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AP Calculus ABCourse OverviewThis course covers every concept in the Calculus AB topic outline as it appears in the AP? Calculus Course Description. Students enter this course knowing that it will be challenging and that they are expected to work together to solve problems and communicate about calculus concepts. We explore concepts using the rule of four (VNAG): verbally, numerically, analytically and graphically.Course PlannerUnit 1: Limits and ContinuityGraphical limitsLimits at a point algebraicallyLimits involving infinity algebraicallyLimits of piecewise functions graphically and algebraicallyAppropriate window for a graph on the graphing calculatorUse of table feature of graphing calculator to determine limitsContinuity graphically and as defined by limitsUnit 2: The derivativeDefinition of the derivativeDifferentiability and continuityDerivatives of polynomial functions using the power ruleProduct and quotient RulesDerivatives of trigonometric, exponential and logarithmic functionsThe chain ruleApplications to velocity and accelerationL’Hopital’s Rule for finding the limit of indeterminant formsUnit 3: Applications of the Derivative Part 1 Analysis of graphs using the first and second derivatives Local and Global ExtremaIntervals of increasing and decreasingPoints of inflectionIntervals of concavityMaking the connections between in tables and graphsTangent line to a curve at a point and using this line to approximate the value of the function Use of graphing calculator to find the numerical derivative at a point and the graph of the derivative of a functionIntermediate Value TheoremUnit 4: Applications of the Derivative Part 2Implicit DifferentiationDerivative of inverse trigonometric functionsLogarithmic differentiation and derivative of Related RatesUnit 5: Applications of the Derivative Part 3Rolle’s Theorem and Mean-Value TheoremExtreme-Value TheoremOptimization with and without the graphing calculatorUnit 6: AntiderivativesBasic integrationIntegration by u-substitutionUnit 7: Applications of IntegrationsSeparable Differential Equations and their use in modeling different types of growthMotion problemsSlope fieldsUnit 8: Numerical approximations and the Fundamental Theorem of CalculusRiemann sums to approximate area using right, left, midpoint and trapezoidDefinite integral as a limit of Riemann sumsFundamental Theorem of Calculus (Part 1)Fundamental Theorem of Calculus (Part 2)Average Value Theorem6. Basic properties of definite integrals 7. Total distance traveled by an object8. Summing rates of changeUnit 9: Applications of the definite integralArea of a regionVolume of a solid with a known cross sectionVolume of solids of revolutionWasher-Disk MethodReview until testTeaching Strategies and Student evaluationI stress communication as a major goal of this class. For many of the students this course is the first math class in which students are expected to ask questions, go to the board to explain their thinking and work with other students to figure out a problem. We have to work on them communicating with one another instead of always coming to me for help. In order to facilitate this, the students are randomly grouped in partners and in groups of four for different assignments. The rule of the groups is that everyone discusses a question before eliciting my help. This has encouraged students to depend on and communicate with each other and form study groups. The students are assigned to present problems to the class individually, with partners and in groups. We also work old AP problems in which one student does one part of the problem and then “passes the pen” to another classmate to do the next part. This allows me to assess what students know and what they can figure out when faced with a new situation. In the beginning this is intimidating for those students who are not accustomed to this expectation, but by the first few months, students are comfortable with “thinking” and writing explanations in written sentences in front of the class. Each unit assessment contains multiple-choice and free response questions that require mathematical ideas to be explained and justified in written sentences. Explaining the “why” is just as important as the “how”. Justification of answers is required in written form. Each unit assessment contains calculator and non-calculator problems. The semester exam and mock AP Exams use multiple choice and free response that follow the format of the AP Exam. Questions from previous AP Exams influence assessment during the year. Teacher ResourcesPrimary Textbook ISBN: 978-1429250757Rowawski and Cannon. Rogawski’s Calculus for AP, 2nd ed. New York: W.H. Freeman and Company, 2012.Technology and softwareEach of my students has a graphing calculator and we use them daily to explore, discover and reinforce calculus concepts. My students use a TI-83 Plus, a TI-84 or a TI-89. If they do not have a graphing calculator, then one is given to them to use for the year. I use a Mobi that allows me to run the computer from anywhere in the room and I when I write on the pad, students can see what I have written. I can hand the pad to a student and he can share his thoughts from his seat. We incorporate the TI SmartView software so that I can demonstrate new features and so that students can share with one another the results of their calculations. I incorporate the software Calculus in Motion to enhance the visual representations of Calculus concepts. This software is very helpful when I introduce the definition of the derivative because it draws the secant line and shows what happens as h approaches zero. The students can actually see the secant line become the tangent line. When teaching volume by known cross sections and by revolution, this software allows the students to see the 3-D object for which the volume is being computed. Each test is designed to have a calculator section and a non-calculator section. Student Activities1. During the limits section the students discuss different limits in Limits War. This activity is similar to “War” when using playing cards. The cards in this activity contain limit problems and the student with the largest value wins the round. The students will use their reasoning skills of theorems and definitions to create pictures of functions to obtain the limit graphically, write out the limit algebraically, make a table of values to find the limit and discuss the method of getting the limit verbally.2. During the derivatives unit, students use graphing calculators to explore how the difference quotient zooms into the same value as x approaches the point of tangency from both sides of the specific x value. 3. During the related rates unit students are put in groups and given a problem to work and present to the class. The students present their conclusion on a single paper that is graded based upon the correctness of the mathematics, the use of correct mathematical notation, their ability to communicate the calculus concepts both orally and written and their ability to correctly answer questions asked from the students. Students are encouraged to write their conclusions in complete sentences.4. As a means of understanding the objectives of the Mean Value Theorem, students work through and use their reasoning skills of the theorem to complete an activity in which they find the “c” guaranteed by the theorem and then on the graphing calculator graph both the secant line and tangent line to that value of c in order to “see” validate that their answer is correct.5. During the discussion about , students are given a graph of and asked to do a sketch of . They then use the graphing capabilities of nDer to determine how well their sketch is compared to the derivative graph from the graphing utility. Students will use their knowledge of the function notation to input the function and its derivative into the graphing utility.6. During the unit involving the relationship between the students are given cards with the three functions and are asked to match them in order. As I walk around and assess the students, I encourage them to use vocabulary such as increasing, decreasing, maximum, minimum, point of inflection, concave up and down, in order to decide which functions represent each derivative. Students must also practice their mathematical notation to describe as well as write down the corresponding functions.7. During the integration unit the students take a partner quiz that contains 12 problems and there are 3 problems they do not know how to integrate. They must reason through the theorems and defintions of integrals and work together discussing the problems and answering the quiz problems together. A great deal of learning and solidifying the different integrating situations occur during this activity. ................
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