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The topical outline for Calculus BC includes all Calculus AB topics. Additional topics are found in paragraphs that are marked with a plus sign (+) or an asterisk (*). The additional topics can be taught anywhere in the course that the instructor wishes. Some topics will naturally fit immediately after their Calculus AB counterparts (+). Other topics may fit best after the completion of the Calculus AB syllabus(*). Although the examination is based on the topics listed in the topical outline, teachers may wish to enrich their courses with additional topics.

The NCSSM Calculus Text, Contemporary Calculus through Applications, emphasizes functions as models. This theme runs throughout the course.

Limits of functions (including one-sided limits)
An intuitive understanding of the limiting process is sufficient for this course.

• Calculating limits using algebra CCTA: 2.6, 2.7
• Estimating limits from graphs or tables of data CCTA: 2.5, Lab 5

Asymptotic and unbounded behavior

• Understanding asymptotes in terms of graphical behavior (From Precalculus)
• Describing asymptotic behavior in terms of limits involving infinity (From Precalculus)
• Comparing relative magnitudes of functions and their rates of change (For example, contrasting exponential growth, polynomial growth, and logarithmic growth) CCTA: Lab 1
• Continuity as a property of functions. The central idea of continuity is that close values of the domain lead to close values of the range CCTA: 2.7
• Understanding continuity in terms of limits CCTA: 2.7
• Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)
• *Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form. CCTA: 4.6

Derivative at a Point

• Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents CCTA: 2.3, 2.4, 4.6
• Tangent line to a curve at a point and local linear approximation CCTA 2.3, Lab 2
• Instantaneous rate of change as the limit of average rate of change CCTA 2.1-2.2
• Approximate rate of change from graphs and tables of values CCTA 2.1

Derivative as a Function

• Corresponding characteristics of graphs of f and f' CCTA 2.1-2.3, Lab 1
• Relationship between increasing and decreasing behavior of f and the sign of f' CCTA 3.1, Lab 6
• The Mean Value Theorem and its geometric consequences CCTA 6 Review
• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa CCTA Chapters 1-3

Second derivatives

• Corresponding characteristics of the graphs of f , f' and f" CCTA 3.1, Lab 6
• Relationship between the concavity of f and the sign of f' CCTA 3.1, Lab 6
• Points of inflection as places where concavity changes CCTA 3.2

Applications of Derivatives

• Analysis of curves, including the notions of monotonicity and concavity CCTA 3.1,3.2
• +Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors CCTA 4.7
• Optimization, both absolute (global) and relative (local) extreme CCTA 3.2-3.4
• Modeling rates of change, including related rates problems CCTA 3.6
• Use of implicit differentiation to find the derivative of an inverse function CCTA 3.7
• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration CCTA 4.4, Lab 8
• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions CCTA 4.1-4.3
• +Numerical solution of differential equations using Euler's method. CCTA 4.3, Lab 7
• +L'Hopital's Rule and its use in determining convergence of improper integrals and series CCTA 2.6, Lab 5, 6.4

Computation of derivatives

• Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions CCTA 2.2-2.6, Lab 1
• Basic rules for the derivative of sums, products, and quotients of functions CCTA 2.5, Lab 3
• Chain rule and implicit differentiation CCTA 2.5, Lab 4, 3.7
• +Derivatives of parametric, polar, and vector functions CCTA 4.7, Lab 9, 6.13, Lab 15

Interpretations and properties of definite integrals

• Definite integral as a limit of Riemann sums CCTA 5.1
• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: CCTA 5.1
• Basic properties of definite integrals (For example, additivity and linearity).

*Applications of integrals
Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve (including a curve given in parametric form). CCTA Chapters 5 - 6

Fundamental Theorem of Calculus

• Use of the Fundamental Theorem to evaluate definite integrals CCTA 5.1
• Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined CCTA 5.5

Techniques of antidifferentiation

• Antiderivatives directly from derivatives of basic functions CCTA 5.1
• +Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only) CCTA 5.5-5.7
• +Improper integrals (as limits of definite integrals) CCTA 6.4

Applications of antidifferentiation

• Finding specific antiderivatives using initial conditions, including applications to motion along a line CCTA 5.6 5.9, Lab 12
• Solving separable differential equations and using them in modeling. In particular, studying the equation and exponential growth CCTA 5.6
• +Solving logistic differential equations and using them in modeling CCTA 5.6, Lab 12

Numerical approximations to definite integrals
Use of Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values CCTA 5.8

*Series of constants

• +Motivating examples including decimal expansion CCTA 7.1
• +Geometric series with applications CCTA 6.4, 7.1
• +The harmonic series CCTA 6.4, 7.1
• +Alternating series with error bound CCTA 7.1
• +Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series CCTA 6.5
• +The ratio test for convergence and divergence CCTA 7.1
• +Comparing series to test for convergence or divergence

*Taylor Series

• +Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve) CCTA 4.8, 7.1
• +The general Taylor series centered at x = a CCTA 7.1
• +Maclaurin series for the functions and CCTA 4.8, 7.1
• +Formal manipulation of Taylor series and shortcuts to computing Taylor series, including differentiation, antidifferentiation, and the formation of new series from known series CCTA 7.2
• +Functions defined by power series and radius of convergence CCTA 7.1-7.3
• +Lagrange error bound for Taylor polynomials

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