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AP Calculus BC (1202320)

Curriculum and Pacing Guide - MathematicsAP Calculus BC (1202320)

(Every day period)

Revised June 2011


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.


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. I use these exercises 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.

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. In my view, 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 of 4 which change 3 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 note Study Island.

IV. Course Outline for AP Calculus BC

The following is from the document, “Curriculum and Pacing Guide – Mathematics: Calculus BC.” 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

http://www.studyisland.com/


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, except as a check during term 2 after the mastery of analytical skills.

Curriculum and Pacing Guide - MathematicsAP Calculus BC (1202320)

Revised June 2011

Students who successfully complete this course will receive one credit AP Calculus BC and will take the AP Calculus BC 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 Lines Increments, Slope, Point-slope Form, Slope-intercept, Standard Form

(Skip Regression Analysis)

(NV)

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

Recognizing Graphing Failure

(TGV) Graphing Features of calculator, Graph Viewing Skills

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

(TG) Graphing and Table Features

* 1.5 Functions and Logarithms

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

Change of Base Formula

(VGN)

* 1.6 Trigonometric Functions Periodic Function, Period, (TVGN)


Trigonometric Functions, Inverse Trig, Even and Odd

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

1 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.1 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.

1 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 Average Rate of Change, Slope of a (VN)


http://www.calculus-help.com/

http://www.calculus-help.com/tutorials/












Lines Curve at a Point, Tangent Line, Normal to a Curve,

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

1 Review1 Test7 Total

Days

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 derivative 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 given along with their corresponding derivatives. Various derivative techniques are developed for natural logarithm functions techniques for rational form functions, including trigonometric.

http://www.calculus-help.com/the-difference-quotient/

http://www.calculus-help.com/the-difference-quotient/


Chapter 3: DerivativesDays Section Topic Concepts Suggested Approaches


(SI) Study Island2 3.1 Derivative of a

FunctionDerivative, 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.ht

ml(SI) 2.a.

2 3.2 Differentiability Non-Differentiability, Local Linearity (VGN)2 3.3 Rules for

DifferentiationDifferentiation 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.

2 3.5 Derivatives of Trigonometric

Functions

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

(SI) 2.d.2 3.6 Chain Rule Derivative of Composite Functions, Chain Rule

(Skip Slopes of Parameterized Curves)(VN)

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

(SI) 2.e.2 3.7 Implicit

DifferentiationImplicit 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.




http://www.calculus-help.com/the-chain-rule/

http://www.calculus-help.com/the-chain-rule/








http://www.ima.umn.edu/~arnold/graphics.html

http://www.ima.umn.edu/~arnold/graphics.html



3 3.9 Derivatives of Exponential and

Logarithmic Functions

Derivative 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.


4. This unit continues the application of derivatives. Also three principle theorems are developed and used – the Extreme Value Theorem, Rolle’sTheorem, 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 intervals concave up and concave down. 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 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 in this unit.

Chapter 4: Applications of DerivativesDays Section Topic Concepts Suggested Approaches


(SI) Study Island2 4.1 Application of Derivatives Absolute Extrema, Local Extreme Values, Critical

Point, Extreme Value Theorem,(VGN)

1 4.2 Mean Value Theorem Mean Value Theorem, Increasing Decreasing Intervals, Antiderivative, Rolle’s Theorem,

Monotonic Functions

(VN)

2 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.3 4.4 Modeling and Optimization Max-Min Problems, Optimization (V)


http://www.unitedstreaming.com/

http://archives.math.utk.edu/visual.calculus/3/graphing.3/index.html



http://www.univie.ac.at/future.media/moe/tests/diff1/ablerkennen.html






(SI) 2.g.1 4.5 Linearization Linear Approximation, Differentials, Absolute

Relative and Percent Change,(Skip Newton’s Method)

(VN)

3 4.6 Related Rates Related Rates (VN)


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 can then be given as a limit to 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 leading to 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, and average-value-of-a-function problems. This unit also includes estimation of plane regions by using trapezoidal approximation.

Chapter 5: The Definite IntegralDays Section Topic Concepts Suggested Approaches


(SI) Study Island2 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.3 5.2 Riemann Sums Definite Integral, Integrability, (T, N) Graphing calculator: table

feature(SI) 3.c.

5 5.3 Definite Integrals and Antiderivatives

Integral Properties, Average Value, Mean Value Theorem for Integrals

(VN)(SI) 3.d.

3 5.4 Fundamental Theorem of Calculus

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

(VGN)(T)




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






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

1 5.5 Trapezoidal Rule Trapezoid Rule(Skip Other Algorithms, Simpson’s Rule and Error

Bounds)

(VN)

1 Review1 Test16 Total days




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. Integral techniques are expanded to include integration by parts, trig identities (including using identities to find integrals of powers of trig functions), and partial fractions. 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

4 6.1 Slope Fields Slope Fields, Differential Equations, 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.

2 6.3 Antidifferentiation by Parts Product Rule in Integral Form, Solving for the Unknown Integral,

Tabular Integration, Inverse Trigonometric and Logarithmic

functions

Supplement from Larson

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

and Compound Interest, Modeling Growth and Decay

(VN)

4 6.5 Logistic Growth Logistic Differential Equations, Logistic Curve, Logistic Growth Model (Skip Partial Fractions)

(VN)


Days



http://www.apcentral.com/



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 diminution shapes also knows as cross sections. Also included are volumes of rotation using the disk, and washer method incorporating the change of axis.

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


(SI) Study Island3 7.1 Integral as Net Change Displacement, Consumption over time,

Net Change from Data(VN)

1 7.2 Areas in the Plane 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/MathAni

m.html(SI) 3.f.

3 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/MathAni

m.html (SI) 3.g.

2 7.4 Lengths of Curves Sine Wave, Length of a smooth curve, Vertical tangents, Corners, and Cusps

Optional 7.5 Applications from Science and Statistics

Work, fluid force, fluid pressure, normal probabilities


Days


http://clem.mscd.edu/~talmanl/MathAnim.html







9. This unit starts with developing the concepts and notation of sequences followed by what it means for the sequence to converge or diverge. Attention is given to types of sequences such as bounded, monotonic and oscillating. Limits are briefly reviewed focusing on indeterminate forms and a reminder of the methods used for improper integrals is also included. Previous math courses have dealt with part of these concepts. The main purpose for the time spent dealing with the above is to set a foundation for the study of Series. The concepts of Series and the notation can now be developed and explained with the goal that students understand a Series is a sequence of partial sums. Whether or not this sequence of partial sums has a limit determines the convergence or divergence of the Series. This is begun by examining geometric and telescoping Series and the decision process used for convergence and divergence. The repeating decimal is a primary example for a convergent geometric Series. Other Series are then addressed such as harmonic and p-series and alternating including alternating series estimation. Various test for convergence and divergence are developed and used including the integral test, p-series convergence, direct comparison, limit comparison, alternating series test, ratio, and root test. Next is a transition to polynomial approximations of elementary functions and the Taylor and MacLaurin Series. The polynomial approximations are built for elementary functions such as sin x, cos x, e^x, and 1/(1-x), first centered at 0 (MacLaurin) and then centered at other values, “c”, (Taylor). Use of the TI – 89 graphing utilities is indispensable at this juncture for students to understand the approximation process. These polynomials are used to find a function approximate value and Taylor’s Theorem (Lagrange remainder) used involving the accuracy of the approximation. We then address general power series of functions and the examination of their domains, convergence, and radii and intervals of convergence. The rest of this unit goes into the manipulation of power series and differentiation and integration of power series.


Chapter 9: Sequences/Infinite Series Days Section Topic Concepts Suggested Approaches


2 8.1 Sequences Displacement, Consumption over time, Net Change from Data

(VN)

4 9.1 Power Series Infinites Series, Partial Sums, Converge, Diverge, Geometric Series, Interval of

Convergence, Power Series, Centered at x=0, centered at x=a

(TVN)

4 9.2 Taylor Series Taylor Polynomial, Maclaurin and Taylor Series, Taylor Series Gernated at

x=a, Maclaurin Series Table

(VGN)

4 9.3 Taylor’s Theorem Truncation Error, Taylor’s Formula, Remainder, La Grange Form, La Grange

Error Bounds, Remainder Estimation

(VN)

3 9.4 Radius of Convergence Convergence, Radius of Convergence, Interval of Convergence, Nth term Test,

Direct Comparison Test, Ratio Test, Absolute Convergence, Telescoping

Series

(VN)

5 9.5 Testing Convergence at Endpoints

Integral Test, P-Series, Harmonic Series, Limit Comparison Test, Alternating

Series Test, Improper Integrals

(VN)


Days


8. In this unit we use L’Hopital’s Rule to calculate limits of fractions whose numerators and denominators both approach zero or are unbounded. We also use L’Hopital’s rule to compare the rates at which functions grow as x becomes large. We also evaluate definite integrals of continuous functions and bounded functions with a finite number of discontinuities on finite close intervals.

Chapter 8: L’Hopital’s Rule, Improper Integrals, and Partial FractionsDays Section Topic Concepts Suggested Approaches


1 8.2 L’Hopital’s Rule L’Hopital’s Rule, Indeterminate Form (VGN)1 8.3 Relative Rates of Growth Transitivity of Growing Rates (VGN)2 8.4 Improper Integrals Infinite Discontinuities, Integral

Comparison Test(VN)


Days


10. In this unit we analyze calculus in three kinds of two variable contexts, parametric, vector, and polar to analyze new kinds of curves. We also analyze motion that does not proceed along a straight line. This can be done using single variable calculus in some different and interesting ways.

Chapter 10: Parametric, Vector and Polar FunctionsDays Section Topic Concepts Suggested Approaches


4 10.1 Parametric Functions Parametric Differentiation Formulas, Arc Length

(VGN)

3 10.2 Vectors in the Plane Two Dimensional Vectors, Direction Angle, Magnitude, Components, Vector Operations, Unit Vector, Parallelogram Representation, Displacement, Distance,

Velocity and Acceleration Vectors

(VN)

4 10.3 Polar Functions Polar Coordinates, Pole, Polar Curves, Conversion Formulas, Area enclosed by

Polars

(VGN)


Days

Total Instructional days: 137The 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

IV. 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.


curriculum and pacing guide · web viewreview of the concepts from algebra and pre-calculus...

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