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Big Idea 1: Limits
Essential Understanding 1.1 Students will understand that the concept of a limit can be used to understand the behavior of functions.
Learning Objective 1.1A(a) Students will be able to express limits symbolically using correct notation and (b) Interpret limits expressed symbolically.
Essential Knowledge 1.1A1 Students will know that given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is
.
Essential Knowledge 1.1A2 Students will know that the concept of a limit can be extended to include one-sided limits, limits at infinity, and infinite limits.
Essential Knowledge 1.1A3 Students will know that a limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.
Learning Objective 1.1B Students will be able to estimate limits of functions.
Essential Knowledge 1.1B1 Students will know that numerical and graphical information can be used to estimate limits.
Learning Objective 1.1C Students will be able to determine limits of functions.
Essential Knowledge 1.1C1 Students will know that limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules
Essential Knowledge 1.1C2 Students will know that the limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions, or the squeeze theorem
Essential Knowledge 1.1C3 Students will know that limits of the indeterminate forms and may be evaluated using L'HĂ´pital's Rule.
Learning Objective 1.1D Students will be able to deduce and interpret behavior of functions using limits.
Essential Knowledge 1.1D1 Students will know that asymptotic and unbounded behavior of functions can be explained and described using limits.
Essential Knowledge 1.1D2 Students will know that relative magnitudes of functions and their rates of change can be compared using limits.
Essential Understanding 1.2 Students will understand that continuity is a key property of functions that is defined using limits.
Learning Objective 1.2A Students will be able to analyze functions for intervals of continuity or points of discontinuity.
Essential Knowledge 1.2A1 Students will know that a function f is continuous at x = c provided that f(c) exists, exists, and
.
Essential Knowledge 1.2A2 Students will know that polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous at all points in their domains.
Essential Knowledge 1.2A3 Students will know that types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.
Learning Objective 1.2B Students will be able to determine the applicability of important calculus theorems using continuity.
Essential Knowledge 1.2B1 Students will know that continuity is an essential condition for theorems such as the Intermediate Value Theorem, the Extreme Value Theorem, and the Mean Value Theorem.
Big Idea 2: Derivatives
Essential Understanding 2.1 Students will understand that the derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.
Learning Objective 2.1A Students will be able to identify the derivative of a function as the limit of a difference quotient.
Essential Knowledge 2.1A1 Students will know that the difference quotients
and express the average rate of change of a function over an interval.
Essential Knowledge 2.1A2 Students will know that the instantaneous rate of change of a function at a point can be expressed by
or , provided that the limit exists. These are common forms of the definition of the derivative and are denoted f '(a).
Essential Knowledge 2.1A3 Students will know that the derivative of f is the function whose value at x is provided this limit exists.
Essential Knowledge 2.1A4 Students will know that for y = f(x), notations for the derivative include , f '(x), and y'.
Essential Knowledge 2.1A5 Students will know that the derivative can be represented graphically, numerically, analytically, and verbally.
Learning Objective 2.1B Students will be able to estimate derivatives.
Essential Knowledge 2.1B1 Students will know that the derivative at a point can be estimated from information given in tables or graphs.
Learning Objective 2.1C Students will be able to calculate derivatives.
Essential Knowledge 2.1C1 Students will know that direct application of the definition of the derivative can be used to find the derivative for selected functions, including polynomial, power, sine, cosine, exponential, and logarithmic functions.
Essential Knowledge 2.1C2 Students will know that specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.
Essential Knowledge 2.1C3 Students will know that sums, differences, products, and quotients of functions can be differentiated using derivative rules.
Essential Knowledge 2.1C4 Students will know that the chain rule provides a way to differentiate composite functions.
Essential Knowledge 2.1C5 Students will know that the chain rule is the basis for implicit differentiation.
Essential Knowledge 2.1C6 Students will know that the chain rule can be used to find the derivative of an inverse function, provided the derivative of that function exists.
Learning Objective 2.1D Students will be able to determine higher order derivatives.
Essential Knowledge 2.1D1 Students will know that differentiating f ' produces the second derivative f '', provided the derivative of f ' exists; repeating this process produces higher order derivatives of f.
Essential Knowledge 2.1D2 Students will know that higher order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include
, f ''(x), and y''. Higher order derivatives can be denoted or f (n)(x).
Essential Understanding 2.2 Students will understand that a function's derivative, which is itself a function, can be used to understand the behavior or the function.
Learning Objective 2.2A Students will be able to use derivatives to analyze properties of a function.
Essential Knowledge 2.2A1 Students will know that first and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection.
Essential Knowledge 2.2A2 Students will know that key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.
Essential Knowledge 2.2A3 Students will know that key features of the graphs of f, f ', and f '' are related to one another.
Learning Objective 2.2B Students will be able to recognize the connection between differentiability and continuity.
Essential Knowledge 2.2B1 Students will know that a continuous function may fail to be differentiable at a point in its domain.
Essential Knowledge 2.2B2 Students will know that if a function is differentiable at a point, then it is continuous at that point.
Essential Understanding 2.3 Students will understand that the derivative has multiple interpretations and applications including those that involve instantaneous rates of change.
Learning Objective 2.3A Students will be able to interpret the meaning of a derivative within a problem.
Essential Knowledge 2.3A1 Students will know that the unit for f '(x) is the unit for f divided by the unit for x.
Essential Knowledge 2.3A2 Students will know that the derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable.
Learning Objective 2.3B Students will be able to solve problems involving the slope of a tangent line.
Essential Knowledge 2.3B1 Students will know that the derivative at a point is the slope of the line tangent to a graph at that point on the graph.
Essential Knowledge 2.3B2 Students will know that the tangent line is the graph of a locally linear approximation of the function near the point of tangency.
Learning Objective 2.3C Students will be able to solve problems involving related rates, optimization, and rectilinear motion.
Essential Knowledge 2.3C1 Students will know that the derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration.
Essential Knowledge 2.3C2 Students will know that the derivative can be used to solve related rates problems, that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known.
Essential Knowledge 2.3C3 Students will know that the derivative can be used to solve optimization problems, that is, finding a maximum or minimum value of a function over a given interval.
Learning Objective 2.3D Students will be able to solve problems involving rates of change in applied contexts.
Essential Knowledge 2.3D1 Students will know that the derivative can be used to express information about rates of change in applied contexts.
Learning Objective 2.3E Students will be able to verify solutions to differential equations.
Essential Knowledge 2.3E1 Students will know that solutions to differential equations are functions or families of functions.
Essential Knowledge 2.3E2 Students will know that derivatives can be used to verify that a function is a solution to a given differential equation.
Learning Objective 2.3F Students will be able to estimate solutions to differential equations.
Essential Knowledge 2.3F1 Students will know that slope fields provide visual clues to the behavior of solutions to first order differential equations.
Essential Understanding 2.4 Students will understand that the Mean Value Theorem connects the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval.
Learning Objective 2.4A Students will be able to apply the Mean Value Theorem to describe the behavior of a function over an interval.
Essential Knowledge 2.4A1 Students will know that if a function f is continuous over the interval [a, b] and differentiable over the interval (a, b), the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval.
Big Idea 3: Integrals and the Fundamental Theorem of Calculus
Essential Understanding 3.1 Students will understand that antidifferentiation is the inverse process of differentiation.
Learning Objective 3.1A Students will be able to recognize antiderivatives of basic functions.
Essential Knowledge 3.1A1 Students will know that an antiderivative of a function f is a function g whose derivative is f.
Essential Knowledge 3.1A2 Students will know that differentiation rules provide the foundation for finding antiderivatives.
Essential Understanding 3.2 Students will understand that the definite integral of a function over an interval is the limit of a Riemann sum over that interval and can be calculated using a variety of strategies.
Learning Objective 3.2A(a) Students will be able to interpret the definite integral as the limit of a Riemann sum and (b) express the limit of a Riemann sum in integral notation.
Essential Knowledge 3.2A1 Students will know that a Riemann sum, which requires the partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.
Essential Knowledge 3.2A2 Students will know that the definite integral of a continuous function f, over the interval [a, b], denoted by
, is the limit of Riemann sums as the widths of the subintervals approach 0. That is,
where xi* is a value in the ith subinterval, and Δxi is the width of the ith subinterval, n is the number of subintervals, and max Δxi is the width of the largest subinterval. Another form of the definition is
, where
and xi* is a value in the ith subinterval.
Essential Knowledge 3.2A3 Students will know that the information in a definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.
Learning Objective 3.2B Students will be able to approximate a definite integral.
Essential Knowledge 3.2B1 Students will know that definite integrals can be approximated for functions that are represented graphically, numerically, algebraically, and verbally.
Essential Knowledge 3.2B2 Students will know that definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions.
Learning Objective 3.2C Students will be able to calculate a definite integral using areas and properties of definite integrals.
Essential Knowledge 3.2C1 Students will know that in some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area.
Essential Knowledge 3.2C2 Students will know that properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals.
Essential Knowledge 3.2C3 Students will know that the definition of the definite integral may be extended to functions with removable or jump discontinuities.
Essential Understanding 3.3 Students will understand that the Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration.
Learning Objective 3.3A Students will be able to analyze functions defined by an integral.
Essential Knowledge 3.3A1 Students will know that the definite integral can be used to define new functions; for example,
Essential Knowledge 3.3A2 Students will know that a if f is a continuous function on the interval [a, b], then , where x is between a and b.
.
Essential Knowledge 3.3A3 Students will know that graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as
.
Learning Objective 3.3B(a) Students will be able to calculate antiderivatives and (b) evaluate definite integrals.
Essential Knowledge 3.3B1 Students will know that the function defined by , is an antiderivative of f.
Essential Knowledge 3.3B2 Students will know that if f is continuous on the interval [a, b] and F is an antiderivative of f, then
.
Essential Knowledge 3.3B3 Students will know that the notation means that F'(x) = f(x), and is called an indefinite integral of the function f.
Essential Knowledge 3.3B4 Students will know that many functions do not have closed form antiderivatives.
Essential Knowledge 3.3B5 Techniques for finding antiderivatives include algebraic manipulation (such as long division and completing the square) and substitution of variables.
Essential Understanding 3.4 Students will understand that the definite integral of a function over an interval is a mathematical tool with many interpretations and applications involving accumulation.
Learning Objective 3.4A Students will be able to interpret the meaning of a definite integral within a problem.
Essential Knowledge 3.4A1 Students will know that a function defined as an integral represents an accumulation of a rate of change.
Essential Knowledge 3.4A2 Students will know that the definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval.
Essential Knowledge 3.4A3 Students will know that the limit of an approximating Riemann sum can be interpreted as a definite integral.
Learning Objective 3.4B Students will be able to apply definite integrals to problems involving the average value of a function.
Essential Knowledge 3.4B1 Students will know that the average value of a function f over an interval [a, b] is .
Learning Objective 3.4C Students will be able to apply definite integrals to problems involving motion.
Essential Knowledge 3.4C1 Students will know that for a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle's displacement over the interval of time, and the definite integral of speed represents the particle's total distance traveled over the interval of time.
Learning Objective 3.4D Students will be able to apply definite integrals to problems involving area and volume.
Essential Knowledge 3.4D1 Students will know that areas of certain regions in the plane can be calculated with definite integrals.
Essential Knowledge 3.4D2 Students will know that volumes of solids with known cross sections, including discs and washers, can be calculated with definite integrals.
Learning Objective 3.4E Students will be able to use the definite integral to solve problems in various contexts.
Essential Knowledge 3.4E1 Students will know that the definite integral can be used to express information about accumulation and net change in many applied contexts.
Essential Understanding 3.5 Students will understand that antidifferentiation is an underlying concept involved in solving separable differential equations. Solving separable differential equations involves determining a function or relation give its rate of change.
Learning Objective 3.5A Students will be able to analyze differential equations to obtain general and specific solutions.
Essential Knowledge 3.5A1 Students will know that antidifferentiation can be used to find specific solutions to differential equations with given initial conditions, including applications to motion along a line, and exponential growth and decay.
Essential Knowledge 3.5A2 Students will know that some differential equations can be solved by separation of variables.
Essential Knowledge 3.5A3 Students will know that solutions to differential equations may be subject to domain restrictions.
Essential Knowledge 3.5A4 Students will know that the function F defined by is a general solution to the differential equation
, and is a particular solution to the equation satisfying F(a) = y0.
Learning Objective 3.5B Students will be able to analyze differential equations to obtain general and specific solutions.
Essential Knowledge 3.5B1 Students will know that the model for exponential growth that arises from the statement "The rate of change of a quantity is proportional to the size of the quantity" is
.
Big ideas: The course is organized around "Big Ideas", which correspond to foundational concepts of calculus: limits, derivatives, and integrals and the Fundamental Theorem of Calculus.
Enduring understandings: Within each big idea are enduring understandings. These are the long-term takeaways related to the big ideas that a student should have after exploring the content and skills. These understandings are expressed as generalizations that specify what a student will come to understand about the key concepts in each course. Enduring understandings are labeled to correspond with the appropriate big idea.
Learning objectives: Linked to each enduring understanding are the corresponding learning objectives. The learning objectives convey what a student needs to be able to do in order to develop the enduring understandings. The learning objectives serve as targets of assessment for each course. Learning objectives are labeled to correspond with the appropriate big idea and enduring understanding.
Essential knowledge: Essential knowledge statements describe the facts and basic concepts that a student should know and be able to recall in order to demonstrate mastery of each learning objective. Essential knowledge statements are labeled to correspond with the appropriate big idea, enduring understanding, and learning objective.
Further clarification regarding the content covered in AP Calculus is provided by examples and exclusion statements. Examples are provided to address potential inconsistencies among definitions given by various sources. Exclusion statements identify topics that may be covered in a first-year college calculus course but are not assessed on the AP Calculus AB or BC Exam.
Big Idea 1: Limits
Many calculus concepts are developed by first considering a discrete model and then the consequences of a limiting case. Therefore, the idea of limits is essential for discovering and developing important ideas, definitions, formulas, and theorems in calculus. Students must have a solid, intuitive understanding of limits and be able to compute various limits, including one-sided limits, limits at infinity, the limit of a sequence, and infinite limits. They should be able to work with tables and graphs in order to estimate the limit of a function at a point. Students should know the algebraic properties of limits and techniques for finding limits of indeterminate forms, and they should be able to apply limits to understand the behavior of a function near a point. Students must also understand how limits are used to determine continuity, a fundamental property of functions.
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