Isaac Newton

(death mask)

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AP CALCULUS AB SYLLABUS

 

Leslie Howe

Farragut High School

Class Policies

Class Notes First Block

Class Notes Third Block

Honor Code

Help for Calculus Students and links to outside resources

 

Countdown to AP

 

 

 

Gotfrid Leibniz

 

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WHAT IS CALCULUS:

 

 

Calculus is the mathematics of change.  Developed by Sir Isaac Newton and Gotfrid Leibniz, calculus provides the answer to the persistent question, "When are we ever going to use this?"  All the skills of elementary mathematics come together to enable the implementation of Calculus concepts to solve and model dynamic real world problems. Instantaneous slope, velocity, acceleration area and volume can all be calculated.

 

Calculus builds on everything!

 

Course Description:

 

The Advanced Placement Program provides an opportunity for secondary school students to pursue and receive credit for college-level course work completed at the secondary school level. Calculus AB uses the most recent College Board Advanced Placement Curriculum outline. This national outline is the summary of concepts needed for preparation for the Advanced Placement Examination. This course, while maintaining strict, traditional mathematical content, will incorporate technology to study limits, derivatives, integrals, and applications. Previous mathematics courses will serve as a foundation for calculus. From algebra and pre-calculus, students should be able to recognize and understand patterns and functions, solve equations, and should be skilled at analyzing functions both algebraically and graphically. From geometry, students should be familiar with figures, areas and volumes. Students should also be able to use and analyze data, find and use prediction equations and generally be proficient using graphing technology.  Calculus is an advanced mathematics course that uses meaningful problems and appropriate technology to develop concepts and applications related to the continuity and discontinuity of functions as well as to differentiation, and integration.

 

Technology Fee $10/AP MATH  FEE  $25.00  TOTAL $35

This $25.00 fee covers the cost of materials that will be given to students to prepare them for the Advanced Placement Test for BC Calculus.   There is also a $10 math technology fee. Make checks payable to FHS.  Bring to class ASAP.

 

 

References and Materials:

 

TEXTBOOK Link: Calculus Graphical, Numerical, Algebraic ( $72.47 )

 

Larson, Hostetler and Edwards, Houghton Mifflin Company, 2002 (classroom set)

TEXTBOOK Link: Link for textbook calculus quizzes

 

 

 

 

LINK:  http://www.collegeboard.com/student/testing/ap/sub_calab.html?calcab

 

AUDIOVISUAL MATERIALS  used in this class may include:

Biographical videos on the lives of mathematicians (Newton, Leibniz, Euler and others)

Videos or selections from videos may be shown on the development of mathematics by topic  (Disney, Nasa, Math Magic, etc.)

 

MATERIALS NEEDED:

Because of the focus of the textbook and the demands of preparation for college and the AP Exam, a graphing calculator is essential to this course. The TI-89 is the recommended calculator.

A 2-3 inch loose-leaf three-ring notebook is necessary for organization of class notes, homework, handouts, tests and quizzes.

 

 


EVALUATION:

Grades will be based on the following:

Tests and Quizzes

(The Farragut High Math Department has a policy of giving both calculator and non-calculator parts of each exam.)

Homework

Board Problems

Authentic Assessment

Class Participation

 

TECHNOLOGY:

This course requires the use of a graphing calculator.  If a student does not have access to a calculator, one will be provided.  The calculator must have capabilities that will allow it to:

  1. Plot the graph of a function within an arbitrary viewing window,
  2. Find the zeros of functions (solve equations numerically),
  3. Numerically calculate the derivative of a function, and
  4. Numerically calculate the value of a definite integral.

 

The student will become proficient with these skills as they solve problems, experiment, interpret results and support conclusions.

 

The subject syllabus is posted on the school web site and contains links to interactive resources on the internet including those that provide visual demonstrations of many topics including limits, related rates, and volumes of revolution.   Teacher created web pages that reinforce essential concepts are also posted.

 

Smart technology is used to post notes and links to relevant supplemental material.  A projected computer and smartboard is used in the classroom, providing visualization and the potential for posting notes and other references on the internet.  Students who are absent will use these resources to keep pace with the class. Internet access is available before school and in the school library for any student without home access.

 

The department computer math lab is also used.  Students use software in the lab to see calculus concepts in action and to reinforce learning.

 

Teaching Strategies:

 

Chapter tests given in class include both “No Calculator” and “Calculator” portions to better prepare the students for the Advanced Placement Examination.

 

Students study problems and present solutions that are described graphically, numerically, analytically, and verbally.  Students are required to justify their answers appropriately and be able to express the connection between the various forms of description.

 

Students practice multiple-choice questions from previously released AP exams.

 

Students practice free-response questions from previous AP exams where they are exposed to the grading rubric.  They are taught to write solutions that are precise and use well-written sentences.

 

Students present their solutions of problems to the class where they are expected to orally justify their analysis and reasoning.

 

Students work in collaborative groups.  Students communicate orally within their groups as they seek consensus on the solution of a problem.  The problems they work on will require them to practice modeling a physical situation with a function, differential equation, or an integral.

 

The students become proficient with the capabilities of the calculator as well as become informed of its limitations.  They use the calculator to solve problems, experiment, interpret results and support conclusions.  Students learn how to graph slope fields from a program or built-in function.  Additionally, students use the built-in features of the calculator.  As an example, they use the zoom feature to analyze local linearity and experiment to see whether a specific point is differentiable.

 

Students are required to do homework problems that are assigned from the primary text and from released exams (free-response and multiple-choice).

 

NOTE:

 

The State of Tennessee requires that each student will complete at least 4.5 minutes of physical activity during the class period or 22.5 minutes of aggregate time over the course of the week during this block.  This notice serves as documentation of such activity, and classroom teachers should be apprised of any physical limitations that would impact this State mandate.

 

Topics:
[Section references from the primary text will appear in brackets after each student performance indicator.]

 

Standard 1.0:  Functions, Graphs, and limits

 

Learning Expectations:

 

Students will:

 

1.1  Analyze the graphs of functions and relations;

1.2  Evaluate the limits of functions (including one-sided limits);

1.3  Analyze asymptotic and unbounded behavior;

1.4  Understand continuity as a property of functions;

 

 

Student Performance Indicators:

 

1.1.a Graph functions on a graphing calculator using the appropriate windows. [1.1 – 1.6]

1.1.b Recognize functions by type:  linear, quadratic, polynomial, rational, exponential, logarithmic, power, roots, absolute value, [1.1 – 1.6]

1.1.c Predict and explain the observed local and global behavior of a function. [4.1 – 4.3]

 

1.2.a Demonstrate an intuitive understanding of the limiting process. [2.1]

1.2.b Calculate limits using algebra. [2.1]

1.2.c Estimate limits from graphs or tables of data. [2.1]

 

1.3.a Demonstrate an understanding of asymptotes in terms of graphical behavior. [2.2]

1.3.b Describe asymptotic behavior in terms of limits involving infinity. [2.2]

1.3.c Compare relative magnitudes of functions and their rates of change.  (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.) [8.2]

 

1.4.a Develop an intuitive understanding of continuity.

(Close values of the domain lead to close values of the range.)  [3.3]

1.4.b Develop and understanding of continuity in terms of limits. [2.4]

1.4.c Develop a geometric understanding of graphs of continuous functions. (Intermediate Value Theorem and Extreme Value Theorem.) [2.3 and 4.1]

 

 

 

 

 

 

 

Standard 2.0:  Derivatives

 

Learning Expectations:

 

Students will:

 

2.1 Develop the concepts of the derivative;

2.2 Have an understanding of the derivative at a point;

2.3 Investigate the derivative as a function;

2.4 Explore second derivatives;

2.5 Apply derivatives;

2.6 Compute derivatives.

 

Student Performance Indicators:

 

2.1.a Compute derivatives graphically, numerically, and analytically. [3.1 – 3.3]

2.1.b Interpret a derivative as an instantaneous rate of change. [2.4]

2.1.c Define the derivative as the limit of the difference quotient. [3.1]

2.1.d Understand the relationship between differentiability and continuity. [3.2]

 

2.2.a Determine the slope of a curve at a point.  Examples to emphasize include points at which there are vertical tangents and points at which there are no tangents. [3.1 – 3.2]

2.2.b Determine tangent lines to a curve at a point and local linear approximations.

[3.1, 4.5]

2.2.c Compute the instantaneous rate of change as the limit of the average rate of change.

[2.8, 3.1, 3.4]

2.2.d Approximate the rate of change from graphs and tables of values.  [3.1]

 

2.3.a Determine the corresponding characteristics of the graphs of f and f ˘.  [3.1, 3.4]

2.3.b Determine the relationship between the increasing and decreasing behavior of f and the sign of f ˘.  [4.2]

2.3.c Investigate the Mean Value Theorem and its geometric consequences.  [4.2]

2.3.d Explore equations involving derivatives.  Verbal descriptions are to be translated into equations involving derivatives and vice versa.  [3.4, 4.4, 4.6]

 

2.4.a Explore the corresponding characteristics of the graphs of f, f ˘, and f ˛.  [4.3]

2.4.b Determine the relationship between the concavity of f and the sign of f ˛.  [4.3]

2.4.c Find the points of inflection as places where the concavity changes.  [4.3]

 

2.5.a Analyze curves, including the notions of monotoncity and concavity.  [4.3]

2.5.b Perform optimization for both absolute (global) and relative (local) extrema.  [4.4]

2.5.c Model rates of change, including related rates problems.  [4.6]

2.5.d Use implicit differentiation to find the derivative of an inverse function.  [3.8]

2.5.e Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.  [3.4]

2.5.f Investigate the geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.  [6.1]

 

2.6.a Attain knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.  [3.3, 3.5, 3.8, 3.9]

2.6.b Apply the basic rules for the derivative of sums, products, and quotients of functions.  [3.3]

2.6.c Apply the chain rule and implicit differentiation.  [3.6, 3.7]

 

Standard 3.0:  Integrals

 

Learning Expectations:

 

Students will:

 

3.1 Discover the interpretations and the properties of the definite integral;

3.2 Apply integrals;

3.3 Discover the Fundamental Theorem of Calculus;

3.4 Explore techniques of antidifferentiation;

3.5 Apply antiderivatives;

3.6 Compute numerical approximations to definite integrals.

 

Student Performance Indicators:

 

3.1.a Compute Riemann sums using left, right, and midpoint evaluation points.  [5.1]

3.1.b Determine the definite integral as a limit of Riemann sums over equal subdivisions. [5.2]

3.1.c Determine the definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:    [7.1]

3.1.d Explore the basic properties of definite integrals.  (Examples include additivity and linearity.)  [6.1]

 

3.2.a Use appropriate integrals in a variety of applications to model physical, biological, or economic situations.  [7.5]

3.2.b Use knowledge and techniques for solving applications and adapt this knowledge to solve similar application problems.  [7.1 – 7.5]

3.2.c Use the integral of a rate of change to give accumulated change.  [7.1 – 7.5]

3.2.d Represent a Riemann sum as a definite integral.  [5.4]

3.2.e Find the area of a region.  [5.2, 5.3, 7.2]

3.2.f Find the volume of a solid with known cross sections.  [7.3]

3.2.g Find the average value of a function.  [5.3]

3.2.h  Find the distance traveled by a particle along a line.  [7.1]

 

3.3.a Use the Fundamental Theorem to evaluate definite integrals.  [5.3, 5.4]

3.3.b Use the Fundamental Theorem to represent a particular antiderivatives, and the analytic and graphical analysis of functions so defined.  [5.3, 5.4]

 

3.4.a Explore how antiderivatives follow directly from derivatives of basic functions. [6.1]

 

3.5.a Find specific antiderivatives using initial conditions, including applications to motion along a line.  [6.1, 7.1]

3.5.b Solve separable differential equations and using them in modeling.  In particular, studying the equation y ‘ = ky and exponential growth.  [6.1, 6.4]

 

3.6.a Use Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. [5.1, 5.2, 5.5]

 

 

 

Section Summary and Pacing Guide for Calculus AB

 

Sections from the primary text that must be covered:

 

Chapter 2:  Sections 1, 2, 3, and 4

Chapter 3:  Sections 1, 2, 3, 4, 5, 6, 7, 8, and 9

Chapter 4:  Sections 1, 2, 3, 4, and 6

Chapter 5:  Sections 1, 2, 3, 4, and 5

Chapter 6:  Sections 1, 2, and 4

Chapter 7:  Sections 1, 2, and 3

 

Pacing guide:

Day

Section

Topic

1

Quiz

Quiz: Sections 1.1 – 1.6. Skip section 1.4

2

2.1

Rates of Change and Limits

3

2.2

Limits Involving Infinity

4

Quiz

Quiz: Limits

5

2.3

Continuity

6

2.4

Rates of Change and Tangent Lines

7

Quiz

Quiz: Chapter 2

8

3.1

Derivative of a Function

9

 

Derivative of a Function (continued)

10

3.2

Differentiability

11

 

Review

12

Test

Test: Chapter 2 and Sections 3.1 – 3.2

13

3.3

Rules of Differentiation

14

3.4

Velocity and Other Rates of Change

15

 

Velocity and Other Rates of Change (continued)

16

3.5

Derivatives of Trigonometric Functions

17

 

Derivatives of Trigonometric Functions (continued)

18

3.6

Chain Rule

19

 

Chain Rule (continued)

20

 

Review

21

Quiz

Quiz: 3.4 – 3.6

22

3.7

Implicit Differentiation

23

 

Implicit Differentiation (continued)

24

3.8

Derivatives of Inverse Trigonometric Functions

25

3.9

Derivatives of Exponential and Logarithmic Functions

26

 

Derivatives of Exponential and Logarithmic Functions (continued)

27

 

Concept Connections

28

 

Review

29

Test

Test: Chapter 3

30

4.1

Extreme Values of Functions, Optional - Section 4.5

31

4.2

Mean Value Theorem

32

4.3

Connecting Derivatives to the Graph of a Function

33

 

Connecting Derivatives to the Graph of a Function (continued)

34

 

Concept Connections

35

Quiz

Quiz: 4.1 – 4.3

36

4.4

Modeling and Optimization

37

 

Modeling and Optimization (continued)

38

4.6

Related Rates

39

 

Related Rates (continued)

40

 

Related Rates (continued)

41

 

Review

42

Test

Test: Chapter 4

43

5.1

Estimating with Finite Sums

44

5.2

Definite Integrals

45

5.3

Definite Integrals and Antiderivatives

46

 

Concept Connections

47

5.4

Fundamental Theorem of Calculus

48

 

Concept Connections

49

5.5

Trapezoidal Rule (Skip Simpson’s Rule)

50

 

Review

51

Test

Test: Chapter 5

52

6.1

Antiderivatives and Slope Fields

53

6.2

Integration by Substitution

54

 

Integration by Substitution (continued)

55

6.4

Exponential Growth and Decay

56

 

Exponential Growth and Decay (continued)

57

Quiz

Quiz: Chapter 6

58

7.1

Integral as Net Change

59

7.2

Areas in a Plane

60

7.3

Volumes

61

 

Volumes (continued)

62

 

Volumes (continued)

63

Quiz

Quiz 7.1-7.3

64

 

Review

65