AP Calculus BC Syllabus Teaching Philosophy: Our AP Calculus courses are rigorously taught, with very high expectations for both the Instructor and the students. The Instructor s key goal is to enable students to understand the why behind all topics and to facilitate students ability to visualize the web that exists to connect various concepts. The Instructor will place an emphasis on applications of each topic and sub-topic introduced, in order to further provide students with a need for the techniques learned, as well as a more fundamental need for the language of Calculus. The Instructor will put into effect the Rule of Four on a daily basis within the classroom that all students must be able to understand and explain the graphical, numerical, verbal and analytical approach to each and every problem, as well as make connections between these varied approaches. Furthermore, it is imperative that the Instructor teaches each lesson with the big picture in mind that it is his/her role to prepare all students for college and beyond. It is the goal of this course to foster each student so that they will not only master the concepts of Calculus, but will also become adept in analytical thought, to a point where they understand that no challenge is too large. Teaching Strategies: All lessons should be taught with the goal of creating in students a need for using Calculus to solve problems, as well as to incorporate practical applications on nearly all topics and techniques learned. On a nearly daily basis, released AP problems should be incorporated into the classroom in a variety of ways, including, but not limited to: cooperative-learning assignments, individual challenges, lesson examples and homework questions, as well as through assessment on tests and quizzes. Students will be led to approach each problem with the Rule of Four in mind each problem will be considered graphically, numerically, verbally and analytically. Discussion of problems is essential in the classroom, and cooperative-learning techniques will be used whenever possible to further facilitate this. Students must also be proficient in the use of the TI-89, or an equivalent graphing calculator, in order to gain meaningful insight from this course. Assesment: 75% of a student s grade will be based on major assessments (tests and projects see three sample projects, attached) and 25% will be minor assessments (homework, quizzes, participation, etc.) Assessments will place an emphasis on applications of calculus to problem solve. The majority of tests will consist strictly of released AP questions, both multiple choice and free response, accompanied by other similar critical and analytical thinking problems. As such, the grade scale is modeled after the scale created by Dan Kennedy (see Kennedy s website mail.baylorschool.org/~dkennedy/assessment), with grades being assessed in such a way to encourage thinking outside the box and limit students fear of failure. In this way, even the cream of the cream of high school students come to understand the rigorous nature of a college Calculus course, as well as the reality of their grade expectations on college level exams. Additionally, this allows students to adjust to being faced with extremely challenging Calculus problems, the likes of which they ve never before seen, furthering their analytical thought processes and better preparing them for the AP exam. Primary Textbook: Calculus: 7 th Edition, Larson, Hostetler, Edwards; Houghtin Mifflin Technology: All students have a TI-89 or equivalent graphing calculator for use in class, at home, and on the AP exam. Students will use their graphing calculator extensively throughout the course. Supplemental Materials: Textbooks: Calculus, Stewart; Calculus, Anton; Calculus, Demana-Waits-Kennedy Printed Resources: Multiple Choice AP Calculus Review, Lin McMullin, D&H Marketing; Teaching AP Calculus, 2 nd Edition, Lin McMullin; Released AP Exams, 1976-present; Textbook Supplementals, Finney Computer Programs: Winplot; Calculus In Motion; Geometer s Sketchpad, Mathematica; Geogebra; Knowmia; Educreations
Course Outline: The course will preferably be offered in the fall semester of a 4x4 block schedule (85 minute blocks). This will provide the teacher with approximately 90 instructional days to complete the material. Supplemental AP review will be conducted regularly before/after school during the Spring semester. Chapter 1: Prerequisites for Calculus (2 days) Elementary functions: linear, power, exponential, logarithmic, trigonometric/inverse trigonometric Chapter 2: Limits and Continuity (4 days) a) Rates of change b) Limits at a point c) Limits involving infinity d) Properties of limits e) Instantaneous rates of change Special mention should be made of the different approaches to limits. Students are given a well grounded graphical, analytical and numerical approach. Chapter 3: The Derivative (7 days) a) Definition of the derivative b) Differentiability: 1) local linearity 2) numerical derivatives using the calculator 3) differentiability and continuity c) Derivatives of algebraic functions d) Rules for differentiation: sum, product, quotient e) The Chain Rule f) Derivatives of trigonometric functions g) Applications to velocity and acceleration h) Implicit differentiation i) Derivatives of inverse trigonometric functions j) Derivatives of exponential and logarithmic functions Chapter 4: Applications of Derivatives (8 days) a) Extreme values: 1) local (relative) extrema 2) global (absolute) extrema b) Using the derivative: 1) Mean Value Theorem and Rolle s Theorem 2) Increasing and decreasing functions c) Analysis of graphs using the first and second derivatives 1) Critical values 2) First derivative test for extrema 3) Concavity and points of inflection 4) Second derivative test for extrema d) Optimization problems e) Linearization models f) Related rates
Chapter 5: Definite Integral (10 days) a) Approximate areas under a curve : Riemann sums and Trapezoidal rule b) Using a definite integral to find exact area c) Fundamental Theorem of Calculus d) Average value of a function Chapter 6: Differential Equations and Integration Techniques (4 days) a) Antiderivatives and the indefinite integral b) Integration by substitution c) Differential equations d) Slope fields e) Euler s Method f) Exponential growth and decay g) Logistic Growth Chapter 7: Applications of the Definite Integral (7 days) a) Integral as net change b) Particle motion c) Areas in the plane d) Volumes by Rotation, by Known Cross-Section e) Arc Length Chapter 8: Integration Techniques and L Hopital s Rule (7 days) a) Integration by parts b) Integration by partial fractions c) Integration by trigonometric substitution d) Integration of improper integrals e) Application of L Hopital s rule to indeterminate limits Chapter 9: Infinite Series (14 days) a) Sequences and Geometric Series b) Tests for convergence and divergence: 1) nth Term Test 2) Direct Comparison test 3) Limit Comparison Test 4) Ratio Test 5) Integral Test 6) Alternating Series Test c) Power Series: Radius and Interval of Convergence; Term by term differentiation and integration of power series d) Taylor & Maclaurin Series and Polynomials e) Lagrange form of the remainder
Chapter 10: Parametric, Vector and Polar Functions (7 days) a) Parametric Functions: 1) Derivative at a point 2) Second derivative of y with respect to x 3) Length of a curve b) Vectors: using vectors to describe motion in the plane c) Polar coordinates and pole graphs: 1) Slope, horizontal & vertical tangent lines 2) Area, length of a curve Tests and other educational opportunities (20 days) Tests alone account for ten days, while other educational opportunities, such as cooperative learning assignments (see attached samples at syllabus end), help students work with their peers and solidify their understanding of calculus concepts. Students also gain the experience of taking sample multiple choice tests, as well as past free-response questions which strictly follow the AP format of fifty minutes for calculator use, fifty-five minutes for non calculator use and forty five minutes for free-response questions. After these sample tests have been taken, valuable learning moments ensue with students often providing more than one correct method of solving a problem. Review for AP Exam (up until the AP Exam) After these topics are covered, students are given an opportunity to review for the AP Exam by working through released freeresponse and sample multiple choice questions. This is a very productive time with students mastering the material and sharing their solutions to problems with each other
Geometric Series Work the following problems with your teammate(s) and write up your solutions neatly, clearly and carefully. All members of each team should understand and be able to explain the solutions. 1. A certain ball has the property that each time it falls from a height of h feet onto a hard, level surface, it rebounds up to a height of rh feet, where 0 < r < 1. Suppose that the ball is released from a height of H feet. Assuming that the ball bounces indefinitely, find the total distance the ball travels. 2. A right triangle ABC is given with A and AC b. CD is drawn perpendicular to AB, DE is drawn perpendicular to BC, EF AB, and this process is continued indefinitely, as in the figure. Find the total length of all the perpendiculars CD DE EF FG in terms of b and. 1 2 3. Start with the closed interval [0,1]. Note that this interval has length 1. Remove the open interval,. That leaves the two intervals 3 3 1 2 0, and 3,1, from which we remove the open middle third of each. Four intervals remain and again we remove the middle open third of 3 each of them. We continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step. The resulting set is called the Cantor set. (a) Show that the total length of all the intervals that are removed is 1. (b) Despite the fact that we started with an Interval of length 1 and removed intervals whose total length equals 1, the Cantor set contains infinitely many numbers. Give examples of at least four numbers in the Cantor set.
Centers of Mass Work the following problem with your teammate(s) and write up (and sketch) your solutions neatly, clearly and carefully. All members of each team should understand and be able to explain the solutions. You are a sculptor! Your newest work will be featured in the front of our school building for all to see. You have decided to create your work out of extremely thin sheet metal and you plan to sculpt it in such a way that you ll rest an intriguingly shaped flat metal sheet on top of the point of a cone. The only thing you have left to design is the metal sheet that must be balanced on the cone. You must make a mock-up of your piece-de-resistance your metal sheet! Use the cardstock to sketch your shape, and then cut it out. For ease of what is yet to come, please leave two opposite sides as straight lines, parallel to each other. The other two sides should be smooth (ie: differentiable), and may take whatever shape you like! After you have designed your metal sheet and made a model of it, you must find the precise point on the sheet that must be placed on the point of the cone to achieve perfect balance! You have at your disposal the following items: graph paper, Mathematica, your graphing calculator, your notebook and textbook, and your Calculus teacher. When you think you have determined your balance point, mark it on your model and we will try it to achieve balance!
Newton s Law of Cooling Work the following problem with your teammate(s) and write up your solutions neatly, clearly and carefully. All members of each team should understand and be able to explain the solutions. Professor Suluclac stops at Starbucks each day on her way to work for her morning coffee you would not want to sit in her classroom if she missed her daily shot of caffeine! Unfortunately, the local Starbucks has installed a new coffee machine that has caused problems for good old Professor Suluclac. You see, the new machine makes the coffee very, very hot, and if Suluclac takes a sip right away, she burns her tongue, and she cannot talk to teach her class! However, if she waits too long to drink her coffee, it is too cold for her taste and she cannot drink it and again, she cannot teach her class (no caffeine = no Suluclac). Suppose that when Professor Suluclac buys her coffee, its temperature is 110 Fahrenheit. The ideal temperature at which she prefers to drink her coffee is 92 F. If the temperature in her classroom is 68 F, determine the amount of time Suluclac must wait before she can partake in her caffeine rush. You may use your graphing calculator, your notebook and your textbook, as well as your Calculus teacher in your pursuit to save Professor Suluclac and her morning Java.