AP® Calculus AB Syllabus Course Overview Our school’s goal is to educate and graduate our students so that they are prepared for both the work force and/or post secondary studies. The main objective for our AP Calculus course is to provide a firm foundation in Calculus so that if they choose they will be more than prepared for future math courses. We stress that hard work is expected of each student. We cover everything listed in the Calculus AB outline including integration by parts. Occasionally, if we do not have many “snow days” or if we have time after the AP Exam, we try and cover some additional topics that are covered in the BC outline. Students begin their preparation for AP Calculus the previous year in our PreCalculus course. There we begin an intensive review of previous courses and investigation of trigonometry. Limits and continuity are thoroughly examined graphically, numerically, and analytically so that students become accustomed to exploring topics in a variety of manners and making conclusions based on data. More is expected in Calculus. Students are expected to participate in/use: 1. independent study open-ended investigations exploration of the history of mathematics 2. self-directed research and learning writing assignments reading assignments problem solving experiences 3. appropriate use of graphing calculators and computers The date of the AP exam is mentioned and included on the student syllabus on the first day of school. Hard work is stressed, expected, and rewarded. Parents receive a letter with details of the course and expectations the first day. Both the student and the parent sign this document. Our school is composed of two semesters with three six weeks in each. Basically the course is divided up in the following manner: 1st six weeks: Limits, Graphs, and General Differentiation 2nd six weeks: Differentiation and Some Applications 3rd six weeks: Applications of Differentiation; the Definite Integral 4th six weeks: Integration and Some Applications 5th six weeks: Further Applications of Integrations, Transcendental Functions, Differential Equations and Preparation for future Calculus Courses th 6 six weeks: Review for AP Exam
Course Planner First Semester When students sign up for AP Calculus and have been enrolled in our PreCalculus course, they are used to keeping a detailed folder and investigating ideas graphically,
numerically, and analytically. We expect them to know about limits and continuity because of that prerequisite work and the folder which they should review as needed throughout the year. Therefore, we begin Calculus with only a brief review (less than a week) of limits, terminology, and analysis of graphs. We then review as needed throughout the year these same topics as well as others when they relate to another topic of study in AP Calculus AB. The following topics usually appear in this order, but we change the arrangement of and/or time spent on topics depending on student needs and holidays/school calendar arrangement. The Derivative Average Rate of Change Instantaneous Rate of Change Dropping a Ball or other Object Using a Motion Detector Secant Lines and Tangent Lines, Drawing Tangent Lines Definition of the Derivative The Difference Quotient and the Alternate Method Tangent Line Equations Differentiability and Continuity Derivative Rules and “Short Cuts” (Product and Quotient Rules, Chain Rule) Approximations with Tangent Line Equations Graphing the Derivative Finding the Derivative of Trig Functions Trig “Short Cuts” for Finding Derivatives Implicit Differentiation Applications of the Derivative Related Rates Extreme Values Local (Relative) and Global (Absolute) Using the Derivative Mean Value Theorem and Rolle’s Theorem Increasing and Decreasing Functions In-depth Analysis of Graphs using the First and Second Derivatives Critical Values, First Derivative Test, Concavity, Points of Inflection, Second Derivative Test, Limits, Asymptotes, etc. Graphing Project Optimization Problems Packaging Problem Derivatives of Distance and Velocity CBR or CBL activity similar to “Match It, Graph It!”- which is used in some of our Algebra Classes earlier in the students’ careers, but now at a higher level The Definite Integral and Applications Riemann Sums – Area Approximations
Left, Right, Mid-point, Trapezoidal, Simpson’s Rule Independent Project or Group Project - Driving Activity using an Approximation for the Area under the Speed Curve and “Discovering” its Meaning Sigma Sum Review Infinite Partitions to find Area The Definite Integral Fundamental Theorem of Calculus Antiderivatives U-Substitution Average Value Fundamental Theorem of Calculus – Part Two Area Between Curves Second Semester The Definite Integral continued… Volume Known cross-sections, Disk/Washer Method, Shell Method Bundt Cake Project and/or Play Doh Models Summing rates of Change Particle Motion using Derivatives and Antiderivatives appropriately Transcendental Functions, Differential Equations, and Further Explorations Inverses and Derivatives – Discovering Relationships Natural Logs, Logarithms, Exponential Functions Differentiation and Integration involving Natural Logs, Logarithms, Exponential Functions Integration by Parts Slope Fields Applications and Modeling, Exponential Growth and Decay Inverse Trig Partial Fractions (if time or after AP Exam) L’Hopital Begin Review for AP Exam (approximately 4 weeks or a little more before exam)
Student Activities We usually use a variety of activities to help the students better understand the new ideas and how they relate to previous studies. The textbook contains a multitude of problems to explore which we do use as our primary source. But we also use projects and worksheets from a variety of sources. Some worksheets have been found at various conferences, in workbooks or on the Internet while others are teacher-made. Some projects are independent studies for the students and some involve groups or the entire class. Sometimes guided discoveries using worksheets or class discussions are used in order for students to discover a “short cut” or new idea on their own. The graphing calculator is used to help speed up discoveries with the use of its tables and
graphing capabilities. PowerPoint presentations along with Geometer’s Sketchpad and other graphics found on the Internet are used also. The following are some examples of activities that are usually used every year: Motion Detectors: We use some type of motion detector equipment (LabPro, CBL, or CBR – whichever may be available) in order to see relationships between average rate and instantaneous rate of change as well as displacement, velocity, and acceleration in general. The graphical relationships between them and the difference between velocity and speed as well as displacement and distance traveled can be easily explored. Students still enjoy watching themselves try to graph a given scenario on a CBR by moving in front of the detector. Some students never did this in earlier Algebra or science classes so they really like it. Velocity versus time requires them to think a little more than just simply distance versus time. Graphing Project: After learning about first derivatives, critical points, second derivatives, etc., each student is given a list containing a variety of functions in which they must graph using the tools that we’ve learned along with previous studies involving limits and asymptotes, zeroes, etc. They are allowed to check their work on the graphing calculators, but the work should be completed without them. They are given at least two weeks to work on this outside of class. This allows them lots of independent practice with all the shortcuts and new rules. Optimization/Packaging Project: Usually each student is given a sheet of paper a day or so before we begin optimization problems involving surface area and volume. They are told to try and make a box with the largest volume box possible. We then compare the found volumes and see who is the “volume champ” and discuss the methods used to make the box. We then show how derivative tests can be used to solve such problems. At times in our school’s history, we have taken a more in-depth look and actually looked at real life objects such as soft drink cans and glass “Ball” containers used in canning. Students wrote letters to companies and found out why the container was used even if at times the real life object in question was not found to be the “optimal design” to us. This discussion leads to many real world situations such as cost and engineering. Driving Activity: After students have learned how to make area approximations, they are given a sheet to record data and then graph speed versus time over a 20 minute drive. Students are urged to work in groups of three (a driver, timer, recorder) in order to be safe and to stay under the speed limit. Parents as well as friends (Calculus or other) can aid in the recording of data. Calculus students should then find the area under the curve and compare it to the odometer reading and make a new discovery that we discuss later in class.
Area and Volume: Students can use software programs such as Geometer’s Sketchpad and some accompanying programs to see how increasing the number of partitions can find the area under a curve for any function that they type into it. When it is time to learn about volumes, our school has a set of 3-d models for several types of cross-sections given a variety of bases. At times we build our own models using Play-Doh or the like. We also have several holiday and wedding decorations that unfold to produce bells, pumpkins, and such that help demonstrate the revolving of a curve around an axis to produce a 3-dimensional object. Many students have a hard time drawing and visualizing these problems so these types of activities help students. Bundt Cake Project: The above activities help visualize the 3-D object better, but the best motivator for calculating volume is food and especially cake! Students make bundt cakes, cut a piece, trace its silhouette onto graph paper and use calculators to find the regression equation that best fits their piece. Then the students use the shell method and find the volume of the Bundt cake/pan. Students then build a box of the same volume and fill it up with rice. The rice from the box is then poured into the cake pan and is compared to the height of the cake piece to see if it is accurate. Students must write about the experiment and we discuss our findings in class. And of course, we all get to eat a variety of cakes. Students really like this activity and it can be done independently or in groups. Because the cake pans are not always the same and neither is the amount of batter, we do get different answers. I know that some teachers have used Hershey Kisses and other types of treats. The school librarian told me that the US Postal Service had a contest about how many Hershey Kisses can fit into a box with given dimensions around Valentine’s Day. We found out about the contest too late and we had already did the Bundt cake project, but it did make a great tie in and topic of discussion. Inverses and Transcendental Functions: When we begin inverses and their relationships, we not only review and practice finding inverse equations, but we let the students discover patterns in the slopes of the tangent lines also. We also let them use the graph f(x) = ln x and discover its derivative through in depth approximations involving tables and the graph of ln x. We then do a similar x
experiment with f(x) = e . Students had previously not been taught the antiderivative of 1/x and knew that using the ‘reverse of the power rule’ would not produce an answer so they really are experimenting to find these new derivative rules. At the end, we summarize findings and then continue on with a more in depth study of derivatives of logs and exponential functions. History of Mathematics: We also usually participate in some type of history of mathematics project. This last year we invited a class from the neighboring elementary school (it was in walking distance so was a free field trip for them) and helped them discover the magic of the
number pi as well as its usefulness. Our Calculus students spent two class periods in our multipurpose room on March 14 and set up and ran different booths from “pi in your face” – face painting to making pi charts based on favorite pies, from trivia questions/scavenger hunts to understanding hat sizes, and then on to pi jewelry based on the non-pattern of the digits and color of beads and “cutting pi”. There were several other activities and then the fifth grade students enjoyed a brief PowerPoint presentation and lots of homemade pies. The Calculus students became teachers and enjoyed it probably more than the fifth graders. We have decided to invite more than one class next year and do more based on this year’s success. We also usually do some type of Mathematician project. This year students made a set of player cards like baseball cards or a movie poster with an accompanying report modeled after a movie preview. Students used historical facts with their creativity. In the past timelines, posters, reports, and such have also been made. We talk about these people in our classes or we use a theorem or process that is named after them, but the students usually do not know much about them. And honestly, these projects if nothing else help the students remember the math especially if the mathematician had some quirky habit, died a martyr, or did something at age 9 that other people can’t do until they are much older if at all. AP Review and Test Practice In the first semester, I begin giving some old released AP Free Response problems for practice. Some we do together and others I assign for them to turn in by a certain date. This allows them to justify answers and refine their communication skills both in writing and orally.
Technology Resources Students are required to have a graphing calculator of their own (most choose the TI83, 84, or 89), but a classroom set of TI-89’s is available. We have access to a CBL, CBR, and some LabPro equipment from Vernier so that at anytime a motion detector or probe is needed for a classroom activity there will not be any problem with availability of equipment. Calculators are used as a tool to experiment, to illustrate and to aid in understanding as well as to speed up the process of finding a root or a graph within a given window. Our teachers have also prepared several PowerPoint presentations to accompany certain units of study that have graphics and real world examples to better facilitate problem visualization and learning. As mentioned earlier we own Geometer’s Sketchpad and some software programs that allow students to see what is actually happening in area approximations and other topics of study. We use these, the Internet, newspaper articles and even movies such as excerpts from Mean Girls (the limit problem) and Stand and Deliver in its entirety to demonstrate problems or provide motivation.
Evaluation Students are used to keeping a notebook and it being checked approximately every two weeks in PreCalculus. Therefore, keeping a notebook in AP Calculus is already expected. These notebooks contain classroom notes, handouts, homework assignments, and explanations. Notebooks are graded every time the students have a test. The notebook usually counts for at least a fourth and usually less than half of the six weeks grade. Notebooks are graded based on points and are dependent on the number and the difficulty of the assignments. Although many college courses do not check homework in this manner, the students will see a direct correlation in their class work and homework efforts when compared to their overall test grades and understanding. Tests are given approximately every two weeks or at the end of a major unit of study and are usually worth 100 points each time. On at least half of the tests, students may use a calculator. However on many of the “calculator tests”, there is still a noncalculator section as well. Depending on the difficulty of the project, projects may count as much as a test grade. Semester Exams use multiple choice questions that follow the format of the AP Exam.
Primary Textbook: Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. 7th ed. Boston: Houghton, Mifflin Company, 2002.
