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