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AP Calculus BC Topic Outline

I. Functions, Graphs, and Limits

Analysis of graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

Limits of functions (including one-sided limits)

  • An intuitive understanding of the limiting process  Pre-Calculus Review 
  • Calculating limits using algebra  Pre-Calculus Review 
  • Estimating limits from graphs or tables of data  Pre-Calculus Review 

Asymptotic and unbounded behavior

  • Understanding asymptotes in terms of graphical behavior  Pre-Calculus Review 
  • Describing asymptotic behavior in terms of limits involving infinity  Pre-Calculus Review 
  • Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth)  Pre-Calculus Review 

Continuity as a property of functions

  • An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)  Pre-Calculus Review 
  • Understanding continuity in terms of limits  Pre-Calculus Review 
  • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)  Pre-Calculus Review (IVT is NEW) 

Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form.  Chapter 10/11 

II. Derivatives

Concept of the derivative

  • Derivative presented graphically, numerically, and analytically  Pre-Calculus Review 
  • Derivative interpreted as an instantaneous rate of change  Pre-Calculus Review 
  • Derivative defined as the limit of the difference quotient  Pre-Calculus Review 
  • Relationship between differentiability and continuity  Pre-Calculus Review 

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.  Pre-Calculus Review 
  • Tangent line to a curve at a point and local linear approximation  Pre-Calculus Review 
  • Instantaneous rate of change as the limit of average rate of change  Pre-Calculus Review 
  • Approximate rate of change from graphs and tables of values  Pre-Calculus Review 

Derivative as a function

  • Corresponding characteristics of graphs of ƒ and ƒ'  Pre-Calculus Review 
  • Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ'  Pre-Calculus Review 
  • The Mean Value Theorem and its geometric interpretation  Pre-Calculus Review (MVT is NEW) 
  • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.  Pre-Calculus Review 

Second derivatives

  • Corresponding characteristics of the graphs of ƒ, ƒ', and ƒ''  Pre-Calculus Review 
  • Relationship between the concavity of ƒ and the sign of ƒ'  Pre-Calculus Review 
  • Points of inflection as places where concavity changes  Pre-Calculus Review 

Applications of derivatives

  • Analysis of curves, including the notions of monotonicity and concavity  Pre-Calculus Review 
  • Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration  Chapter 10/11 
  • Optimization, both absolute (global) and relative (local) extrema  Pre-Calculus Review (Optimization is NEW) 
  • Modeling rates of change, including related rates problems  Pre-Calculus Review (Special Review of this Topic) 
  • Use of implicit differentiation to find the derivative of an inverse function  Pre-Calculus Review 
  • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration  Pre-Calculus Review 
  • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations  Chapter 6 
  • Numerical solution of differential equations using Euler’s method  Chapter 6 
  • L’Hospital’s Rule, including its use in determining limits  Chapter 8  and convergence of improper integrals and series  Chapter 9A 

Computation of derivatives

  • Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric,   Pre-Calculus Review  and inverse trigonometric functions  Chapter 5 
  • Derivative rules for sums, products, and quotients of functions  Pre-Calculus Review 
  • Chain rule and implicit differentiation  Pre-Calculus Review 
  • Derivatives of parametric, polar, and vector functions.  Chapter 10/11 

III. Integrals

Interpretations and properties of definite integrals

  • Definite integral as a limit of Riemann sums  Chapter 4 
  • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:  Chapter 4 
  • Basic properties of definite integrals (examples include additivity and linearity)  Chapter 4 

Applications of integrals - Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in 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 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   Chapter 4  
(including a region bounded by polar curves) 
 Chapter 10/11 , the volume of a solid with known cross sections  Chapter 7 , the average value of a function  Chapter 4 , the distance traveled by a particle along a line  Chapter 4 the length of a curve  Chapter 7  (including a curve given in parametric form)  Chapter 10/11  and accumulated change from a rate of change.  Chapter 4 

Fundamental Theorem of Calculus

  • Use of the Fundamental Theorem to evaluate definite integrals  Chapter 4 
  • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined  Chapter 4 

Techniques of antidifferentiation

  • Antiderivatives following directly from derivatives of basic functions  Chapter 4 
  • Antiderivatives by substitution of variables (including change of limits for definite integrals)  Chapter 4 ,  parts  Chapter 8 , and simple partial fractions (nonrepeating linear factors only)  Chapter 8 
  • Improper integrals (as limits of definite integrals)  Chapter 8 

Applications of antidifferentiation

  • Finding specific antiderivatives using initial conditions, including applications to motion along a line  Chapter 4 
  • Solving separable differential equations and using them in modeling (including the study of the equation y' = ky and exponential growth)  Chapter 6 
  • Solving logistic differential equations and using them in modeling  Chapter 6 

Numerical approximations to definite integrals

  • Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Chapter 4 

IV. Polynomial Approximations and Series

Concept of series - A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence and divergence.  Chapter 9A 

Series of constants

  • Motivating examples, including decimal expansion.  Chapter 9A 
  • Geometric series with applications.  Chapter 9A 
  • The harmonic series.  Chapter 9A 
  • Alternating series with error bound.  Chapter 9A 
  • 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.  Chapter 9A 
  • The ratio test for convergence and divergence.  Chapter 9A 
  • Comparing series to test for convergence or divergence.  Chapter 9A 

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).  Chapter 9B 
  • Maclaurin series and the general Taylor series centered at x = a.  Chapter 9B 
  • Maclaurin series for the functions ex , sin x, cos x, and 1/(1 - x)  Chapter 9B 
  • Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.  Chapter 9B 
  • Functions defined by power series.  Chapter 9B 
  • Radius and interval of convergence of power series.  Chapter 9B 
  • Lagrange error bound for Taylor polynomials.  Chapter 9B