The Courses
An AP course in calculus consists of a full high school academic year of work that is comparable to calculus courses in colleges and universities. It is expected that students who take an AP course in calculus will seek college credit, college placement, or both, from institutions of higher learning.
Calculus AB and Calculus BC are primarily concerned with developing the students' understanding of the concepts of calculus and providing experience with its methods and applications. The courses emphasize a multi-representational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections among these representations also are important.
Calculus BC is an extension of Calculus AB rather than an enhancement; common topics require a similar depth of understanding. Both courses are intended to be challenging and demanding.
Broad concepts and widely applicable methods are emphasized. The focus of the courses is neither manipulation nor memorization of an extensive taxonomy of functions, curves, theorems, or problem types. Thus, although facility with manipulation and computational competence are important outcomes, they are not the core of these courses.
Through the use of the unifying themes of derivatives, integrals, limits, approximation, and applications and modeling, the course becomes a cohesive whole rather than a collection of unrelated topics. These themes are developed using all the functions listed in the prerequisites.
Goals
- Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.
- Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems.
- Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems.
- Students should understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
- Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.
- Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral.
- Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions.
- Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
- Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.
Prerequisites
Before studying calculus, all students should complete four years of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the values of the trigonometric functions of the numbers: 0, π/6, π/4, π/3, π/2 and their multiples.
Topic Outline for Calculus AB
- Functions, Graphs, and Limits
Analysis of graphs, Limits of functions (including one-sided limits); Asymptotic and unbounded behavior; Continuity as a property of functions
- Derivatives
Concept of the derivative; Derivative at a point; Derivative as a function; Second derivatives; Applications of derivatives; Computation of derivatives
- Integrals
Interpretations and properties of definite integrals; Applications of integrals; Fundamental Theorem of Calculus; Techniques of antidifferentiation; Applications of antidifferentiation; Numerical approximations to definite integrals
Topic Outline for Calculus BC
- Functions, Graphs, and Limits
Analysis of graphs, Limits of functions (including one-sided limits); Asymptotic and unbounded behavior; Continuity as a property of functions; Parametric, polar, and vector functions
- Derivatives
Concept of the derivative; Derivative at a point; Derivative as a function; Second derivatives; Applications of derivatives; Computation of derivatives
- Integrals
Interpretations and properties of definite integrals; Applications of integrals; Fundamental Theorem of Calculus; Techniques of antidifferentiation; Applications of antidifferentiation; Numerical approximations to definite integrals
- Polynomial Approximations and Series
Concept of series; Series of constants; Taylor series
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* Denotes college credit granted via the St. John's College Extension Program.
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