ࡱ> TVS t6bjbj 4Leet.NN!!!!!!!!!!D!1v"""""S'S'S'11313131313131$(35W1!S'&S'S'S'W1!!""l1(((S'^!"!"11(S'11((7/00"\'g0x11010\6o(|\6,0\6!0((S'S'S'W1W1(S'S'S'1S'S'S'S'\6S'S'S'S'S'S'S'S'S'N n : AP Calculus AB Syllabus 2016-2017 Mike Lanzarone Rm 226 lanzarone_michael@asdk12.org Course Description The only constant in the world is change. Calculus broadly defined is the study of how functions change and, as such, is the most applicable math to the world around us. In order to explore these changes mathematics needed new tools: the limit, derivative, and integral. ܽƵapps will study limits at points and at infinity and how this translates to the continuity of the function. ܽƵapps will study derivatives and the relationship between the derivative and the function. Finally, students will study some of the techniques of integration and develop an understanding of the relationship between the integral and the function. The application of the concepts will be heavily emphasized in order to more closely match a situation in the real world. Since real world situations can arise in multiple ways, students will interact with the functions in multiple ways: Graphically graph of function is available Numerically solutions of the function are provided (think data collection) Analytically the equation of the function is known Written/Verbally information about the situation is in words Finally, students will be expected to justify and defend their conclusions using concrete evidence both in written and verbal formats. Grading Categories Assignments: 20% - Homework, in-class assignments, class participation Assessments: 80% - Quizzes, exit-slips, group tests, unit tests, finals Homework Homework is a way to practice the material explored and introduced in class. Without applying the material through practice theres very little chance of actually learning the material. Its expected that students complete their homework. During class I will walk around the room and spot check each students progress on the assignment from the previous night. Absences It is the students responsibility to get any notes or materials from any class missed. When asking for the materials or notes students need to select an appropriate time: before or after class, before or after school, or during lunch. If the excused absence falls on a test or quiz day, students will be given a reasonable amount of time to make up the assessment either at lunch or before or after school. If the absence is unexcused any assessment or assignment will not be able to be made up. In other words, if you skip class you cannot get points for anything that was done that day. Remember it is your responsibility to make sure that your absences get excused! Tutoring I will be available every day at lunch and after school most days from 2:00-3:00. Please get help if you need it early and often! Study Groups While it is not required, it is highly recommended that you form a study group with a handful of other students in the class. These should be people you can learn from and who you are able to stay on task with. AP Stuff AP Calculus A/B Test: ________________________ I feel that the best way to prepare for the AP test is to continually practice AP problems. As such these type of questions will be embedded in tests, quizzes, assignments, warm-ups, etc. As we get closer to the test we will take numerous mock-AP tests in class. These will be real AP tests from years past. Technology Requirement ܽƵapps will use a graphing calculator on a daily basis in order to explore topics and expand the difficulty and thus novelty of problems they can solve in the course. The graphing calculator will also aid students in supporting conclusions to solutions obtained without a calculator. ܽƵapps should already be comfortable with graphing a function, setting an arbitrary window, and finding the zeros of a function. I will use a TI-80's series calculator or a TI- nSpire. Course Outline Chapter 1: Limits and Their Properties (3 weeks) Content and/or skills taught: Use a graphing calculator to find the limit of a function from tables and the graph. Use properties of limits to calculate the limit of a sum, difference, product and quotient of functions. Determine when a limit does not exist. Evaluate one-sided limits. Use the definition of continuity to determine whether a function is continuous at a point or on an interval. Verify the Intermediate Value Theorem for a specific function. Find the points of discontinuity for a function. Understand the concept of infinite limits and use the concept of infinite limits to find vertical asymptotes of functions. Major Assignments and/or Assessments: Using the concept of limits describe, both verbally and in writing, the relationship between continuity and limits. Compare and contrast one and two-sided limits. Chapter Test and quizzes. Chapter 2: Differentiation (5 weeks) Content and/or skills taught: Find the slope of a tangent line and write the equation of a tangent line using the limit process. Define and compute the derivative of a function using the limit of the difference quotient. Demonstrate the concepts of average rate of change and instantaneous rate of change of a function. *ܽƵapps will use a toothpick and trace the curve of a graph. They will orally identify when the instantaneous slope is the same as the average slope. Define the derivative of a function in a variety of ways including slope of the tangent line, rate of change of the function and instantaneous velocity. Demonstrate an understanding of the relationship between differentiability and continuity of a function. Develop and use differentiation rules to compute derivatives of functions, including the trigonometric functions. Use the Chain Rule to differentiate composite functions. Find the derivative of implicitly defined functions. Compute successive derivatives of functions. Use the derivative to find the slope of the tangent to a curve (if it exists) and to write equations of tangent and normal lines to a curve at a given point. Use the graphing calculator to compute the numerical derivative of a function. Solve problems involving related rates of change. Major Assignments and/or Assessments: Graph and verbally describe the limit process of finding a derivative. Using the concept of limits describe, both verbally and in writing, the relationship between the slope of a secant line and the derivative of a function at a point. Chapter Test and quizzes. Chapter 3: Applications of Differentiation (6 weeks) Content and/or skills taught: Apply the Extreme Value Theorem to find the maximum and minimum values of a function on a closed interval. State and apply Rolle's Theorem and the Mean Value Theorem. Use the First Derivative Test to find the intervals on which a function is increasing and decreasing and to determine relative extrema of a function. Use the Second Derivative Test to determine intervals of concavity of a function and to locate inflection points. Use the Second Derivative Test to analyze relative extrema of a function. Use information about intervals of increase and decrease, relative extrema, intervals of concavity, and inflection points to sketch the graph of a function. *ܽƵapps will participate in a matching game where they will have to match graphs of functions to the graph of the appropriate derivative. ܽƵapps will work in pairs and have to justify their matches verbally to the instructor. If a match is questionable, they will have to use the calculator to verify if they are correct. Use the graph of a given function f and its first and second derivatives to explore corresponding characteristics. Calculate limits at infinity and use the concept of limits at infinity to determine horizontal asymptotes of functions. Use derivatives to find both absolute and relative extrema and to solve optimization problems. Interpret the derivative in problems involving speed, velocity and acceleration. *ܽƵapps will work on AP Free Response questions in a group setting. Together, they will solve speed/velocity/acceleration problems and justify their answers in written form. The instructor will focus on showing students how to justify their solutions adequately. Use Differentials to obtain linear approximations. Major Assignments and/or Assessments: Extensive analysis of curves. Demonstrate an understanding of the relationship between the equation of a function and the general shape of the curve. Written and verbal interpretation of sign chart information. Chapter Test and quizzes. Chapter 4: Integration (5 weeks) Content and/or skills taught: Compute simple anti-derivatives using basic integration rules. Use the anti-derivative to solve problems involving motion along a straight line when given initial conditions. Understand the concept of area under a curve using a Riemann sum over equal subdivisions. Compute Riemann sums using left endpoints, right endpoints, and midpoints as evaluation points. Use the limit of a Riemann sum to calculate a definite integral. Use the First Fundamental Theorem of Calculus to evaluate definite integrals. Calculate the antiderivatives using substitution of variables and change of limits. Use the graphing calculator to compute definite integrals numerically. Use the Mean Value Theorem for integrals to find the average value of a function on an interval. Use the Second Fundamental Theorem of Calculus to find derivatives. Use the Trapezoidal Rule to approximate area under a curve. Major Assignments and/or Assessments: Extensive in-class discussion of the relationship between acceleration, velocity and acceleration. Discussion regarding the relationship between the two parts of the Fundamental Theorem of Calculus. Chapter Test and quizzes. Chapter 5: Logarithmic, Exponential and Other Transcendental Functions. (4 weeks) Content and/or skills taught: Review properties of the natural log function and the exponential function. Define the natural log function as the area under the curve, from 1 to x. Differentiate exponential and logarithmic functions. Find the derivative of the inverse of a function. Use logarithmic differentiation to find derivative of complicated functions. Find derivatives of inverse trigonometric functions. Evaluate integrals yielding inverse trig functions Major Assignments and/or Assessments: Extensive discussion and interpretation of the concepts of inverses and their graphs, with emphasis on natural logs and exponential functions. Chapter tests and quizzes. Chapter 6: Differential Equations (2 weeks) Content and/or skills taught: Differentiate and integrate general exponential and logarithmic functions. Find general and particular solutions to differential equations using separation of variables. Construct a slope field to show the geometric interpretation of a differential equation and use a slope field to show the solution curves for a differential equation. *ܽƵapps will be drawing slop fields of various differential equations. They will have to match the slope field to its function and also verify their results with a calculator. Use definite integrals to find the area under a curve. Use definite integrals to find the area between two curves. Major Assignments and/or Assessments: Chapter tests and quizzes. Chapter AP: Review and Practice for the AP Test (6 weeks) Content and/or skills taught: Review the first year calculus topics. Practice taking AP tests. Post-AP Test Unit: Calculus Projects and Outreach (2 weeks) Textbook Title: Calculus of a Single Variable Publisher: Houghton Mifflin Company Published Date: 2006 Author: Larson, Hostetler, Edwards Signature Page Please read the following statement and print and sign your name. 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