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 tetbook used for this class is Calculus: Graphical, Numerical, Algebraic, 3 rd 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 eam 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 Tetbook: 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 Eponential 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 Eponential and Logarithm Functions 172-180
Second Nine-Weeks 4 days Derivatives (Chapter 3) 3.9 Derivatives of Eponential 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 Etreme 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 Eponential 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 tet) 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 eam. The teacher works as a coach, with students working as a team to develop the skills necessary to achieve success on the AP eam. The teacher offers numerous after school study sessions, including several before semester eams. Technology and Computer Software The teacher and students use the TI-89 graphing calculator, as well as the TI-83 or TI-84 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: http://www.apevs.com/apeui/ is incorporated to enrich understanding. Student Activities (utilizing the graphing calculator): In PreCalculus, students learn to find an appropriate viewing window to produce a complete graph of a function, find the zeros of a function, evaluate a function at a point, and determine symmetry. These topics are reinforced early in AP Calculus. Students need to practice using their calculator to solve multiple-choice and free response questions. A favorite calculator activity is one that helps strengthen the students understanding of etrema and concavity. I produce the graph of the derivative on the calculator and then ask information about the graph of the original function. Students make the connection between the behaviors of the graphs of f, f, and f.
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 eposure 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. LAB: LIMITS OF FUNCTIONS 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 feel 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. 4 1 1. Consider the function f defined by f()= 1 a) By successive evaluation of f at = 1.8, 1.9, 1.99, 1.999, and 1.9999, what do you think happens to the values of f as increases towards 2?
b) Do a similar evaluation of f for values of slightly greater than 2. Comment on your results. As a shorthand, we will describe these results by writing lim f( ) 15 2 c) Evaluate f (2) Comment on this answer. 2. Use the function f as above; consider what happens as approaches 1. a) Study this situation numerically as you did in parts a and b above. What are your conclusions? In particular, what is lim f( )? 1 b) What is f(1)? What is the difference in the 2 situations? c) Factor f() and simplify. Now substitute 1 for. Is this consistent with your findings in part a? 3. Using your calculator, determine the values of these limits. 4 1 ; 1 1 a) lim g ( ) where g( ) 10; 1 1 10 14 ; 1 3 27 b) lim 3 3 sin(10 ) c) lim 0 d) lim 1 1 0 4. Limits can sometimes fail to eist. Investigate the following limits and eplain why you think each does not eist. Use your calculator to graph functions as needed. What are these different from the functions previously considered? 1 1 a) lim b) lim c) lim 2 2 2 0 2 2 d) lim tan 2 Student Evaluation e) 1 lim sin 0 Each nine weeks, a student s grade is computed using homework, quizzes, and tests as categories. Each grade represents approimately 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 eam, and the remaining 10 percent is the final eam. Quizzes and tests vary depending on the teacher, but all follow closely the curriculum of the designated tetbook and that of the College Board. Many tests are half calculator and half non-calculator. The semester and final eams incorporate calculator portions and non-calculator portions, as well as multiple choice and free-response. Throughout the course students are given frequent eposure to questions from previous AP eams. The students also are made familiar with scoring of the free response questions with frequent practice. Teacher Resources: Primary Tetbook; Finney, Demana, Waits, and Kennedy. AP edition Calculus: Graphical, Numerical, Algebraic. 3 rd ed. Boston: Pearson Prentice Hall, 2007. A technology resource manual that accompanied the selected tetbook 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 tetbook and are used frequently in class as a visual aid for enhancing learning.