OCG - AP Calculus BC

South Carolina College and Career- Ready (SCCCR) Calculus Standards:

Limits and Continuity
The student will:
C.LC.1 Understand the concept of a limit graphically, numerically, analytically, and contextually.
Estimate and verify limits using tables, graphs of functions, and technology.
Calculate limits, including one-sided limits, algebraically using direct substitution, simplification, rationalization, and the limit laws for constant multiples, sums, differences, products, and quotients.
Calculate infinite limits and limits at infinity. Understand that infinite limits and limits at infinity provide information regarding the asymptotes of certain functions, including rational, exponential and logarithmic functions.
C.LC.2 Understand the definition and graphical interpretation of continuity of a function.
Apply the definition of continuity of a function at a point to solve problems.
Classify discontinuities as removable, jump, or infinite. Justify that classification using the definition of continuity.
Understand the Intermediate Value Theorem and apply the theorem to prove the existence of solutions of equations arising in mathematical and real- world problems.

Derivatives
The student will:
C.D.1 Understand the concept of the derivative of a function geometrically, numerically, analytically, and verbally.
Interpret the value of the derivative of a function as the slope of the corresponding tangent line.
Interpret the value of the derivative as an instantaneous rate of change in a variety of real-world contexts such as velocity and population growth.
Approximate the derivative graphically by finding the slope of the tangent line drawn to a curve at a given point and numerically by using the difference quotient.
Understand and explain graphically and analytically the relationship between differentiability and continuity.
Explain graphically and analytically the relationship between the average rate of change and the instantaneous rate of change.
Understand the definition of the derivative and use this definition to determine the derivatives of various functions.
C.D.2 Apply the rules of differentiation to functions.
Know and apply the derivatives of constant, power, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
Use the constant multiple, sum, difference, product, quotient, and chain rules to find the derivatives of functions.
Understand and apply the methods of implicit and logarithmic differentiation.
C.D.3 Apply theorems and rules of differentiation to solve mathematical and real-world problems.
Explain geometrically and verbally the mathematical and real-world meanings of the Extreme Value Theorem and the Mean Value Theorem.
Write an equation of a line tangent to the graph of a function at a point.
Explain the relationship between the increasing/decreasing behavior of f and the signs of f′. Use the relationship to generate a graph of f given the graph of f′, and vice versa, and to identify relative and absolute extrema of f.
Explain the relationships among the concavity of the graph of f, the increasing/decreasing behavior of f′ and the signs of f′′. Use those relationships to generate graphs of f, f′, and f′′ given any one of them and identify the points of inflection of f.
Solve a variety of real-world problems involving related rates, optimization, linear approximation, and rates of change.

Integrals
The student will:
C.I.1 Understand the concept of the integral of a function geometrically, numerically, analytically, and contextually.
Explain how the definite integral is used to solve area problems.
Approximate definite integrals by calculating Riemann sums using left, right, and midpoint evaluations, and using trapezoidal sums.
Interpret the definite integral as a limit of Riemann sums.
Explain the relationship between the integral and derivative as expressed in both parts of the Fundamental Theorem of Calculus. Interpret the relationship in terms of rates of change.
C.I.2 Apply theorems and rules of integration to solve mathematical and real-world problems.
Apply the Fundamental Theorems of Calculus to solve mathematical and real-world problems.
Explain graphically and verbally the properties of the definite integral. Apply these properties to evaluate basic definite integrals.
Evaluate integrals using substitution.

South Carolina College- and Career-Ready Mathematical Process Standards

A mathematically literate student can:

Make sense of problems and persevere in solving them.
Relate a problem to prior knowledge.
Recognize there may be multiple entry points to a problem and more than one path to a solution.
Analyze what is given, what is not given, what is being asked, and what strategies are needed, and make an initial attempt to solve a problem.
Evaluate the success of an approach to solve a problem and refine it if necessary.

Reason both contextually and abstractly.
Make sense of quantities and their relationships in mathematical and real-world situations.
Describe a given situation using multiple mathematical representations.
Translate among multiple mathematical representations and compare the meanings each representation conveys about the situation.
Connect the meaning of mathematical operations to the context of a given situation.

Use critical thinking skills to justify mathematical reasoning and critique the reasoning of others.
Construct and justify a solution to a problem.
Compare and discuss the validity of various reasoning strategies.
Make conjectures and explore their validity.
Reflect on and provide thoughtful responses to the reasoning of others.

Connect mathematical ideas and real-world situations through modeling.
Identify relevant quantities and develop a model to describe their relationships.
Interpret mathematical models in the context of the situation.
Make assumptions and estimates to simplify complicated situations.
Evaluate the reasonableness of a model and refine if necessary.

Use a variety of mathematical tools effectively and strategically.
Select and use appropriate tools when solving a mathematical problem.
Use technological tools and other external mathematical resources to explore and deepen understanding of concepts.

Communicate mathematically and approach mathematical situations with precision.
Express numerical answers with the degree of precision appropriate for the context of a situation.
Represent numbers in an appropriate form according to the context of the situation.
Use appropriate and precise mathematical language.
Use appropriate units, scales, and labels.

Identify and utilize structure and patterns.
Recognize complex mathematical objects as being composed of more than one simple object.
Recognize mathematical repetition in order to make generalizations.
Look for structures to interpret meaning and develop solution strategies.

Other Standards:
The College Board Topic Outline for AP Calculus BC:

AP Calculus BC includes all topics included in AP Calculus AB with additional topics. These additional topics are noted with an asterisk (*).

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.
• Calculating limits using algebra.
• Estimating limits from graphs or tables of data.

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

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.)
• Understanding continuity in terms of limits.
• 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.

II. Derivatives

Concept of the derivative
• Derivative presented graphically, numerically, and analytically.
• Derivative interpreted as an instantaneous rate of change.
• Derivative defined as the limit of the difference quotient.
• Relationship between differentiability and continuity.

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

Derivative as a function
• Corresponding characteristics of graphs of ƒ and ƒ’.
• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.
• The Mean Value Theorem and its geometric interpretation­.
• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

Second derivatives
• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ ’’.
• Relationship between the concavity of ƒ and the sign of ƒ ’’.
• Points of inflection as places where concavity changes.

Applications of derivatives
• Analysis of curves, including the notions of monotonicity and concavity.
•* Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration.
Optimization, both absolute (global) and relative (local) extrema.
• Modeling rates of change, including related rates problems­.
• Use of implicit differentiation to find the derivative of an inverse function.
• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
• *Numerical solution of differential equations using Euler’s method.
• *L’Hospital’s Rule, including its use in determining limits and convergence of improper integrals and series¬

Computation of derivatives
• Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
• Derivative rules for sums, products, and quotients of functions.
• Chain rule and implicit differentiation.
• *Derivative of parametric, polar, and vector functions

III. Integrals

Interpretations and properties of definite integrals
• Definite integral as a limit of Riemann sums.
• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
∫_a^b▒〖f^' (x)dx=f(b)-f(a)〗
• Basic properties of definite integrals (examples include additivity and linearity).

*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 (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, the length of a curve (including a curve given in parametric form), and accumulated change from a rate of change.

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

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

Applications of antidifferentiation
• Finding specific antiderivatives using initial conditions, including applications to motion along a line.
• Solving separable differential equations and using them in modeling (including the study of the equation y’ = ky and exponential growth).
• *Solving logistic differential equations and using them in modeling.

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.

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.

*Series of constants
• *Motivating examples, including decimal expansion.
• * Geometric series with applications.
• * The harmonic series.
• * Alternating series with error bound.
• * 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.
• * The ratio test for convergence and divergence.
• * 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).
• * Maclaurin series and the general Taylor series centered at x = a.
• * Maclaurin series for the functions: e^x, sin(x), cos(x), and 1/(1-x).
• * Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.
• * Functions defined by power series.
• * Radius and interval of convergence of power series.
• * Lagrange error bound for Taylor polynomials.

Required Instructional Materials and Resources:

• Larson, Ron and Bruce Edwards. Calculus of a Single Variable AP* Ninth Edition. Belmont: Brooks/Cole Cengage Learning, 2010.
• Stewart, Single Variable Calculus Early Transcendentals Seventh Edition

 

Technology Requirement/Resources:

Students MUST have access to graphing calculators both during and outside of class time.  Calculators will be used to conduct explorations, analyze and interpret results, justify and explain results, and provide visual enhancement so that students are prepared for calculator use on the AP Calculus AB exam to:

1. Graph a function within an arbitrary viewing window
2. Find the zeros of a function
3. Numerically calculate the derivative of a function
4. Numerically calculate the value of a definite integral       

 

Supplemental Materials and Resources:

• Bock, David.  AP Calculus:  400 Flash Cards.  New York: Barron’s Educational Series, 2014.
• Cade, Sharon, Rhea Caldwell and Jeff Lucia.  Fast Track to a 5:  Preparing for the AP* Calculus AB and BC Examinations. Belmont: Brooks/Cole Cengage Learning, 2010.
• Hockett, Shirley O. and David Bock. Barron’s How to Prepare for the AP Advanced Placement Exam 12th Edition. New York: Barron’s Educational Series, 2013.
• Lederman, David. Multiple-choice & Free-response Questions in Preparation for the AP Calculus (AB) Examination 9th Edition. D&S Marketing Systems, Inc., 2003.
• Ma, William. 5 Steps to a 5:  AP* Calculus AB 2014-2015. New York:  McGraw-Hill Education, 2013.
• Ruby, Tamara Lefcourt, James Sellers, Lisa Korf, Jeremy Van Horn, and Mike Munn. Kaplan AP Calculus AB &BC 2006 Edition. Simon & Schuster, 2006.
• Schwartz, Stu.  AB Calculus Manual. Retrieved June 23, 2014 from http://www.mastermathmentor.com/calc/ABcalc.ashx.
• Schwartz, Stu. Calculus on the Wall. Retrieved June 23, 2014 from http://www.mastermathmentor.com/mmm/calc-OnTheWall-intro.ashx.
• Schwartz, Stu.  RU Ready For Some Calculus? A Precalculus Review. Retrieved June 2, 2014 from http://www.mastermathmentor.com/calc/RUReady.ashx.
• The College Board. (2014). AP Central – The AP Calculus AB Exam: Released Test Items. Retrieved June 23, 2014 from http://apcentral.collegeboard.com/apc/members/exam/exam_information/1997.html.

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