Unit | Timeframe (Number of Classes) | Big Ideas (Statements or Essential Questions) | Major Learning Experiences from Unit |
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Functions and Models
| AB: 6 BC: 8 | Summary of essential themes of all pre-calculus topics essential to AP Calculus.
| Understanding different representations of parent functions and their transformations Graphing and calculating trigonometric functions and their derivatives. Interpreting and graphing exponential and logarithmic functions. Learning use of a graphing calculator.
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Limits
| AB: 6 BC: 6 | Can change occur at an instant? How does knowing the value of a limit, or that a limit does not exist, help you to make sense of interesting features of functions and their graphs? How do we close loopholes so that a conclusion about a function is always true?
| Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant. Represent limits analytically using correct notation. Interpret limits expressed in analytic notation. Estimate limits of functions 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. 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. 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. Interpret the behavior of functions using limits involving infinity Explain the behavior of a function on an interval using the Intermediate Value Theorem.
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The Derivative
| AB: 14 BC: 14 | How can a state determine the rate of change in high school graduates at a particular level of public investment in education (in graduates per dollar) based on a model for the number of graduates as a function of the state’s education budget? Why do mathematical properties and rules for simplifying and evaluating limits apply to differentiation? If you knew that the rate of change in high school graduates at a particular level of public investment in education (in graduates per dollar) was a positive number, what might that tell you about the number of graduates at that level of investment? If pressure experienced by a diver is a function of depth and depth is a function of time, how might we find the rate of change in pressure with respect to time?
| 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 familiar functions. Interpret a limit as a definition of a derivative. Calculate derivatives of products and quotients of differentiable functions. 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.
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Applications of Differentiation
| AB: 11 BC: 10 | How are problems about position, velocity, and acceleration of a particle in motion over time structurally similar to problems about the volume of a rising balloon over an interval of heights, the population of London over the 14th century, or the metabolism of a dose of medicine over time? Since certain indeterminate forms seem to actually approach a limit, how can we determine that limit, provided it exists? How might the Mean Value Theorem be used to justify a conclusion that you were speeding at some point on a certain stretch of highway, even without knowing the exact time you were speeding? § What additional information is included in a sound mathematical argument about optimization that a simple description of an equivalent answer lacks?
| 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. Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.
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Integration
| AB: 12 BC: 10 | Given information about a rate of population growth over time, how can we determine how much the population changed over a given interval of time? If compounding more often increases the amount in an account with a given rate of return and term, why doesn’t compounding continuously result in an infinite account balance, all other things being equal? How is integrating to find areas related to differentiating to find slopes?
| 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. 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: (a) Determine indefinite integrals. (b) Evaluate definite integrals.
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Applications of INtegration
| AB: 11 BC: 10 | How is finding the number of visitors to a museum over an interval of time based on information about the rate of entry similar to finding the area of a region between a curve and the x-axis?
| 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
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Techniques of INtegration
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BC: 7 | Given information about a rate of population growth over time, how can we determine how much the population changed over a given interval of time? If compounding more often increases the amount in an account with a given rate of return and term, why doesn’t compounding continuously result in an infinite account balance, all other things being equal? How is integrating to find areas related to differentiating to find slopes?
| For integrands requiring integration by parts: (a) Determine indefinite integrals. bc only (b) Evaluate definite integrals. bc only For integrands requiring integration by linear partial fractions: (a) Determine indefinite integrals. bc only (b) Evaluate definite integrals. bc only Evaluate an improper integral or determine that the integral diverges. bc only
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Differential Equations
| AB: 5 BC: 6 | How can we derive a model for the number of computers, C, infected by a virus, given a model for how fast the computers are being infected, dC dt , at a particular time?
| 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
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Infinite Sequences and Series
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BC: 21 | How can the sum of infinitely many discrete terms be a finite value or represent a continuous function?
| 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 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
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Parametric
| BC: 10 | How can we model motion not constrained to a linear path? How does the chain rule help us to analyze graphs defined using parametric equations or polar functions?
| Calculate derivatives of parametric functions. bc only Determine the length of a curve in the plane defined by parametric functions, using a definite integral. bc only Calculate derivatives of vector-valued functions. 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 Calculate derivatives of functions written in polar coordinates. bc only Calculate areas of regions defined by polar curves using definite integrals. bc only Calculate areas of regions defined by polar curves using definite integrals. bc only
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