Curriculum and Pacing Guide · Web viewThese prerequisite courses include algebra I and II, geometry, and precalculus - which consist of the study of transcendental functions and

Curriculum and Pacing Guide - MathematicsAP Calculus AB (1202310)

(Every day period)

Revised June 2011

AP Calculus AB (1202310)

I. AP Calculus AB Course Overview and Philosophy

In developing and applying the topics of differential and integral calculus, students have the opportunity to solidify all their previous high school math experience, since all the major ideas from the prerequisite math courses are necessary for calculus success. These prerequisite courses include algebra I and II, geometry, and precalculus - which consist of the study of transcendental functions and their graphs, with a strong emphasis on trigonometry and its applications. Consequently, there are high expectations which emphasize concepts reinforced through development and application rather than just procedure. The goal is that the successful student of Advanced Placement Calculus will validate his/her success with adequate performance on the AP Exam and place out of a comparable college course. However, beyond this, it is desired that the successful student appreciates the high level of their mathematical success and at the same time realizes that this accomplishment is not an end in itself, but a beginning point for many other opportunities in various fields of study and work. The hope is that through the study of this course, students gain a genuine appreciation for the intricacies and beauty of mathematics, as well as proficiency with the subject matter of calculus.

II. Use of Technology

Students learn to use the TI-89 graphing calculator in the following ways:1. The use of tables to analyze data and behavior of functions.2. The use of graphs to determine behavior of functions and their limits.3. The use of various types of solution capabilities of the calculator for various functions.4. The use of the symbolic manipulation capabilities of the calculator in order to determine information about functions for which they have no analytical techniques, including numerical derivatives and definite integrals.

AP Calculus AB (1202310) III. Strategies for Success

Since communication, both verbal and written is a major component of the course, the main classroom focus, besides guided practice for instruction of techniques, is cooperative group work for learning concepts and problem solving.

1. Almost every section of our adopted text, which is an AP Edition textbook, includes problem sets of Standardized Test Questions, Explorations, and Extensions. These exercises are used after students have done an assignment on the section, beginning at Chapter 1. Students discuss and defend their answers before submitting one set of answers per group. Student interaction is a necessity if they are to know the depth to which they understand a concept. Having to explain to each other and to the class gives the opportunity to do this, whether they are questioning or explaining.

2. After 2 or 3 sections of each Chapter, the text includes a Quick Quiz for A.P. Preparation. These along with College board released exam items are invaluable in giving students the “A P Experience” throughout the course.

3. Students accumulate a set of note cards on important theorems as they progress through the course. In review for the AP test, students are randomly given a theorem to present and explain on the spot, so they know them well for justifications in free response items.

4. The goal for the AP Calculus student is to think creatively in solving problems that apply “familiar concepts in unfamiliar settings” and those which combine concepts. Also, students must learn to solve multi-part problems in which they often have to explain not only what they have done, but the theory behind why it works. For this, the group setting is essential for brain-storming to acquire attack skills on such problems and to gain confidence in solving problems that seem totally unfamiliar.

5. Students gradually become independent problem-solvers through the following progression. When the AP free response questions are first presented, students are in groups which change multiple times during the first term. At this time, questions are used a few at a time and grouped by the topic being studied. During the second term, students work with one partner, and then finally, for the last two weeks before the AP test, each works alone. At this time, the questions are presented in their original test format, beginning with the most recent and working back in time. One week before the actual exam, students also have the opportunity to take a complete released exam, and then given the opportunity to score it, according to the released rubric, and discuss the solutions.

6. District will provide licenses for all students to access Study Island website located at http://www.studyisland.com/. AP Calculus can be found under the US Programs tab on the left side of the webpage. We will use SI in the suggested approaches column to denote Study Island.

IV. Course Outline for AP Calculus AB

The following is from the document, “Curriculum and Pacing Guide – Mathematics: Calculus AB.” This document is relatively dynamic in that is regularly reviewed and updated. The columns in the pacing of each unit provide the general time line, sections

of the text book, topics, and major concepts covered along with suggested approaches and activities for some of the concepts. The TI-89 graphing calculator is an integral part of concept investigation for this course and one is provided to the students who do not own one. However, it is not used for its algebraic manipulation capabilities during term 1; then during term 2, after the mastery of analytical skills, it is used for checking and determining numerical derivatives and definite integrals.

AP Calculus AB (1202310)

Curriculum and Pacing Guide - MathematicsAP Calculus AB (1202310)

Revised June 2011

Students who successfully complete this course will receive one credit AP Calculus AB and will take the AP Calculus AB Exam.

1. Review of the concepts from algebra and pre-calculus concerning relations /functions will probably be necessary to some extent. These include linear, trigonometric, exponential, logarithmic, the generic functions, and piecewise functions. Topics relative to these are; slopes, domain and range, symmetry, intercepts, intersections, and the graphs of these relations/functions. The graphing calculator (TI-89) is used to reinforce these concepts, referring to the solving, table, and graphing capabilities of the calculator.

Chapter 1: Prerequisites for Calculus

DaysSection Topic Concepts Suggested Approaches

(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical

1 1.1 Lines Increments, Slope, Point-slope Form, Slope-intercept, Standard Form

(Skip Regression Analysis)

(NV)

1 1.2 Functions and graphs Function, Even and Odd Functions, Composition, Piecewise Functions,

Recognizing Graphing Failure

(TGV) Graphing Features of calculator, Graph Viewing Skills

1 1.3 Exponential Functions Exponential Function, Exponential Growth and Decay, Half Life

(TG) Graphing and Table Features

1 1.5 Functions and Logarithms

One to One, Logarithms Functions, Inverse Functions, Identity Functions,

Change of Base Formula

(VGN)

2 1.6 Trigonometric Functions Periodic Function, Period, Trigonometric Functions, Inverse

Trig, Even and Odd

(TVGN)

1 Review

1 Test8 Total Days

AP Calculus AB (1202310) 2. The concepts of a limit, as the independent variable approaches a particular value, are addressed numerically, analytically, and graphically. This involves developing the ideas of the existence of a limit. The issues are, if the limit exist, what is it, and if it does not, why not? This necessitates the understanding of “broken graph” oscillating, and asymptotic behaviors. Examination of limits, including one sided limits, is done using the various algebra techniques for the types of functions, the properties of limits and special techniques for rational trig related functions. Continuity and the Intermediate Value theorem and their applications are also part of this unit. Areas of application in this unit are finding equations of tangent lines (and normal lines) at a point and beginning motion problems.

Chapter 2: Limits and ContinuityDays Section Topic Concepts Suggested Approaches

(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical (SI) Study Island

2 2.1 Rates of Change and Limits Average v. Instantaneous Speed, Properties of Limits, One sided v.

Two sided Limits, Sandwich Theorem

(T) www.calculus-help.com(T) table features of calculator(T) trace features of calculator

(T) Chapter 1: Lessons 1-3 http://www.calculus-help.com/tutorials/

Supplement Algebraic Limit Techniques from Larson

(SI) 1.a. / 1.b.2 2.2 Limits Involving Infinity Horizontal Asymptote, Properties of

Limits as x approaches infinity, Vertical Asymptote, End Behavior

Model

(T) Chapter 1: Lesson 4 http://www.calculus-help.com/tutorials/

(T) www.calculus-help.com(G) Graphical(SI) 1.c / 1.d.

2 2.3 Continuity Continuity at a Point, Properties of Continuous Functions, Composition of

Continuous Functions, Discontinuities, Intermediate Value

Theorem, Jump Discontinuity

(T) Chapter 1: Lesson 5-6 http://www.calculus-help.com/tutorials/

(T) www.calculus-help.com(G) Graphical

(VN)(SI) 1.c.

2 2.4 Rates of Change and Tangent Lines

Average Rate of Change, Slope of a Curve at a Point, Tangent Line,

Normal to a Curve,

(VN)(T) http://www.calculus-help.com/the-difference-

quotient/2 Review and Quiz1 Test11 Total

Days

AP Calculus AB (1202310)

3. The derivative is developed in this unit which involves the geometric interpretation of the tangent line at a point, leading to the limit definition of the derivative of a function. The limit definition is used both to find derivatives at a point and to develop the basic differentiation rules. Basic derivative rules are used for first, second, and higher order derivatives and also for implicit derivatives. The derivative is also investigated in the relationship between position, velocity, acceleration and jerk. Inverse functions are addressed with particular attention given to the natural exponential function as the inverse of the natural logarithm function. The derivatives of the natural exponential function and the natural log function are given. Logarithmic and exponential functions of any base are also investigated along with their corresponding derivatives. Various derivative techniques are developed for natural logarithm functions as needed for techniques for rational form functions, including trigonometric functions.

Chapter 3: Derivatives AP Calculus AB (1202310) Days Section Topic Concepts Suggested Approaches

(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical

(SI) Study Island3 3.1 Derivative of a Function Derivative, Derivative at a Point, Notation,

Relationship between the Graphs of F and F Prime, One-Sided Derivatives

(VGN) (T) http://www.ima.umn.edu/~arnold/graphics.html

(SI) 2.a.2 3.2 Differentiability Non-Differentiability, Local Linearity (VGN)3 3.3 Rules for Differentiation Differentiation Rules, Higher Order Derivatives (VN)

(T) Chapter 2: Lesson 2-4 http://www.calculus-help.com/tutorials/

(T) www.calculus-help.com (SI) 2.b. / 2.c. / 2.h.

3 3.4 Velocity and Other Rates of Change

Instantaneous Rates of Change, Instantaneous Velocity, Speed, Acceleration, Linear Motion,

Free-Fall Motion

(VG)(SI) 2.i.

3 3.5 Derivatives of Trigonometric Functions

Derivative of Trig Functions, Jerk, (VG) Graphical(T) www.calculus-help.com

(SI) 2.d.1 Mid-Chapter Test3 3.6 Chain Rule Derivative of Composite Functions, Chain Rule,

Finding dy/dx Parametrically(Skip Derivatives of Parametric Defined

Functions)

(VN) (T) http://www.calculus-help.com/the-chain-rule/

(SI) 2.e.

2 3.7 Implicit Differentiation Implicit Function, Implicit Differentiation, Power Rule for Rational Powers

(VN)(SI) 2.k.

2 3.8 Derivatives of Inverse Trigonometric Functions

Inverse, Derivatives of Inverse Trigonometric Functions

(VGN) Supplement Theorem 5.9 From Larson for Derivative of Inverse Functions

(SI) 4.c.3 3.9 Derivatives of Exponential

and Logarithmic FunctionsDerivative of Exponential Functions base e and base a, Derivative of Log Functions base e and

base a

(VN) Supplement from Larson Implicit Differentiation with Exponentials and Logarithmic

(SI) 4.a. / 4.b.1 Review (SI) 2.f.1 Test27 Total

AP Calculus AB (1202310) 4. This unit continues the study of derivatives by investigating their applications. Also three principle theorems are developed and used – the Extreme Value Theorem, Rolle’s Theorem, and the Mean Value Theorem. First and second derivatives are used to determine for a given function the critical values, intervals of increase and decrease, relative maxima and minima, points of inflection, and concavity intervals. This application is done with and without graphing calculators. Included with this application is the examination of the relationships of the graphs of a function, the graph of its 1st derivative, and the graph of its 2nd derivative and their relationship through the use of tables. The very useful and important derivative application of solving optimization problems, as well as linear approximations, differentials, and related rates are also investigated in this unit.

Chapter 4: Applications of DerivativesDays Section Topic Concepts Suggested Approaches

(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical

(SI) Study Island4 4.1 Application of Derivatives Absolute Extrema, Local Extreme Values, Critical

Point, Extreme Value Theorem,(VGN)

2 4.2 Mean Value Theorem Mean Value Theorem, Increasing Decreasing Intervals, Antiderivative, Rolle’s Theorem,

Monotonic Functions

(VN)

4 4.3 Connecting f prime and f double prime with the

graph of f

First Derivative Test, Definition of Concavity, Concavity Intervals, Point of Inflection, Second

Derivative Test,

(G)http://www.univie.ac.at/future.media/moe/tests/diff1/

ablerkennen.html(G)http://archives.math.utk.edu/

visual.calculus/3/graphing.3/index.html (T) www.unitedstreaming.com :

“applications of derivatives”(SI) 2.j.

4 4.4 Modeling and Optimization

Max-Min Problems, Optimization (V)(SI) 2.g.

3 4.5 Linearization Linear Approximation, Differentials, Absolute Relative and Percent Change,

(Skip Newton’s Method)

(VN)

6 4.6 Related Rates Related Rates (VN)

2 Review and Quiz1 Test26 Total

AP Calculus AB (1202310)

5. The definite integral is developed by first examining estimates of the areas of plane regions as sums of rectangles constructed by using a partitioning of an interval and the right, left, midpoint or any point of the partition. The definition of a definite integral is then given as a limit of an infinite Riemann Sum, the exact area of the plane region. The Fundamental Theorem of Calculus is developed along with the Mean Value Theorem for Integrals and the Average Value of a function on an interval. The Second Fundamental Theorem is also given. The main applications here are areas of simple plane regions by approximation, including the analysis of whether the approximation is an over or under estimate. This unit also includes estimation of plane regions by trapezoidal approximation.

Chapter 5: The Definite IntegralDays Section Topic Concepts Suggested Approaches

(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical

(SI) Study Island3 5.1 Estimating with Finite

SumsDistance Traveled, Rectangular Approximation

Method(VN)http://

designatedderiver.wikispaces.com/Integrals+Def+and+Ind

(Click on: Calc-SM-Approximating Under the Curve. Docx)

(SI) 3.e.4 5.2 Riemann Sums Definite Integral, Integrability, (T, N) Graphing calculator: table

feature(SI) 3.c.

6 5.3 Definite Integrals and Antiderivatives

Integral Properties, Average Value, Mean Value Theorem for Integrals

(VN)(SI) 3.d.

4 5.4 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus Part 1 and 2, Connection to Area

(VGN)(T)

http://clem.mscd.edu/~talmanl/HTML/FTOC.html

3 5.5 Trapezoidal Rule Trapezoid Rule(Skip Other Algorithms, Simpson’s Rule and Error

Bounds)

(VN)

2 Review and Quiz1 Test23 Total

AP Calculus AB (1202310) 6. This unit introduces slope fields and the solving of differential equations, leading to the concept of an antiderivative. Indefinite integrals are solved through various techniques including u du substitution and pattern recognition. Exponential growth and decay model is developed from integration of separation of variables. The Logistic Growth model is also included.

Chapter 6: Differential Equations and Mathematical ModelingDays Section Topic Concepts Suggested Approaches

(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical (SI) Study Island

3 6.1 Slope Fields Slope Fields, Differential Equations (Skip Euler’s Method)

(VN)(G) www.apcentral.com :

Teacher Resource: Slope fields(SI) 3.b. / 4.d.

3 6.2 Antidifferentiation by Substitution

Properties of Indefinite Integrals, Indefinite Integrals, Leibniz Notation, u du substitution

(VN)(SI) 5.a.

1 6.3 Antidifferentiation by Parts Integral forms whose antiderivatives are inverse trig

functions (Skip Antidifferentiation by Parts)

Supplement from Larson

4 6.4 Exponential Growth and Decay Separable Differential Equation, Exponential Change, Continuous

and Compound Interest, Modeling Growth and Decay

(VN)

3 6.5 Logistic Growth Logistic Differential Equations, Logistic Curve, Logistic Growth Model (Skip Partial Fractions)

(VN)

2 Review and Quiz1 Test16 Total

AP Calculus AB (1202310)

7. This unit involves the interpretation of the integral as an accumulator and applications of finding areas and volumes. The definite integral is used to find the areas of regions between curves using all types of functions. These are areas on an interval, areas between curves including curves with more than two intersections, also incorporating change of axis. The next application is volume beginning with three dimensional shapes of uniform cross sections. Next included are volumes of rotation using the disk, and washer method, also incorporating the change of axis.

Chapter 7: Applications of Definite Integrals Days Section Topic Concepts Suggested Approaches

(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical

(SI) Study Island4 7.1 Integral as Net Change Displacement, Consumption over

time, Net Change from Data(VN)

3 7.2 Areas in the Planes Area between Curves, Area Enclosed by Intersecting Curves, Integrating in respect to y, Integration by Geometric

Formula

(VGN)(T) Calculus in Motion (Fee)

(T) http://clem.mscd.edu/~talmanl/MathAnim.html

(SI) 3.f.

7 7.3 Volume Cross Sections, Washers(Skip Shells)

(VGN)(T) Calculus in Motion (Fee)

Supplement from Larson for Disk and Washer Methods

(T) http://clem.mscd.edu/~talmanl/MathAnim.html

(SI) 3.g.2 Review and Quiz1 Test17 Total

Days

Total Instructional days: 128

The remaining time is for AP Test Review consisting of discussion of release multiple choice and free response questions.

V. Major Text: Taken from: Finney, Damana, Waits and Kennedy. Calculus: Graphical, Numerical, Algebraic (AP Edition), 3rd Ed. Upper Saddle River, New Jersey, 2007 Finney, Damana, Waits and Kennedy. Calculus: A Complete Course, 2nd Ed. Upper Saddle River, New Jersey, 2000

VI. Supplementary Materials: Calculus in Motion (Must purchase program and must also have geometry sketchpad). Larson, Ron, Robert P. Hostettler, and Bruce H. Edwards. Calculus with Analytic Geometry. 7th ad. Boston , New York: Houghton Mifflin, 2002 College Board AP Calculus Released Exams. Study Island: http://www.studyisland.com/. AP Calculus can be found under the US Programs tab on the left side of the webpage. We will use SI in the suggested approaches column to note Study Island. District will provide licenses for all students.