Grades 9, 10, 11, 12

Mathematical Practices

Course Content

Implementing Mathematical Processes

Connecting Representations

Justification

Communication and Notation

Limits and Continuity

Differentiation: Definition and Fundamental Properties

Differentiation: Composite, Implicit, and Inverse Functions

Contextual Applications of Dif ferentiation

Analytical Applications of Differentiation

Integration and Accumulation of Change

Differential Equations

Applications of Integration

Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Infinite Sequences and Series

Determine expressions and values using mathematical procedures and rules.

Identify the question to be answered or problem to be solved (not assessed).

Identify key and relevant information to answer a question or solve a problem (not assessed).

Identify an appropriate mathematical rule or procedure based on the classification of a given expression (e.g., Use the chain rule to find the derivative of a composite function).

Identify an appropriate mathematical rule or procedure based on the relationship between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process, anti-differentiation) to solve problems.

Apply appropriate mathematical rules or procedures, with and without technology.

Explain how an approximated value relates to the actual value.

Translate mathematical information from a single representation or across multiple representations.

Identify common underlying structures in problems involving different contextual situations.

Identify mathematical information from graphical, numerical, analytical, and/or verbal representations.

Identify a re-expression of mathematical information presented in a given representation.

Identify how mathematical characteristics or properties of functions are related in different representations.

Describe the relationships among different representations of functions and their derivatives.

Justify reasoning and solutions.

Apply technology to develop claims and conjectures (not assessed).

Identify an appropriate mathematical definition, theorem, or test to apply.

Confirm whether hypotheses or conditions of a selected definition, theorem, or test have been satisfied.

Apply an appropriate mathematical definition, theorem, or test.

Provide reasons or rationales for solutions and conclusions.

Explain the meaning of mathematical solutions in context.

Confirm that solutions are accurate and appropriate.

Use correct notation, language, and mathematical conventions to communicate results or solutions.

Use precise mathematical language.

Use appropriate units of measure.

Use appropriate mathematical symbols and notation (e.g., Represent a derivative using f'(x), y', and dy/dx).

Use appropriate graphing techniques.

Apply appropriate rounding procedures.

Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.

Interpret limits expressed in analytic notation.

Estimate limits of functions.

Determine the limits of functions using limit theorems.

Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.

Justify conclusions about continuity at a point using the definition.

Determine intervals over which a function is continuous.

Determine values of x or solve for parameters that make discontinuous functions continuous, if possible.

Interpret the behavior of functions using limits involving infinity.

Explain the behavior of a function on an interval using the Intermediate Value Theorem.

Determine average rates of change using difference quotients.

Represent the derivative of a function as the limit of a difference quotient.

Determine the equation of a line tangent to a curve at a given point.

Estimate derivatives.

Explain the relationship between differentiability and continuity.

Calculate derivatives of products and quotients of differentiable functions.

Interpret a limit as a definition of a derivative.

Calculate derivatives of compositions of differentiable functions.

Calculate derivatives of implicitly defined functions.

Calculate derivatives of inverse and inverse trigonometric functions.

Calculate derivatives of inverse and inverse trigonometric functions.

Determine higher order derivatives of a function.

Interpret the meaning of a derivative in context.

Calculate rates of change in applied contexts.

Interpret rates of change in applied contexts.

Calculate related rates in applied contexts.

Interpret related rates in applied contexts.

Approximate a value on a curve using the equation of a tangent line.

Determine limits of functions that result in indeterminate forms.

Justify conclusions about functions by applying the Mean Value Theorem over an interval.

Justify conclusions about functions by applying the Extreme Value Theorem.

Justify conclusions about the behavior of a function based on the behavior of its derivatives.

Calculate minimum and maximum values in applied contexts or analysis of functions.

Interpret minimum and maximum values calculated in applied contexts.

Determine critical points of implicit relations.

Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.

Interpret the meaning of areas associated with the graph of a rate of change in context.

Approximate a definite integral using geometric and numerical methods.

Interpret the limiting case of the Riemann sum as a definite integral.

Represent the limiting case of the Riemann sum as a definite integral.

Represent accumulation functions using definite integrals.

Calculate a definite integral using areas and properties of definite integrals.

Evaluate definite integrals analytically using the Fundamental Theorem of Calculus.

Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.

For integrands requiring substitution or rearrangements into equivalent forms:

For integrands requiring integration by parts:

For integrands requiring integration by linear partial fractions:

Evaluate an improper integral or determine that the integral diverges. (BC only)

Interpret verbal statements of problems as differential equations involving a derivative expression.

Verify solutions to differential equations.

Estimate solutions to differential equations.

Determine general solutions to differential equations.

Determine particular solutions to differential equations.

Interpret the meaning of a differential equation and its variables in context.

Determine general and particular solutions for problems involving differential equations in context.

Interpret the meaning of the logistic growth model in context. (BC only)

Determine the average value of a function using definite integrals.

Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion.

Interpret the meaning of a definite integral in accumulation problems.

Determine net change using definite integrals in applied contexts.

Calculate areas in the plane using the definite integral.

Calculate volumes of solids with known cross sections using definite integrals.

Calculate volumes of solids of revolution using definite integrals.

Determine the length of a curve in the plane defined by a function, using a definite integral. (BC only)

Calculate derivatives of parametric functions. (BC only)

Calculate derivatives of vector-valued functions. (BC only)

Calculate areas of regions defined by polar curves using definite integrals. (BC only)

Determine the length of a curve in the plane defined by parametric functions, using a definite integral. (BC only)

Determine a particular solution given a rate vector and initial conditions. (BC only)

Determine values for positions and rates of change in problems involving planar motion. (BC only)

Determine whether a series converges or diverges. (BC only)

Approximate the sum of a series. (BC only)

Represent a function at a point as a Taylor polynomial. (BC only) Approximate function values using a Taylor polynomial. (BC only)

Determine the error bound associated with a Taylor polynomial approximation. (BC only)

Determine the radius of convergence and interval of convergence for a power series. (BC only)

Represent a function as a Taylor series or a Maclaurin series. (BC only)

Interpret Taylor series and Maclaurin series. (BC only)

Represent a given function as a power series. (BC only)

Determine indefinite integrals.

Evaluate definite integrals.

Determine indefinite integrals. (BC only)

Evaluate definite integrals. (BC only)

Determine indefinite integrals. bc only

Evaluate definite integrals. (BC only)