OCG - AP Calculus AB

• Algebra 1
• Geometry
• Algebra 2
• Precalculus

Limits and Continuity The student will:
C.LC.1	Understand the concept of a limit graphically, numerically, analytically, and contextually. a.    Estimate and verify limits using tables, graphs of functions, and technology. b.    Calculate limits, including one-sided limits, algebraically using direct substitution, simplification, rationalization, and the limit laws for constant multiples, sums, differences, products, and quotients. c.    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. a.    Apply the definition of continuity of a function at a point to solve problems. b.    Classify discontinuities as removable, jump, or infinite. Justify that classification using the definition of continuity. c.    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. a.    Interpret the value of the derivative of a function as the slope of the corresponding tangent line. b.    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. c.    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. d.    Understand and explain graphically and analytically the relationship between differentiability and continuity. e.    Explain graphically and analytically the relationship between the average rate of change and the instantaneous rate of change. f.     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. a.    Know and apply the derivatives of constant, power, trigonometric, inverse trigonometric, exponential, and logarithmic functions. b.    Use the constant multiple, sum, difference, product, quotient, and chain rules to find the derivatives of functions. c.    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. a.    Explain geometrically and verbally the mathematical and real-world meanings of the Extreme Value Theorem and the Mean Value Theorem. b.    Write an equation of a line tangent to the graph of a function at a point. c.    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. d.    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. e.    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. a.    Explain how the definite integral is used to solve area problems. b.    Approximate definite integrals by calculating Riemann sums using left, right, and midpoint evaluations, and using trapezoidal sums. c.    Interpret the definite integral as a limit of Riemann sums. d.    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. a.    Apply the Fundamental Theorems of Calculus to solve mathematical and real-world problems. b.    Explain graphically and verbally the properties of the definite integral. Apply these properties to evaluate basic definite integrals. c.    Evaluate integrals using substitution.

 

Other Standards: 

The College Board Topic Outline for AP Calculus AB:

 

1. 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).

 

1. 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.
• 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.

 

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.

 

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:
• 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, 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 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).

 

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).

 

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.

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.

High School Mathematics