AP Calculus AB .k12.va.us



AP Calculus AB

Syllabus

Course Overview:

At this school, we cover everything in the Calculus AB topic outline as it appears in the AP Calculus Course Description, including partial fractions, integration by parts, and L’Hopital’s Rule. The textbook used for this class is Calculus: Graphical, Numerical, Algebraic, 3rd ed. AP edition, By Finney, Demana, Waits, Kennedy. The main objectives for the course are that students will be properly prepared for success on the AP exam and in subsequent mathematics courses. There is an attempt to balance mastery of concepts, critical thinking, and integration of technology.

Course Planner:

Below is the sequence of our AP Calculus course.

|High School Mathematics AB | |

|AP Calculus Course Outline – 47 minute periods | |

|Textbook: Calculus: Graphical, Numerical, Algebraic, by Finney, et al. 2007 |

| | | |

| | | |

|SECTION |TITLE |PAGES |

|  |  |  |

|  |FIRST SEMESTER |  |

| |First Nine-Weeks | |

|5 days | Prerequisites for Calculus (Chapter 1) |  |

|1.1 | Lines |1-11 |

|1.2 | Functions and Graphs |12-21 |

|1.3 | Exponential Functions |22-29 |

|1.4 | Parametric Equations |30-36 |

|1.5 | Functions and Logarithms |37-45 |

|1.6 | Trigonometric Functions |46-55 |

|  |  |  |

|11 days | Limits and Continuity (Chapter 2) |  |

|2.1 | Rates of Change and Limits |56-69 |

|2.2 | Limits Involving Infinity |70-77 |

|2.3 | Continuity |78-86 |

|2.4 | Rates of Change and Tangent Lines |87-94 |

|  |  |  |

|30 days | Derivatives (Chapter 3) |  |

|3.1 | Derivative of a Function |99-108 |

|3.2 | Differentiability |109-115 |

|3.3 | Rules for Differentiation |116-126 |

|3.4 | Velocity and Other Rates of Change |127-140 |

|3.5 | Derivatives of Trigonometric Functions |141-147 |

|3.6 | Chain Rule |148-156 |

|3.7 | Implicit Differentiation |157-164 |

|3.9 | Derivatives of Exponential and Logarithm Functions |172-180 |

| |Second Nine-Weeks | |

|4 days |Derivatives (Chapter 3) | |

|3.9 | Derivatives of Exponential and Logarithm Functions |172-180 |

|3.8 | Derivatives of Inverse Trigonometric Functions |165-171 |

|  |  |  |

|32 days | Applications of Derivatives (Chapter 4) |  |

|4.6 | Related Rates |246-255 |

|4.1 | Extreme Values of Functions |187-195 |

|4.2 | Mean Value Theorem |196-204 |

|4.3 | Connection f' and f'' with the graph of f |205-218 |

|4.4 | Modeling and Optimization |219-232 |

| 8.1 |L’Hopital’s Rule | |

|  | SHOE BOX PROJECT |  |

|5 days |The Definite Integral (Chapter 5) |  |

|5.1 | Estimating with Finite Sums |263-273 |

|5.2 | Definite Integrals |274-284 |

| | | |

|2 days |EXAM REVIEW |  |

| | | |

|1 day |FIRST SEMESTER EXAM |  |

|  |  |  |

|  |SECOND SEMESTER |  |

| |Third Nine-Weeks | |

|18 days |The Definite Integral (Chapter 5) |  |

|5.3 | Definite Integrals and Antiderivatives |285-293 |

|5.4 | Fundamental Theorem of Calculus |294-305 |

|5.5 | Trapezoidal Rule |306-315 |

|  |  |  |

|13 days |Differential Equations and Mathematical Modeling (Chapter 6) |  |

|6.1 | Antiderivatives and Slope Fields |320-330 |

|6.2 | Integration using Chain Rule |331-340 |

|6.3 | Integration by Parts |341-348 |

|6.4 | Exponential Growth and Decay |350-361 |

|6.5 | Logistic Growth |362-372 |

|  |  |  |

|14 days |Applications of Definite Integrals (Chapter 7) |  |

|7.1 | Integral as Net Change |378-389 |

|7.2 | Areas in the Plane |390-398 |

|7.3 | Volumes |399-411 |

|7.5 | Applications from Science and Statistics |419-429 |

|  | FOOD VOLUME PROJECT |  |

|  |  |  |

|  |Fourth Nine-Weeks |  |

|10 days |Motion (throughout the text) | |

|2 days |Derivatives of Inverse Functions |165-170 |

|13 days |AP REVIEW |  |

|1 day  |AP EXAM |  |

|  |  |  |

|16 days |Preparation for AP Calculus 2 |  |

|7.3 | Shell Method |399-411 |

|7.4 |Lengths of Curves |412-418 |

|8.4 | Partial Fractions |371 and more  |

|  |  |  |

|3 days |POST AP EXAM PROJECTS |  |

|  | MODEL OF A VOLUME BY CROSS-SECTION |  |

|  |  |  |

|2 days |EXAM REVIEW |  |

|  |  |  |

|1 day |FINAL EXAM |  |

Teaching Strategies

For a number of students entering Calculus AB, they have had honors level classes before and are prepared for the rigor of a more advanced class. An electronic webpage maintained by the teacher shows the pacing guide for the year, clearly indicating the target date of the AP exam. The teacher works as a coach, with students working as a team to develop the skills necessary to achieve success on the AP exam. The teacher offers numerous after school study sessions, including several before semester exams.

Technology and Computer Software

The teacher and students use the TI-89 graphing calculator in class. Nearly all students have one of these calculators.

A number of powerpoint presentations are incorporated in the class utilizing an ActivInspire Board. The presentations provide aid in teaching many of the calculus concepts, such as the definition of the limit. Also, the AP collegeboard website: is incorporated to enrich understanding.

Student Evaluation

Each nine weeks, a student’s grade is computed using homework, quizzes, and tests as categories. Each grade represents approximately 60 percent of the test average, 30 percent of the quiz average and 10 percent of the homework average. For the final grade, 80 percent of the grade is the cumulative average of each marking period, 10 percent represents the semester exam, and the remaining 10 percent is the final exam. Quizzes and tests vary depending on the teacher, but all follow closely the curriculum of the designated textbook and that of the College Board. Many tests are half calculator and half non-calculator. The semester and final exams incorporate calculator portions and non-calculator portions, as well as multiple choice and free-response. Throughout the course students are given frequent exposure to questions from previous AP exams. The students also are made familiar with scoring of the free response questions with frequent practice.

Teacher Resources:

Primary Textbook;

Finney, Demana, Waits, and Kennedy. AP edition Calculus: Graphical, Numerical, Algebraic. 3rd ed. Boston: Pearson Prentice Hall, 2007.

A technology resource manual that accompanied the selected textbook is utilized for the TI-83 and TI-89. Students are strongly recommended to purchase either a TI-83 or TI-89. Power point presentation slides are also part of the package with the textbook and are used frequently in class as a visual aid for enhancing learning.

Student Activities:

The following activity was developed to strengthen the student’s understanding of the limits of functions.

The main objectives in using this lesson is to help students to develop an intuitive understanding of the nature of limits, lay the foundation for the use of limits in calculus and to practice evaluating limits graphically, numerically, and algebraically.

In the lab the student studies the behavior of a function f near a specified point. As they go through the lab they will need to find the limit numerically (using tables on the TI-83 or TI-89 calculator), graphically, or algebraically. The students work in groups of 3 or 4 and all groups have TI-83 and TI-89 owners in them so the students have exposure to both.

As the students work through the problems, there are a number of higher-order questions that require the students to write up their conclusions as a group.

*See next page for Limits Lab to satisfy the question of using graphing calculators to experiment.

LAB: LIMITS OF FUNCTIONS [pic]

Goals:

• To develop an intuitive understanding of the nature of limits

• To lay the foundation for the use of limits in calculus

• To evaluate limits graphically, numerically, and algebraically

Procedure

In this lab, we will study the behavior of a function f near a specified point. While this may be a straightforward process, it can also be very subtle. In some instances in calculus the process for finding a limit must be applied carefully. By gaining an intuitive fell for the idea of limits, you will be laying a solid foundation for success in calculus.

As you go through this lab, evaluate the limits by the method given: numerically (using table on calculator), graphically, or algebraically. You might also want to try to use another method as a check and to see how all 3 methods can be used.

1. Consider the function f defined by f(x)=[pic]

a) By successive evaluation of f at x= 1.8, 1.9, 1.99, 1.999, and 1.9999, what do you think happens to the values of f as x increases towards 2?

b) Do a similar evaluation of f for values of x slightly greater than 2. Comment on your results. As a shorthand, we will describe these results by writing [pic]

c) Evaluate f (2) Comment on this answer.

2. Use the function f as above; consider what happens as x approaches 1.

a) Study this situation numerically as you did in parts a and b above. What are your conclusions? In particular, what is [pic]?

b) What is f(1)? What is the difference in the 2 situations?

c) Factor f(x) and simplify. Now substitute 1 for x. Is this consistent with your findings in part a?

3. Using your calculator, determine the values of these limits.

a) [pic] where [pic]

b) [pic]

c) [pic]

d) [pic]

4. Limits can sometimes fail to exist. Investigate the following limits and explain why you think each does not exist. Use your calculator to graph functions as needed. What are these different from the functions previously considered?

a) [pic] b) [pic] c) [pic]

d)[pic] e) [pic]

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