AP Calculus syllabi - Edl€¦ · Section 1 is the AP Calculus AB Course Syllabus, Section 2 is the AP Calculus AB Pacing Plan, and Section 3 is the AB Calculus AB Topic Outline AP

AP CALCULUS HANDBOOK

AT

CATHEDRAL HIGH

2013-2014

MR. MIKE TRAFECANTY

Chapter 1

Twenty-nine juniors are taking this course in the

2013/2014 school year. This is the first Math and

Science Academy group; moreover, these Academy

students will be taking their first University level math

course. Section 1 is the AP Calculus AB Course Syllabus,

Section 2 is the AP Calculus AB Pacing Plan, and Section

3 is the AB Calculus AB Topic Outline

AP CALCULUS AB-PERIOD 3

Overview

Advanced Placement Calculus AB is a University Level course taught at Cathedral High School. The class officially meets three times a week for two 80-minute classes and a 40-minute class, and students are required to attend additional study sessions. University credit is obtained only by passing the required May AP Exam; moreover, not all Universities accept passed AP exams for credit. In recent years, ninety percent of Cathedral students who take both the AB and BC exams on a two-year schedule have received eligible credit for one or two University of California Calculus courses. AP Calculus AB is an Honors course and contributes an extra 5 grade points per semester for every “C” grade or higher. This course satisfies the “a-g” requirements for the UC and Califor-nia State system. Mr. Trafecanty maintains a California Single Subject Math Credential and is a nationally certified AP Calculus AB and BC teacher.

2

SECTION 1

AP Calculus AB Syllabus

The TI-84 is required for AP Calculus. Forty percent of the exams will require the use of this calculator

Course Sequence Leading to AP Calculus AB

Algebra I, Geometry, Algebra II, and Precalculus are the courses that students have passed with a “B” or better in or-der to be eligible for AP Calculus AB.

Teaching Philosophy and Strategies

CALCULUS AND A CLASSICAL EDUCATION

A core understanding of History, the Arts, English, Lan-guages, Christianity, Math and Science are requirements for success in a University environment; moreover, these under-standings are the most important ingredient for a whole per-son who can critically analyze the massive amounts of infor-mation in our modern, global society.

TRUTHS and RULES in Mr. Trafecanty’s class

1) All people have an immeasurable worth and value re-gardless of their ability in Calculus.

RULE I: All students and teachers are to be treated with re-spect and are equals on a human level.

2) All people are loved by God and saved by Jesus Christ with no special reserved spaces for passing Calculus or attend-ing the “Great University”.

RULE II: Love your classmates as you love yourself.

3) All people must MAKE CHOICES that will affect their destiny and unfortunately sometimes the destiny of others both good and bad including choices of pursuing a Christian education with integrity. No human being has a prepro-grammed future disregarding their choices. We can all be a “Hero” or a “Villain” with equal success.

RULE III: All students are to both respect Christianity and to have established goals.

RULE IV: All students are to do their work on time, profes-sionally, and with the intent to both practice and learn the material taught.

SPECIAL NOTE 1: Success, as defined by Christians, CAN-NOT be about money/property or personal happiness or in-stant gratification as these can be achieved without work, love, or other people.

3

SPECIAL NOTE 2: Attending a University is more impor-tant than which. Great Universities have developed heroes and villains with equanimity and most have trouble identify-ing the true qualities of a good person least of all defining what a good Christian would be.

MR. TRAFECANTY’S EXPECTATIONS

1) STUDENT’S MUST BE MOTIVATED AND DE-VELOP A PURPOSE FOR WHY they are in CALCULUS.

2) A STUDENT’S MIND is the MOST IMORTANT TOOL in the classroom. Although I will use many of the newest technologies to present the topics including the IPAD, Calculus textbook, web based video, smart boards, comput-ers and the TI84+, the best way to learn calculus is to sit down and do the problems without any other tools than the mind. Many colleges do not allow the use of technology in the first courses of Calculus. Class time is devoted to work-ing on Calculus problems.

3) A STUDENT’S CHRISTIAN MODELING IS THE MOST IMPORTANT CHARACTER TRAIT. I will ap-peal to a student’s inner sense of right and wrong, and burn-ing desire to be really alive. I hope that my two hundred

days of arguments and insistence on students defense of their work have a lasting affect on developing life-long learners whose relationships with God and others has primary impor-tance and whose skills and understanding allow them to think critically.

4) IT’S ALL ABOUT THE CALCULUS. The driving force for all lesson planning and discussions is the calculus that is listed on the next few pages The connection to science, eco-nomics, and research is intentional.

Homework

Homework assignments are assigned EVERY class day and due the next class. Generally, problems are assigned from the text and are related to any topic presented in class that day or a prior class. Questions on the homework are answered the next day of class NO LATE HOMEWORK even if absent. Work is posted using the app Notability and will be submitted electronically. See Chapter 3 - Homework Policies

Notebooks

Students will be required to take notes during class. Notes will be written in a spiral notebook. Students will be re-quired to use a formal note-taking process.

Assessment

4

All Calculus exams and quizzes are used to assess student learning. Students take written quizzes at least twice a week and take unit exams at the end of each unit. Students will also be quizzed orally in groups and individually to deter-mine their individual progress in learning how to verbalize calculus. They also participate in class and are selected at random to display answers to others. Students prepare and present Calculus topics in class.

AP Exam

Students are exposed to AP questions throughout the year. The class discusses how problems are answered on the AP exam, and they work on SAMPLE AP problems. Students take partial and full AP practice exams throughout the entire second semester in order to enhance their confidence in tak-ing exams and thoroughly demonstrate their learning. All stu-dents are required to take an AP Calculus exam in May (there is an extra exam fee on the tuition bill).

Required Materials

Calculus, 3rd Edition by Finney, Demana, Waits, Kennedy

TI84 calculator, iPAD

Spiral Notebook with pockets

Grading

89.5% - A, 79.5%- 89.49% - B, 69.5% - 79.49% - C,

59.5% - 69.49% - D

40% Exams 20% Quizzes 20% Final

15% HW 10% iPad Protocol

****Special Notes****

1) iPads are required for class and points will be lost if they are not brought to class every day.

2) The first quiz covers the Summer Reading Book, “Mon-ster” which will be administered on Monday August 19

3) iPad Protocol: 10% of the student’s final grade will be de-termined by iPad Protocol (appropriate integration of the iPad in the classroom). This may include presentations created/delivered on the iPad, bringing iPads charged to class, appropriate usage during class, successful navigation of apps, electronic books and materials, and online re-sources, homework submitted electronically. Online re-sources may include Edmodo, PowerSchool, Khan Acad-

5

emy, the school website, Google Docs. iPads must be prop-erly registered for use at Cathedral and students must fol-low all school/ teacher-specific iPad policies to receive credit in this course.

Extra Credit/Make-up Work

No Extra Credit and No Make-up quizzes, Finals or Home-work.

A Make-up Unit Exam can only be taken on a scheduled Make-up Exam day as determined by the teacher

6

First Semester

Introductions August 16

Calculus Prerequisites (August 19 - September 20 )

I. Lines (August 19 - 23)

a. Increments, slope

b. Parallel and perpendicular

c. Equations of lines

d. Graphing lines

e. Regression analysis

II. Functions and Graphs (August 26 - 30)

a. Domain and Range

b. Viewing and interpreting graphs

c. Even and Odd functions – symmetry

d. Piece-wise functions, composite functions

IPAD INFORMATION

❖ All classes will be using an iPad❖ AP Calculus AB will be using the iPad on a daily

basis for Homework, Quizzes, Exams, Presentations, Surveys, Edmodo, Powerschool, and many other applications.

7

SECTION 2

AP Calculus AB Timeline

III. Exponential Functions (September 2 - 3)

a. Exponential Growth and Decay

b. The natural number

IV. Functions and Logarithms (September 5-10)

a. One-to-One Functions and Inverses

b. Logarithmic Functions, Properties, and Applications

V. Trigonometric Functions (September 11 - 20)

a. Radian Measure

b. Graphs of Trigonometric Functions

c. Inverse Trigonometric Functions

Limits and Continuity (September 23 - October 18))

I. Rates of Change and Limits (Sept. 23 - Sept. 27)

a. Average and Instantaneous Speed

b. Definition of Limit

c. Properties of Limits

d. Limits of Polynomial and Rational Functions

e. One-sided and Two-sided Limits

f. Sandwich Theorem

II. Limits Involving Infinity (Sept. 30 - Oct. 4)

a. Horizontal and Vertical Asymptotes

b. End Behavior Models

III. Continuity (Oct. 7 - Oct. 11)

a. Continuity and Discontinuity at a Point

b. Continuous Functions and Properties

c. Composite of Continuous Functions

d. The Intermediate Value Theorem

IV. Rates of Change and Tangent Lines (Oct. 14 -18)

a. Average Rates of Change

b. Secant, Tangent, and Normal Lines to a Curve

Derivatives (Oct. 21 - Nov. 29)

I. Derivative of a Function (Oct. 21 - 25)

a. Definition of Derivative and Derivative at a Point

b. Relationships between the Graphs of f and f ’

c. Graphing the Derivative from Data

d. One-sided Derivatives

II. Differentiability (Oct. 28 - Nov. 1)

a. Existence, Linearity, and Continuity

b. NDER – numerical derivative

c. Intermediate Value Theorem for Derivatives

8

III. Rules for Differentiation (Nov. 4 - 5)

a. Power Rule, Product Rule, Quotient Rule

b. Second and Higher Order Derivatives

IV. Velocity and Other Rates of Change (Nov. 7 -12)

a. Instantaneous Rate of Change and Velocity

b. Speed and Acceleration

c. Modeling Vertical and Horizontal Motion

d. Derivatives in Economics

V. Derivatives of Trigonometric Functions (Nov. 14 - 18)

a. Derivative of Sine, Cosine and Tangent

b. Definition of Jerk

c. Derivative of cotangent, secant, cosecant

VI. Chain Rule (Nov. 19)

a. Derivative of a Composite Function

b. “Outside-Inside” Rule

c. Power Chain Rule

VII.Implicit Differentiation (Nov. 21)

a. Implicitly Defined Functions

b. Process for Finding the Derivative Implicitly

c. Higher Order Implicit Differentiation

VIII. Derivatives of Inverse Trig Functions (Nov 25)

IX. Derivatives of Exponential and Log Functions (Nov. 26)

Applications of Derivatives Part I (Dec. 2 - 13)

I. Extreme Values of Functions (Dec. 2 and 3)

a. Absolute Extreme Values

b. Extreme Value Theorem

c. Local Extreme Values

d. Critical Points

II. Mean Value Theorem (Dec. 5)

a. Mean Value Theorem for Derivatives

b. Physical Interpretation

c. Increasing and Decreasing Functions

d. Definition of Antiderivative

III. Connecting f ’ and f ’’ with the Graph of f (Dec. 9 - 13)

a. Using the First Derivative Test for Local Extrema

b. Applying the Concavity Test

c. Point of Inflection

d. Using the Second Derivative Test for Local Extrema

FINAL EXAM Semester 1

9

Second Semester

Applications of Derivatives Part II (January 7 - 17)

IV. Modeling and Optimization (Jan. 7 - 10)

a. Strategy for Solving Max-Min Problems

b. Examples from Economics

c. Modeling Discrete Phenomena

V. Related Rates (Jan. 13 - 17)

a. Finding Related Rate Equations and Strategies

b. Simulating Related Motion

The Definite Integral (January 20 - Feb. 21))

I. Estimating with Finite Sums (January 20 - 24)

a. Distance Traveled

b. Rectangular Approximation Method

c. Volume of a Sphere

II. Definite Integrals (Jan. 27 - 31)

a. Defined as a Limit of Riemann Sums

b. Terminology and Notation

c. Definite Integral and Area Under a Curve

d. Using NINT

e. Discontinuous Integrable Functions

III. Definite Integrals and Antiderivatives (Feb. 3 - 7)

a. Properties of Definite Integrals

b. Average (Mean) Value of a Function

c. The Mean Value Theorem for Definite Integrals

d. Finding an Integral Using Antiderivatives

IV. Fundamental Theorem of Calculus (Feb. 10 - 14)

a. Fundamental Theorem and its Application

b. Fundamental Theorem with Chain and Variable limits

c. Graphing the Variable Limit Integral

d. Integral Evaluation Theorem

e. Total Area Analytically, Graphically, and Numerically

V. Trapezoidal Rule (Feb. 17 - 21)

a. Trapezoidal Approximations

b. Simpson’s Rule

Differential Equations and Modeling (Feb. 24 - March 14)

I. Antiderivatives and Slope Fields (Feb . 24 - 28)

a. Solving Initial Value Problems

b. Slope Field or Directional Field

c. Antiderivatives and Indefinite Integrals

d. Integral Formulas

10

e. Using the Fundamental Theorem

f. Applications

II. Integration by Substitution (March 3 - 7)

a. Power Rule for Integration

b. Graphical Support for Integration

c. Trigonometric Integral Formulas

d. Substitution in Definite and Indefinite Integrals

e. Separable Differential Equations

III. Exponential Growth and Decay (March 10 - 14)

a. Law of Exponential Change

b. Applications

Applications of Definite Integrals (March 17 - April 4)

I. Integral as Net Change (March 17 - March 21

a. Linear Motion and Interpreting Velocity of a Function

b. Finding Position from Displacement

c. Calculating Total Distances

d. Modeling the Effects of Acceleration

e. Applications of Work

II. Areas in the Plane (March 24 - March 28)

a. Area Between Curves and Applications

b. Integrating with Respect to y

c. Geometry Formulas Used to Save Time in Areas

III. Volumes (March 31 - April 4)

a. Volume as an Integral

b. Volume by Slicing

c. Square and Circular Cross Sections

d. Solid of Revolution

e. Washer Cross Sections

f. Other Cross Sections

L’Hopital’s Rule (April 7 - 11)

I. L’Hopital’s Rule both forms

a. Indeterminate Form 0/0

b. One-Sided Limits

c. Other Indeterminate Forms of Product, Quotient, and Difference

Summary Review of All Topics and AP Exam Preparation (April 14- May 6))

Preparation for BC Calculus (May 12 - May 30)

11

AP Calculus AB Course Outline

(Incorporated by the AP Calculus Development Committee-adapted)

Functions, Graphs, and Limits

1. Analysis of graphs

a) Comparing analytical information with graphs

b) Being able to predict a functions local and global be-havior

2. Limits of functions (including one-sided limits)

a) Calculating limits using algebra

b) Estimating limits from graphs or tables of data

c) Finding limits by explanation such as the Sandwich Theorem

3. Asymptotic and unbounded behavior

a) Understanding asymptotes in terms of graphs

b) Describing asymptotes in terms of limits involving in-finity

c) Comparing relative magnitudes of functions and their rates of change

4. Continuity as a property of functions

a) Understanding continuity in terms of limits

b) Geometric understanding of graphs of continuous functions

(The Intermediate Value Theorem and Extreme Value Theorem)

Derivatives

1. Concept of the derivative

a) Derivative defined as the limit of the difference quo-tient

b) relationship between differentiability and continuity

2. Derivative at a point

a) Slope of a curve at a point

b) Tangent line to a curve at a point and local linear ap-proximation

c) Instantaneous rate of change as the limit of average rate of change

d) Approximate rate of change from graphs and tables of values

3. Derivative as a function

a) Corresponding characteristics of graphs of f and f ’

b) Relationships between the increasing and decreasing of f and the sign of f ’.

12

c) The Mean Value Theorem and its geometric conse-quences.

d) Equations involving derivatives.

e) Graphing the derivative and using NDER on the cal-culator.

4. Second Derivative

a) Corresponding characteristics of graphs of f , f ’, and f ’’

b) Relationships between the concavity of f and the sign of f ’.

c) Points of inflection as places where concavity changes.

5. Application of derivatives

a) Analysis of curves, including the notions of monoton-icity and concavity

b) Optimization, both absolute (global) and relative (lo-cal) extrema

c) Modeling rates of change, including related rate prob-lems

d) Use of implicit differentiation to find the derivative of an inverse function

e) Interpretation of derivative as a rate of change in var-ied applied contexts

1) Velocity, speed, and acceleration

f) Geometric interpretation of differential equations via slope fields and the

relationship between slope fields and derivatives of implicitly defined functions

g) L’Hospitals Rule and its use in determining conver-gence of improper integrals

6. Computation of derivatives

a) Knowledge of derivatives of basic functions, includ-ing xr , exponential,

logarithmic, and trigonometric functions.

b) Basic rules for the derivative of sums, products, and quotients of functions.

c) Chain rule and implicit differentiation.

Second Semester

Integrals

1. Riemann sums

a) Concept of a Riemann sum over equal subdivisions

b) Computation of Riemann sums using left, right, and midpoint evaluation

points.

2. Interpretations and properties of definite integrals

13

a) Definite integral as a limit of Riemann sums

b) Definite integral of the rate of change of a quantity over an interval interpreted

as the change of the quantity over the interval

c) Basic properties of definite integrals

3. Applications of integrals

a) areas, volumes (by slicing), average value of a func-tion, linear distance traveled

4. Fundamental Theorem of Calculus

a) Use of the Fundamental Theorem to evaluate defi-nite integrals.

b) Use of the Fundamental Theorem to represent a par-ticular antiderivative,

and the analytical and graphical analysis of functions so defined.

5. Techniques of antidifferentiation

a) Antiderivatives following directly from derivatives of basic functions

b) Antiderivatives by substitution of variables (including change of limits for

definite integrals)

6. Applications of antidifferentiation

a) Finding specific antiderivatives using initial condi-tions, including applications

to motion along a line.

b) Solving separable differential equations and using them in modeling.

7. Numerical approximations to definite integrals

a) Use of Riemann sums and the Trapezoidal Rule to approximate definite

integrals of functions represented algebraically, geo-metrically, and by

tables of values.

14

(Incorporated by the AP Calculus Development Committee-adapted)

Functions, Graphs, and Limits


a) Comparing analytical information with graphs

b) Predicting a functions local and global behavior




c) Finding limits by explanation such as the Sandwich Theo-rem



b) Describing asymptotes using limits involving infinity

NOTES IN AP CALCULUS

❖ Students are required to use note-taking skills in class. These notes must be maintained in a four-section binder.

15

SECTION 3

AP Calculus AB Course Outline

c) Comparing relative magnitudes of functions and their rates of change



b) Geometric understanding of graphs of continuous functions

Derivatives











b) Relationships between the increasing and decreasing of f and the sign of f ’.



e) Graphing derivatives and using NDER on the TI84


a) Corresponding characteristics of graphs of f, f ’, f ’’

b) Relationships between the concavity of f and the sign of f ’.




b) Optimization, both absolute (global) and relative (lo-cal) extrema

c) Modeling rates of change, including related rate prob-lems

d) Use of implicit differentiation to find the derivative of an inverse function

e) Interpretation of derivative as a rate of change in var-ied applied contexts: Velocity, speed, and acceleration

f) Geometric interpretation of differential equations via slope fields and the relationship between slope fields and de-rivatives of implicitly defined functions

16

g) L’Hospitals Rule and its use in determining conver-gence of improper integrals


a) Knowledge of derivatives of basic functions, includ-ing power, exponential, logarithmic, and trigonometric func-tions.



Second Semester - Integration

1. Riemann sums


b) Computation of Riemann sums using left, right, and midpoint evaluation points.



b) Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval



a) areas, volumes (by slicing), average value of a func-tion, linear distance traveled



b) Use of the Fundamental Theorem to represent a par-ticular antiderivative, and the analytical and graphical analy-sis of functions so defined.



b) Antiderivatives by substitution of variables (including change of limits for definite integrals)


a) Finding specific antiderivatives using initial condi-tions, including applications to motion along a line.



a) Use of Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented alge-braically, geometrically, and by tables of values.

17

Chapter 2

AP Calculus BC is a second year University level

Calculus course taught at Cathedral High School. In the

2013/2014 academic year, there will be 14 seniors in the

course. In the past two years, 90% of Cathedral students

taking the AP Calculus BC class passed one or more AP

Calculus exams qualifying for one or two semesters of

Calculus credit at the University of California System.

Section I of this chapter is the syllabus. Section II is the

Timeline of topics and the Outline. This course begins

on August 16 with introductions and ends during Senior

Semester Exams May 19 - 21. The Timeline is purposely

undated since the course must follow the pace of the

students and accelerates with their understanding and

decelerates with their lack thereof.

AP CALCULUS BC- PERIOD 5

Overview

Advanced Placement Calculus BC is a University Level course taught at Cathedral High School. Although the class officially meets three times a week for two 80-minute classes and a 40-minute class, students are required to attend extra study sessions as scheduled. University credit is obtained only by passing the required May AP Exam; moreover, not all Universities accept passed AP exams for credit. In recent years, ninety percent of Cathedral students who take both the AB and BC exams on a two-year schedule have received eligible credit for one or two University of California Calculus courses. AP Calculus BC is an Honors course and contributes an extra 5 grade points per semester for every “C” grade or higher. This course satisfies the “a-g” requirements for the UC and California State system. Mr. Trafecanty maintains a California Single Subject Math Creden-tial and is certified to teach AP Calculus AB and BC.

AP CENTRAL WEBSITE

❖ AP Calculus students are expected to sign up for an account with AP Central. The site allows students to see example exams and obtain their AP scores electronically

19

SECTION 1

AP Calculus BC Syllabus

Course Sequence Leading to AP Calculus BC

Algebra I, Geometry, Algebra II, Precalculus, and AP Calcu-lus AB are the courses that students have passed with a “B” or better in order to be eligible for AP Calculus BC.

Teaching Philosophy and Strategies

CALCULUS AND A CLASSICAL EDUCATION

A core understanding of History, the Arts, English, Lan-guages, Christianity, Math and Science are requirements for success in a University environment; moreover, these under-standings are the most important ingredient for a whole per-son who can critically analyze the massive amounts of infor-mation in the environments we currently live.

TRUTHS and RULES in Mr. Trafecanty’s class

1) All people have an immeasurable worth and value re-gardless of their ability in Calculus.

RULE I: All students and teachers are to be treated with re-spect and are equals.

2) All people are loved by God and saved by Jesus Christ with no special reserved spaces for passing Calculus or attend-ing the “Great University”.

RULE II: Love your classmates as you love yourself.

3) All people must MAKE CHOICES that will affect their destiny and unfortunately sometimes the destiny of others both good and bad including choices of pursuing a Christian education with integrity. No human being has a prepro-grammed future disregarding their choices. We can all be a “Hero” or a “Villain” with equal success.

RULE III: All students are to both respect Christianity and to have established goals.

RULE IV: All students are to do their work on time, profes-sionally, and with the intent to both practice and learn the material taught.

SPECIAL NOTE 1: Success, as defined by Christians, CAN-NOT be about money/property or personal happiness or in-stant gratification as these can be achieved without work, love, or other people.

20

SPECIAL NOTE 2: Attending a University is more impor-tant than which. Great Universities have developed heroes and villains with equanimity and most have trouble identify-ing the true qualities of a good person least of all defining what a good Christian would be.

MR. TRAFECANTY’S EXPECTATIONS

1) STUDENT’S MUST BE MOTIVATED AND DE-VELOP A PURPOSE FOR WHY they are in CALCULUS.

2) A STUDENT’S MIND is the MOST IMPORTANT TOOL in the classroom. Although I will use many of the newest technologies to present the topics including the IPAD, Calculus textbook, web based video, smart boards, comput-ers and the TI84+, the best way to learn calculus is to sit down and do the problems without any other tools than the mind. Many colleges do not allow the use of technology in the first courses of Calculus. Class time is devoted to work-ing on Calculus problems.

3) A STUDENT’S CHRISTIAN MODELING IS THE MOST IMPORTANT CHARACTER TRAIT. I will ap-peal to a student’s inner sense of right and wrong, and burn-ing desire to be really alive. I hope that my two hundred

days of arguments and insistence on students defense of their work have a lasting affect on developing life-long learners whose relationships with God and others has primary impor-tance and whose skills and understanding allow them to think critically.

4) IT’S ALL ABOUT THE CALCULUS. The driving force for all lesson planning and discussions is the calculus that is listed on the next few pages The connection to science, eco-nomics, and research is intentional.

Homework

Homework assignments are assigned EVERY class day and due the next class. Generally, problems are assigned from the text and are related to any topic presented in class that day or a prior class. Questions on the homework are answered the next day of class NO LATE HOMEWORK even if absent. Work is posted using the app Notability and will be submitted electronically. See Chapter 3 - Homework

Notebooks

Students will be required to take notes during class. Notes will be written in a spiral notebook. Students will be re-quired to use a formal note-taking process.

21

assignments are assigned EVERY class day and due the next class. Problems are assigned from the text and are related to any topic presented in class that day or a prior class. Ques-tions on the homework are answered the next day of class NO LATE HOMEWORK even if absent. Work is posted on-line and through IBOOKS.

Assessment

All Calculus exams and quizzes are used to assess student learning. Students take written quizzes at least twice a week and take unit exams at the end of each unit. Students will also be quizzed orally in groups and individually to deter-mine their individual progress in learning how to verbalize calculus. They also participate in class and are selected at random to display answers to others. Students prepare and present Calculus topics in class.

AP Exam

Students are exposed to AP questions throughout the year. The class discusses how problems are answered on the AP exam, and they work on SAMPLE AP problems. Students take partial and full AP practice exams throughout the entire second semester in order to enhance their confidence in tak-ing exams and thoroughly demonstrate their learning. All stu-

dents are required to take an AP Calculus exam in May (there is an extra exam fee on the tuition bill).

Required Materials

Calculus, 3rd Edition by Finney, Demana, Waits, Kennedy

TI84 calculator, iPAD

4 section 3 ring-binder – HW LIST, HW , NOTES, PAPER

Grading

89.5% - A, 79.5%- 89.49% - B, 69.5% - 79.49% - C, 59.5% - 69.49% - D Below 59.5% is an F grade. D and F grades do not qualify for Honors Credit and are not counted as units achieved for the University of California System.

40% Exams 20% Quizzes 15% Final

15% HW 10% iPad Protocol

****Special Notes-New to 2013/2014****

1) iPads are required for class and points will be lost if they are not brought to class every day.

22

2) The first quiz covers the Summer Reading Book, “Mon-ster” which will be administered on Monday August 19

3) iPad Protocol: 10% of the student’s final grade will be de-termined by iPad Protocol (appropriate integration of the iPad in the classroom). This may include presentations created/delivered on the iPad, bringing iPads charged to class, appropriate usage during class, successful navigation of apps, electronic books and materials, and online re-sources, homework submitted electronically. Online re-sources may include Edmodo, PowerSchool, Khan Acad-emy, the school website, Google Docs. iPads must be prop-erly registered for use at Cathedral and students must fol-low all school/ teacher-specific iPad policies to receive credit in this course.

Extra Credit/Make-up Work

No Extra Credit and No Make-up quizzes, Finals or Home-work.

A Make-up Unit Exam can only be taken on a scheduled Make-up Exam day as determined by the teacher

23

First Semester

Functions, Graphs, and Limits (August 19 - 23)





c) Finding limits by explanation such as the Sandwich Theorem



b) Describing asymptotes with limits involving infinity

c) Comparing relative magnitudes of functions and rates of change



THE ONLINE VIDEO LEARNING WEBSITE

❖ Students in AP Calculus will maintain accounts, monitored by the teacher, on Khan Academy.

❖ Students can use the Khan Academy to review or learn new material.

24

SECTION 2

AP Calculus BC Timeline and Outline

b) Geometric understanding of graphs of continuous functions. (The Intermediate Value Theorem and Extreme Value Theorem)

5. Parametric, polar, and vector functions

Derivatives (7 weeks – 1 week per section with a Review Week)











b) Relationship between increasing and decreasing f and f ’.



e) Graphing the derivative and using NDER on the cal-culator.


a) Corresponding characteristics of graphs of f , f ’, and f ’’

b) Relationships between the concavity of of f and the sign of f ’.




b) Analysis of planar curves n parametric, polar, and vec-tor form, including velocity and acceleration vectors.

c) Optimization, both absolute (global) and relative (lo-cal) extrema

d) Modeling rates of change, including related rate prob-lems

e) Use of implicit differentiation to find derivatives of in-verse functions

f) Interpretation of derivative as a rate of change in var-ied applied contexts

1) Velocity, speed, and acceleration

25

g) Geometric interpretation of differential equations via slope fields and the relationship between slope fields and de-rivatives of implicitly defined functions

h) L’Hospitals Rule and its use in determining conver-gence of improper integrals and series

i) Derivatives of parametric, polar, and vector function


a) Knowledge of derivatives of basic functions, includ-ing power, exponential, logarithmic, and trigonometric func-tions.



d) Numerical solution of differential equations using Euler’s method

Integrals (8 weeks – 1 week per section)

1. Riemann sums


b) Computation of Riemann sums using left, right, and midpoint evaluation points.



b) Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval



a) areas, volumes (by slicing), average value of a func-tion, linear distance traveled, volumes (shell method)

4. Appropriate integrals are used for

a) finding the area of a region bounded by polar curves

b) the length of a curve



b) Use of the Fundamental Theorem to represent a par-ticular antiderivative, and the analytical and graphical analy-sis of functions.



b) Antiderivatives by substitution of variables (including change of limits for definite integrals)

c) Antiderivatives by parts and simple fractions

d) Improper integrals as limits of definite integrals

26


a) Finding specific antiderivatives using initial condi-tions, including applications to motion along a line.


c) Solving logistic differential equations and using them in modeling numerical approximations to definite integrals.


a) Use of Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented alge-braically, geometrically, and by tables of values.

Second Semester

Polynomial Approximation and Series (9 weeks – 4 weeks per section)

1. Concept of Series

a) sequence of partial sums

b) convergence and divergence in terms of limits of the sequence of partial sums

c) Motivating examples including decimal approxima-tion

d) Geometric series

e) The harmonic series

f) alternating series with error bound

g) terms of series as areas of rectangles and relationship to improper integrals, integral test and testing the conver-gence of p-series

h) ratio test for convergence

i) Comparing series to test for convergence

2. Taylor Series

a) Taylor polynomial approximation with graphical dem-onstration of convergence

b) Maclaurin series and the general Taylor series cen-tered at x = a

c) Maclaurin series for exponential, sine, cosine, and

1/(1 – x)

d Formal manipulation of Taylor series and shortcuts to computing including substitution, differentiation, antidifferen-tiation, and the formation of new series from known series.

e) Funtions defined by power series

f ) Radius and interval of convergence of power series

g) Lagrange error bound for Taylor polynomials

Preparation and Review (6 weeks)

27

Chapter 3

In this section you will find school and classroom policies

and procedures in more elaborate detail such as iPad

procedures, homework policies, and parent handouts.

POLICIES, PROCEDURES &

OTHER MATERIALS

Homework is generally assigned every class day. It is submitted online through the Notability App.

Homework may not be made up even if a student is absent. It must be submitted to the teacher by email. Homework must be done using the Notability App.

Mr. Trafecanty’s email: [email protected] POLICIES

❖ Homework must be on the Notability App.

29

SECTION 1

Homework Policies

mailto:[email protected]

mailto:[email protected]

10% of the student’s final grade will be determined by iPad Pro-tocol (appropriate integration of the iPad in the classroom). This may include presentations created/delivered on the iPad, bringing iPads charged to class, appropriate usage during class, successful navigation of apps, electronic books and materials, and online resources, homework submitted electronically. On-line resources may include Edmodo, PowerSchool, Khan Acad-emy, the school website, Google Docs. iPads must be properly registered for use at Cathedral and students must follow all school/ teacher-specific iPad policies to receive credit in this course.

IPAD PROTOCOL

❖ 10% of grades in all classes is allocated for student proper usage of iPads

30

SECTION 2

iPad

31

Student'iPad'User'Rules'

In'Classroom' Consequences'iPad%in%backpack%until%told%to%take%out% Inform%student%of%loss%of%daily%participation%points.%%

AND'1st%Shut%it%down%and%put%away%until%the%end%of%period.%2nd%Confiscate%until%the%end%of%period.%3rd%Teacher%e?mail/call%parent%Last'Resort%Send%student%to%Dean%of%Students.%

Student%plays%games/messaging.% Inform%student%of%loss%of%daily%participation%points.%%AND'1st%Shut%it%down%and%put%away%until%the%end%of%period.%2nd%Confiscate%until%the%end%of%period.%3rd%Teacher%e?mail/call%parent%Last'Resort%Send%student%%to%Dean%of%Students%

Student%on%wrong%site%in%class.% Inform%student%of%loss%of%daily%participation%points.%%AND'1st%Shut%it%down%and%put%away%until%the%end%of%period.%2nd%Confiscate%until%the%end%of%period.%3rd%Teacher%e?mail/call%parent%Last'Resort%Send%student%%to%Dean%of%Students%

Touching/sharing%/locking%/disabling%another%student’s%iPad%

Loss%of%daily%participation%points.%AND'Send%Student%to%Dean%of%Students%

Not%charged% Loss%of%daily%participation%points.%AND'Teacher%e?mail/call%parent%

%

School'wide' Consequences%Student%accesses%inappropriate%sites%% 1st%violation:%Internet%access%will%be%limited%

2rd%violation%:%Internet%access%will%be%blocked%for%the%rest%of%the%year%

Bypassing%CHS%Firewalls% 1st%violation:%Internet%access%will%be%limited%2rd%violation%:%:%Internet%access%will%be%blocked%for%the%rest%of%the%year%

Pictures,%Video%and/or%audio%recording%teacher%or%staff%%without%permission%

Suspension%AND%Camera%restriction%

Inappropriate%pictures,%Video%and/or%audio%recording%

Suspension%AND%Camera%restriction%

Transfer%of%or%receiving%pictures%of%homework,%class%assignments,%quizzes%or%tests%to%other%students.%

Zero%on%assignment,%quiz%or%test%AND%Suspension%AND%Camera%restriction%

%

Polya’s Problem Solving Techniques

In 1945 George Polya published the book How To Solve It which quickly became his most prized publication. In this book he iden-tifies four basic principles of problem solving.

Polya’s First Principle: Understand the problem

This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems sim-ply because they don’t understand it fully, or even in part.

• Do you understand all the words used in stating the problem?

• What are you asked to find or show?

• Can you restate the problem in your own words?

• Can you think of a picture or diagram that might help you understand the problem?

• Is there enough information to enable you to find a solution?

POLYA’S PROBLEM SOLVING

❖ UPEC - Understand, Plan, Execute and Check

32

SECTION 3

Solving Word Problems

Polya’s Second Principle: Devise a plan

Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choos-ing a strategy increasingly easy. A partial list of strategies is included:

• Guess and check

• Make an orderly list

• Eliminate possibilities

• Use symmetry

• Consider special cases

• Use direct reasoning

• Solve an equation

• Look for a pattern • Draw a picture • Solve a simpler problem • Use a model • Work backwards • Use a formula • Be ingenious

Polya’s Third Principle: Execute the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the nec-essary skills. Persist with the plan that you have chosen. If it

continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by profession-als.

Polya’s Fourth Principle: Check or Look back

Polya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn’t. Doing this will enable you to predict what strategy to use to solve future problems.

So starting on the next page, here is a summary, in the mas-ter’s own words, on strategies for attacking problems in mathematics class. This is taken from the book, How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.

1. UNDERSTAND THE PROBLEM

• What is the unknown? What are the data? What is the condition?

• Is it possible to satisfy the condition? Is the condi-tion sufficient to deter- mine the unknown? Or is it insuffi-cient? Or redundant? Or contradictory?

• Draw a figure. Introduce suitable notation.

• Separate the various parts of the condition. Can you write them down?

2. PLAN

• Find the connection between the data and the un-known. You may be obliged to consider auxiliary problems if

33

an immediate connection cannot be found. You should ob-tain eventually a plan of the solution.

• Have you seen it before? Or have you seen the same problem in a slightly different form?

• Do you know a related problem? Do you know a theorem that could be useful?

• Look at the unknown! Try to think of a familiar problem having the same or a similar unknown.

• Here is a problem related to yours and solved be-fore. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

• Could you restate the problem? Could you restate it still differently? Go back to definitions.

• If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to deter-mine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?

• Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions

involved in the problem?

3. EXECUTE THE PLAN

• Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

4. CHECK

• Can you check the result? Can you check the argument? • Can you derive the solution differently? Can you see it at a glance? Can you use the result, or the method, for some other problem?

34

Cathedral High School Math Department

Textbook Rental Agreement 2013-2014

Textbook: Calculus by Demana and Finney

Rental: $15

Replacement Fee: $75

I ______________________ have agreed to rent the Calculus book for the 2013-2014 school year. It is my responsibility to take care to preserve the structural integrity and functionality of the textbook. I must return the book in good re-usable condi-tion including but not limited to: properly labeled and num-bered, attached cover, free from writing or other markings on or inside of the book. If the teacher and/or Dean of Studies deter-mine that the book is not re-usable I must replace the book or pay the $75 replacement fee.

BOOK RENTAL

❖ The Calculus book is rented through the school starting with juniors this year (2013-2014). The rental fee is $15. The same book is used for AP Calculus AB and AP Calculus BC and must be rented each year.

❖ The rental agreement form is on the right.

35

SECTION 4

RENTAL AGREEMENT

Documents

AP Calculus syllabi - Edl€¦ · Section 1 is the AP Calculus AB Course Syllabus, Section 2 is the AP Calculus AB Pacing Plan, and Section 3 is the AB Calculus AB Topic Outline AP