.3.1 Defining Accumulation Functions with Integrals8.3.2 Interpreting Graphs and Tables of Accumulation Functions8.3.3 Net Change as the Integral of a Rate of Change8.3.4 Net Change in Real-World Applied Contexts8.3.5 Setting Up Integrals for Accumulation and Net Change8.4 Finding the Area Between Curves Expressed as Functions of x0/08.4.1 Area Under a Single Curve as a Definite Integral8.4.2 Area Between Two Functions of x8.4.3 Using Graphs and Intersection Points to Set Up Area Integrals8.4.4 Applied Area Problems Using Functions of x8.5 Finding the Area Between Curves Expressed as Functions of y0/08.5.1 Describing Regions Using Functions of y8.5.2 Area Formula with Right Function Minus Left Function8.5.3 Choosing Whether to Integrate with Respect to x or y8.6 Finding the Area Between Curves That Intersect at More Than Two Points0/08.6.1 Splitting Regions with Multiple Intersections into Subregions8.6.2 Using the Integral of the Absolute Value of a Difference8.6.3 Applied Area Problems with Changing Top and Bottom Curves8.7 Volumes with Cross Sections: Squares and Rectangles0/08.7.1 Solids with Known Square or Rectangular Cross Sections8.7.2 Volume Integrals for Square Cross Sections8.7.3 Rectangular Cross Sections and Applications8.8 Volumes with Cross Sections: Triangles and Semicircles0/08.8.1 Volume of Solids with Triangular Cross Sections8.8.2 Volume of Solids with Semicircular Cross Sections8.8.3 Volumes with Other Geometric Cross Sections8.8.4 Viewing Cross-Sectional Volume as Accumulated Area8.9 Volume with Disc Method: Revolving Around the x- or y-Axis0/08.9.1 Solids of Revolution Around Coordinate Axes8.9.2 Disc Method Formulas for Principal Axes8.9.3 Setting Up and Evaluating Disc Method Integrals8.10 Volume with Disc Method: Revolving Around Other Axes0/08.10.1 Revolving Regions Around Horizontal or Vertical Lines8.10.2 Writing Disc Method Integrals for Shifted Axes8.10.3 Comparing Different Lines of Revolution Using the Disc Method8.11 Volume with Washer Method: Revolving Around the x- or y-Axis0/08.11.1 Ring-Shaped Cross Sections and the Washer Idea8.11.2 Washer Method Volume Formula for Principal Axes8.11.3 Applying the Washer Method Around Coordinate Axes8.12 Volume with Washer Method: Revolving Around Other Axes0/08.12.1 Washers for Revolution Around Shifted Lines8.12.2 Washer Integrals for Horizontal or Vertical Lines8.12.3 Interpreting Graphs and Diagrams in Washer Problems8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC only)0/08.13.1 Arc Length as Accumulated Change in Position8.13.2 Using a Definite Integral to Find Arc Length8.13.3 Distance Traveled Along a Planar Curve (BC Only)
