**AP CALCULUS AB**

Defining limits and using limit notation: Limits and continuity

Estimating limit values from graphs: Limits and continuity

Estimating limit values from tables: Limits and continuity

Determining limits using algebraic properties of limits: limit properties: Limits and continuity

Determining limits using algebraic properties of limits: direct substitution: Limits and continuity

Determining limits using algebraic manipulation: Limits and continuity

Selecting procedures for determining limits: Limits and continuity

Determining limits using the squeeze theorem: Limits and continuity

Exploring types of discontinuities: Limits and continuity

Defining continuity at a point: Limits and continuity

Confirming continuity over an interval: Limits and continuity

Removing discontinuities: Limits and continuity

Connecting infinite limits and vertical asymptotes: Limits and continuity

Connecting limits at infinity and horizontal asymptotes: Limits and continuity

Working with the intermediate value theorem: Limits and continuity

**Differentiation: definition and basic derivative rules**

Defining average and instantaneous rates of change at a point: Differentiation: definition and basic derivative rules

Defining the derivative of a function and using derivative notation: Differentiation: definition and basic derivative rules

Estimating derivatives of a function at a point: Differentiation: definition and basic derivative rules

Connecting differentiability and continuity: determining when derivatives do and do not exist: Differentiation: definition and basic derivative rules

Applying the power rule: Differentiation: definition and basic derivative rules

Derivative rules: constant, sum, difference, and constant multiple: introduction: Differentiation: definition and basic derivative rules

Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule: Differentiation: definition and basic derivative rules

Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x): Differentiation: definition and basic derivative rules

The product rule: Differentiation: definition and basic derivative rules

The quotient rule: Differentiation: definition and basic derivative rules

Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions: Differentiation: definition and basic derivative rules

**Differentiation: composite, implicit, and inverse functions**

The chain rule: introduction: Differentiation: composite, implicit, and inverse functions

The chain rule: further practice: Differentiation: composite, implicit, and inverse functionsImplicit differentiation

Differentiation: composite, implicit, and inverse functionsDifferentiating inverse functions

Differentiation: composite, implicit, and inverse functionsDifferentiating inverse trigonometric functions

Differentiation: composite, implicit, and inverse functions

Selecting procedures for calculating derivatives: strategy

Differentiation: composite, implicit, and inverse functions

Selecting procedures for calculating derivatives multiple rules

Differentiation: composite, implicit, and inverse functions

Calculating higher-order derivatives

Differentiation: composite, implicit, and inverse functions

Further practice connecting derivatives and limits

**Contextual applications of differentiation**:

Interpreting the meaning of the derivative in context: Contextual applications of differentiation

Straight-line motion: connecting position, velocity, and acceleration: Contextual applications of differentiation

Rates of change in other applied contexts (non-motion problems): Contextual applications of differentiation

Introduction to related rates: Contextual applications of differentiation

Solving related rates problems: Contextual applications of differentiation

Approximating values of a function using local linearity and linearization: Contextual applications of differentiation

Using L’Hôpital’s rule for finding limits of indeterminate forms

**Applying derivatives to analyze functions**:

Using the mean value theorem: Applying derivatives to analyze functions

Extreme value theorem, global versus local extrema, and critical points: Applying derivatives to analyze functions

Determining intervals on which a function is increasing or decreasing: Applying derivatives to analyze functions

Using the first derivative test to find relative (local) extrema: Applying derivatives to analyze functions

Using the candidates test to find absolute (global) extrema: Applying derivatives to analyze functions

Determining concavity of intervals and finding points of inflection: graphical: Applying derivatives to analyze functions

Determining concavity of intervals and finding points of inflection: algebraic: Applying derivatives to analyze functions

Using the second derivative test to find extrema: Applying derivatives to analyze functions

Sketching curves of functions and their derivatives: Applying derivatives to analyze functions

Connecting a function, its first derivative, and its second derivative: Applying derivatives to analyze functions

Solving optimization problems: Applying derivatives to analyze functions

Exploring behaviors of implicit relations: Applying derivatives to analyze functions

Calculator-active practice

**Integration and accumulation of change**:

Exploring accumulations of change: Integration and accumulation of change

Approximating areas with Riemann sums: Integration and accumulation of change

Riemann sums, summation notation, and definite integral notation: Integration and accumulation of change

The fundamental theorem of calculus and accumulation functions: Integration and accumulation of change

Interpreting the behavior of accumulation functions involving area: Integration and accumulation of change

Applying properties of definite integrals: Integration and accumulation of change

The fundamental theorem of calculus and definite integrals: Integration and accumulation of change

Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule: Integration and accumulation of change

Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals: Integration and accumulation of change

Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals: Integration and accumulation of change

Integrating using substitution: Integration and accumulation of change

Integrating functions using long division and completing the squ

**Differential equations**:

Modeling situations with differential equations: Differential equations

Verifying solutions for differential equations: Differential equations

Sketching slope fields: Differential equations

Reasoning using slope fields: Differential equations

Finding general solutions using separation of variables: Differential equations

Finding particular solutions using initial conditions and separation of variables: Differential equations

Exponential models with differential equations

**Applications of integration**:

Finding the average value of a function on an interval: Applications of integration

Connecting position, velocity, and acceleration functions using integrals: Applications of integration

Using accumulation functions and definite integrals in applied contexts: Applications of integration

Finding the area between curves expressed as functions of x: Applications of integration

Finding the area between curves expressed as functions of y: Applications of integration

Finding the area between curves that intersect at more than two points: Applications of integration

Volumes with cross sections: squares and rectangles: Applications of integration

Volumes with cross sections: triangles and semicircles: Applications of integration

Volume with disc method: revolving around x- or y-axis: Applications of integration

Volume with disc method: revolving around other axes: Applications of integration

Volume with washer method: revolving around x- or y-axis: Applications of integration

Volume with washer method: revolving around other axes: Applications of integration

Calculator-active practice

**AP Calculus BC**

**Limits and continuity**:

Defining limits and using limit notation: Limits and continuity

Estimating limit values from graphs: Limits and continuity

Estimating limit values from tables: Limits and continuity

Determining limits using algebraic properties of limits: limit properties: Limits and continuity

Determining limits using algebraic properties of limits: direct substitution: Limits and continuity

Determining limits using algebraic manipulation: Limits and continuity

Selecting procedures for determining limits: Limits and continuity

Determining limits using the squeeze theorem: Limits and continuity

Exploring types of discontinuities: Limits and continuity

Defining continuity at a point: Limits and continuity

Confirming continuity over an interval: Limits and continuity

Removing discontinuities: Limits and continuity

Connecting infinite limits and vertical asymptotes: Limits and continuity

Connecting limits at infinity and horizontal asymptotes: Limits and continuity

Working with the intermediate value theorem

**Differentiation: definition and basic derivative rules**:

Defining average and instantaneous rates of change at a point: Differentiation: definition and basic derivative rules

Defining the derivative of a function and using derivative notation: Differentiation: definition and basic derivative rules

Estimating derivatives of a function at a point: Differentiation: definition and basic derivative rules

Connecting differentiability and continuity: determining when derivatives do and do not exist: Differentiation: definition and basic derivative rules

Applying the power rule: Differentiation: definition and basic derivative rules

Derivative rules: constant, sum, difference, and constant multiple: introduction: Differentiation: definition and basic derivative rules

Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule: Differentiation: definition and basic derivative rules

Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x): Differentiation: definition and basic derivative rules

The product rule: Differentiation: definition and basic derivative rules

The quotient rule: Differentiation: definition and basic derivative rules

Finding the derivatives of tangent, cotangent, secant, and/or co

**Differentiation: composite, implicit, and inverse functions**:

The chain rule: introduction: Differentiation: composite, implicit, and inverse functions

The chain rule: further practice: Differentiation: composite, implicit, and inverse functions

Implicit differentiation: Differentiation: composite, implicit, and inverse functions

Differentiating inverse functions: Differentiation: composite, implicit, and inverse functions

Differentiating inverse trigonometric functions: Differentiation: composite, implicit, and inverse functions

Selecting procedures for calculating derivatives: strategy: Differentiation: composite, implicit, and inverse functions

Selecting procedures for calculating derivatives: multiple rules: Differentiation: composite, implicit, and inverse functions

Calculating higher-order derivatives: Differentiation: composite, implicit, and inverse functions

Further practice connecting derivatives and limits

**Contextual applications of differentiation**:

Interpreting the meaning of the derivative in context: Contextual applications of differentiation

Straight-line motion: connecting position, velocity, and acceleration: Contextual applications of differentiation

Rates of change in other applied contexts (non-motion problems): Contextual applications of differentiation

Introduction to related rates: Contextual applications of differentiation

Solving related rates problems: Contextual applications of differentiation

Approximating values of a function using local linearity and linearization: Contextual applications of differentiation

Using L’Hôpital’s rule for finding limits of indeterminate forms

**Applying derivatives to analyze functions**:

Using the mean value theorem: Applying derivatives to analyze functions

Extreme value theorem, global versus local extrema, and critical points: Applying derivatives to analyze functions

Determining intervals on which a function is increasing or decreasing: Applying derivatives to analyze functions

Using the first derivative test to find relative (local) extrema: Applying derivatives to analyze functions

Using the candidates test to find absolute (global) extrema: Applying derivatives to analyze functions

Determining concavity of intervals and finding points of inflection: graphical: Applying derivatives to analyze functions

Determining concavity of intervals and finding points of inflection: algebraic: Applying derivatives to analyze functions

Using the second derivative test to find extrema: Applying derivatives to analyze functions

Sketching curves of functions and their derivatives: Applying derivatives to analyze functions

Connecting a function, its first derivative, and its second derivative: Applying derivatives to analyze functions

Solving optimization problems: Applying derivatives to analyze functions

Exploring behaviors of implicit relations: Applying derivatives to analyze functions

Calculator-active practice

**Integration and accumulation of change**:

Exploring accumulations of change: Integration and accumulation of change

Approximating areas with Riemann sums: Integration and accumulation of change

Riemann sums, summation notation, and definite integral notation: Integration and accumulation of change

The fundamental theorem of calculus and accumulation functions: Integration and accumulation of change

Interpreting the behavior of accumulation functions involving area: Integration and accumulation of change

Applying properties of definite integrals: Integration and accumulation of change

The fundamental theorem of calculus and definite integrals: Integration and accumulation of change

Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule: Integration and accumulation of change

Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals: Integration and accumulation of change

Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals: Integration and accumulation of change

Integrating using substitution: Integration and accumulation of change

Integrating functions using long division and completing the square: Integration and accumulation of change

Using integration by parts: Integration and accumulation of change

Integrating using linear partial fractions: Integration and accumulation of change

Evaluating improper integrals

**Differential equations**

Modeling situations with differential equations: Differential equations

Verifying solutions for differential equations: Differential equations

Sketching slope fields: Differential equations

Reasoning using slope fields: Differential equations

Approximating solutions using Euler’s method: Differential equations

Finding general solutions using separation of variables: Differential equations

Finding particular solutions using initial conditions and separation of variables: Differential equations

Exponential models with differential equations: Differential equations

Logistic models with differential equations

**Applications of integration**

Finding the average value of a function on an interval: Applications of integration

Connecting position, velocity, and acceleration functions using integrals: Applications of integration

Using accumulation functions and definite integrals in applied contexts: Applications of integration

Finding the area between curves expressed as functions of x: Applications of integration

Finding the area between curves expressed as functions of y: Applications of integration

Finding the area between curves that intersect at more than two points: Applications of integration

Volumes with cross sections: squares and rectangles: Applications of integration

Volumes with cross sections: triangles and semicircles: Applications of integration

Volume with disc method: revolving around x- or y-axis: Applications of integration

Volume with disc method: revolving around other axes: Applications of integration

Volume with washer method: revolving around x- or y-axis: Applications of integration

Volume with washer method: revolving around other axes: Applications of integration

The arc length of a smooth, planar curve and distance traveled: Applications of integration

Calculator-active practice

**Parametric equations, polar coordinates, and vector-valued functions**

Defining and differentiating parametric equations: Parametric equations, polar coordinates, and vector-valued functions

Second derivatives of parametric equations: Parametric equations, polar coordinates, and vector-valued functions

Finding arc lengths of curves given by parametric equations: Parametric equations, polar coordinates, and vector-valued functions

Defining and differentiating vector-valued functions: Parametric equations, polar coordinates, and vector-valued functions

Solving motion problems using parametric and vector-valued functions: Parametric equations, polar coordinates, and vector-valued functions

Defining polar coordinates and differentiating in polar form: Parametric equations, polar coordinates, and vector-valued functions

Finding the area of a polar region or the area bounded by a single polar curve: Parametric equations, polar coordinates, and vector-valued functions

Finding the area of the region bounded by two polar curves: Parametric equations, polar coordinates, and vector-valued functions

Calculator-active practice

**Infinite sequences and series**

Defining convergent and divergent infinite series: Infinite sequences and series

Working with geometric series: Infinite sequences and series

The nth-term test for divergence: Infinite sequences and series

Integral test for convergence: Infinite sequences and series

Harmonic series and p-series: Infinite sequences and series

Comparison tests for convergence: Infinite sequences and series

Alternating series test for convergence: Infinite sequences and series

Ratio test for convergence: Infinite sequences and series

Determining absolute or conditional convergence: Infinite sequences and series

Alternating series error bound: Infinite sequences and series

Finding Taylor polynomial approximations of functions: Infinite sequences and series

Lagrange error bound: Infinite sequences and series

Radius and interval of convergence of power series: Infinite sequences and series

Finding Taylor or Maclaurin series for a function: Infinite sequences and series

Representing functions as power series

**AP STATICS**

**Analyzing categorical data**

Analyzing one categorical variable: Analyzing categorical data

Two-way tables: Analyzing categorical data

Distributions in two-way tables: Analyzing categorical data

Mosaic plots: Analyzing categorical data

**Displaying and describing quantitative data**

Frequency tables and dot plots: Displaying and describing quantitative data

Histograms and stem-and-leaf plots: Displaying and describing quantitative data

Describing and comparing distributions

**Summarizing quantitative data**

Measuring center in quantitative data: Summarizing quantitative data

More on mean and median: Summarizing quantitative data

Measuring spread in quantitative data: Summarizing quantitative data

More on standard deviation (optional): Summarizing quantitative data

Box and whisker plots

**Modeling data distributions**

Percentiles (cumulative relative frequency): Modeling data distributions

Z-scores: Modeling data distributions

Effects of linear transformations: Modeling data distributions

Density curves: Modeling data distributions

Normal distributions and the empirical rule: Modeling data distributions

Normal distribution calculations

**Exploring bivariate numerical data**

Making and describing scatterplots: Exploring bivariate numerical data

Correlation coefficients: Exploring bivariate numerical data

Least-squares regression equations: Exploring bivariate numerical data

Assessing the fit in least-squares regression

**Study design**

Sampling and observational studies: Study design

Sampling methods: Study design

Types of studies (experimental vs. observational): Study design

Experiments

**Probability**

Randomness, probability, and simulation: Probability

Addition rule: Probability

Multiplication rule: Probability

Conditional probability

**Random variables**

Discrete random variables: Random variables

Continuous random variables: Random variables

Transforming random variables: Random variables

Combining random variables: Random variables

Binomial random variables: Random variables

Binomial mean and standard deviation formulas: Random variables

Geometric random variables

**Sampling distributions**

What is a sampling distribution?: Sampling distributions

Sampling distribution of a sample proportion: Sampling distributions

Sampling distributions for differences in sample proportions: Sampling distributions

Sampling distribution of a sample mean: Sampling distributions

Sampling distributions for differences in sample means

Confidence intervals

Introduction to confidence intervals: Confidence intervals

Confidence intervals for proportions: Confidence intervals

Confidence intervals for means

**Significance tests (hypothesis testing)**

The idea of significance tests: Significance tests (hypothesis testing)

Error probabilities and power: Significance tests (hypothesis testing)

Testing hypotheses about a proportion: Significance tests (hypothesis testing)

Testing hypotheses about a mean

Inference comparing two groups or populations

Confidence intervals for the difference between two proportions: Inference comparing two groups or populations

Testing the difference between two proportions: Inference comparing two groups or populations

Confidence intervals for the difference between two means: Inference comparing two groups or populations

Testing the difference between two means

**Chi-square tests for categorical data**

Chi-square goodness-of-fit tests: Chi-square tests for categorical data

Chi-square tests for relationships

**More on regression**

Inference about slope: More on regression

Transformations to achieve linearity