| AP Calculus AB Course Outline/Syllabus/AP Audit Text: Stewart, James (2021). Calculus: Early Transcendentals Single Variable, (9th edition). Cengage Learning, Inc.  ADDITIONAL RESOURCE: • The College Board AP Released Free Response Questions: 2002-2022 • Other printed or online resources as appropriate TECHNOLOGY: All students are required to have a graphing calculator. The TI-83+ or 84+ will be recommended but students are allowed to purchase the graphing calculator of their preference. (We have a few TI-83+ calculators to check out for those students who cannot afford to purchase their own.) Class time is allocated to teaching students how to use the calculator effectively. When possible, programs collected from AP Workshops, other teachers, and other sources will be linked and transferred to students’ calculators. GENERAL INSTRUCTIONAL GUIDELINES: Various instructional techniques (lecture, group work, guided discovery, and projects) and approaches (graphical, numerical, and analytical) will be used. In working out problems, I will always recommend that students use a second method in order to justify (or correct) the answer they arrived at by a first approach. Sometimes this will be required. Students are always required to support their answers regardless of the approach, primarily in written responses but also, as class time permits, verbally. Once we get through derivatives, students are assigned several free-response questions taken from AP Calculus AB questions released by The College Board as well as various publications dedicated to the AP Calculus course. Unit tests are a combination of multiple choice, short answer, and free-response questions. Work – including justification for answers – must be shown for all questions. After the tests are graded, they are returned to the students for corrections. Justifications for all corrected answers – including multiple choice answers – must be included. We complete the AP Calculus AB course content, as delineated by the Calculus AB Topical Outline in the AP Calculus Course Description, three weeks before the exam. During these three weeks, the students work practice exams so they can get used to the time restrictions on the various parts of the AP Calculus AB exam. They also get some experience using rubrics and grading their solutions to free-response questions. After the exam, we cover topics from the AP Calculus BC course (that were not included in the AB course) by instructional methods similar to those used prior to the AP Exam or by student projects that are presented during finals week. CURRICULUM ORGANIZATION: SUMMER PRECALCULUS REVIEW PACKET: Students who wish to take AP Calculus AB the year following their precalculus course are required to complete a packet of review problems (in algebra, trigonometry, and other precalculus topics such as analytic geometry and elementary functions) during the summer break and submit their work the first week of class. At the end of the first week, a test is given on the prerequisite material. Included with the packet is a suggested time line for completion as well as my email address so that they can ask questions and get help as needed throughout summer break. Unit 1: Functions and Models Key Concepts: • Review of basic ideas concerning functions, their graphs, and ways of transforming and combining them • Representing functions in four ways: by an equation, in a table, by a graph, or in words • Investigation of the main types of functions that occur in calculus and using these functions as mathematical models of real-world phenomena • Using graphing calculators • Exponential Function • Inverse Functions and Logarithms Model Integrated Tasks: A rectangular field is to be enclosed with 260m of fence. Find a mathematical model expressing the area of the field as a function of its length and specify its domain. By plotting on a graphing calculator the graph of the function, estimate to the nearest meter the dimensions of the largest rectangular field that can be enclosed with the 260m of fence. Suggested Time Frame: 3 weeks Unit 2: Limits and Derivatives Key Concepts: • Investigating limits and their properties • Demonstrating knowledge of both the formal definition and the graphical interpretation of the limit of values of functions and the continuity of a function • Demonstrating the understanding and application of the Intermediate Value Theorem • Demonstrating understanding of the formal definition of the derivative of a function at a point • Interpreting derivatives as slopes and rates of change • Estimating derivatives of functions given by tables of values • Graphing derivatives of functions defined graphically • Using early methods for finding tangents using derivatives and rates of each. • Calculating the derivative as a function. Model Integrated Tasks: • If P(x) is a polynomial and Q(x) = x – a, the graph of the function f defined by f(x) = P(x)/Q(x) will have either the line x = a as a vertical asymptote or a hole at the point where x = a. What is the connection between these two geometric concepts and • Show that the Intermediate-zero theorem guarantees that the equation has a root between –2 and 2, and use a graphing calculator to approximate the root to two decimal places. • If a forest can support a squirrel population of 10,000, the rate of growth of the squirrel population is jointly proportional to the number of squirrels present and the difference between 10,000 and the number present. Find a mathematical model expressing the rate of population growth as a function of the number present and determine the domain of the function. Prove the function is continuous on its domain. On a graphing calculator, estimate the size of the squirrel population for the growth rate to be maximum. Suggested Time Frame: 5 weeks Unit 3: Differentiation Rules Key Concepts: • Using rules for finding derivatives to calculate the derivatives of polynomials, rational functions, algebraic functions, exponential functions, logarithmic functions, trigonometric functions and inverse trigonometric functions • Solving problems involving rates of change and the approximation of functions • Determining relates rates of two or more objects. • Using linear approximation and differentials. • Understand rate of change in natural and social sciences. • Interpreting growth and decay functions. Model Integrated Tasks: • A ball is thrown vertically upward from the ground with an initial velocity of 32 ft/sec. Write the equation of motion of the ball, and simulate the motion of the ball on a graphics calculator. Estimate how high the ball will go and how long it takes the ball to reach the highest point. Confirm your answer analytically. Find the instantaneous velocity and the speed of the ball at 0.75 sec and 1.25 sec. Find the speed of the ball when it reaches the ground. • Two particles start at the origin and move along the x-axis. For , their respective position functions are given by . For how many values of t do the particles have the same velocity? Suggested Time Frame: 5 ½ weeks Unit 4: Applications of Differentiation Key Concepts: • Understanding how derivatives affect the shape of a graph of a function • Using derivatives to locate maximum and minimum values of a function • Knowing and applying Rolle’s Theorem, the Mean-Value Theorem, and L’Hospital’s Rule • Using differentiation to sketch, by hand, graphs of functions • Solving a variety of optimization and related rate problems • Finding the antiderivative of a function Model Integrated Tasks: • Use a graphing calculator to estimate the dimensions of the right-circular cylinder of greatest lateral surface area that can be inscribed in a sphere with a radius of 6 in. Confirm your estimates analytically. • If f is a polynomial function, use Rolle’s theorem to show that between any two consecutive roots of the equation f ’(x) = 0 there is at most one root of the equation f(x) = 0. • If f(x) = ax4 + bx3 + cx2 + dx + e, determine a, b, c, d, and e so that the graph of f will have a point of inflection at (1, -1), contain the origin, and be symmetric with respect to the y axis. Support your answer graphically. Suggested Time Frame: 5 weeks Unit 5: Integrals Key Concepts: • Demonstrating understanding of the definition of the definite integral by using Riemann Sums • Using the definition of the definite integral to approximate integrals and to model problems • Approximating the value of a definite integral by the Trapezoidal Rule • Proving and applying the Fundamental Theorem of Calculus Model Integrated Tasks: • The marginal cost function for a particular article of merchandise is given by . If the overhead cost is $10, find the total cost function. • Suppose a ball is dropped from rest and after t seconds its velocity is v feet per second. Neglecting air resistance, express v in terms of t as v = f(t), and find the average value of f on [0, 2]. (Hint: find the value of the definite integral by interpreting it as the measure of the area of a region enclosed by a triangle.) Suggested Time Frame: 4 weeks Unit 6: Applications of the Integral Key Concepts: • Using integrals in problems involving area • Using integrals in problems involving volume by slicing, by disks and washers, and by cylindrical shells • Finding the average value of a function Model Integrated Tasks: • The base of a solid is the region enclosed by the hyperbola and the line x = 4. Find the volume of the solid if all the plane sections perpendicular to the x-axis are squares. Find the volume of the solid if all the plane sections perpendicular to the x-axis are equilateral triangles Suggested Time Frame: 4 weeks Unit 7: Differential Equations Key Concepts: • Graphical and numerical techniques of solving differential equations • Direction fields and Euler’s method • Using differential equations to solve real-world problems Model Integrated Tasks: • Given the slope field for the differential equation , answer the following questions: Calculate dy/dx at the points (3, 5) and (-5, 1); sketch the graph of the particular solution of the differential equation that contains the point (5, 1), then solve the differential equation algebraically and find the particular solution that contains the point (5, 1). • For the given differential equation, use Euler’s method to calculate values of y for the particular solution that contains (1, 9). Use x = 1 and x = .1. Comment on the accuracy of Euler’s method far away from the initial point when you use a relatively large value of x. Suggested Time Frame: 3 weeks Unit 8: Practice Problems for the AP Calculus AB Exam Key Concepts: • Review entire course • Practice timed tests in Multiple Choice and Free-Response formats Suggested Time Frame:3 – 4 weeks Unit 9: Student Projects or Topics in Calculus Beyond the A.P. Exam Key Concepts: Topics covered after the exam, by formal lectures or projects, include (as time allows): • Solving problems involving integration by parts • Evaluating trigonometric integrals • Using trigonometric substitution to evaluate integrals • Integrating rational functions by partial fractions • Evaluating improper integrals • Finding arc length • Graphing curves defined by parametric equations • Determining derivatives and integrals involving parametric equations • Graphing curves in polar form • Finding areas of regions bounded by curves in polar form • Finding lengths of curves in polar form • Determining the convergence or divergence of an infinite sequence • Determining the convergence or divergence of a series using the various tests for convergence and/or divergence • Estimating the sum of a convergent series • Finding the radius and interval of convergence of a power series • Finding the power series representation of a function • Fnding the Taylor or Maclaurin Series for a function • Calculating Taylor polynomials and Taylor series of basic functions, including the remainder term. Suggested Time Frame: 4 weeks Additional Information The course will be taught as a modified flipped classroom. All lessons are videos which students will watch in advance. In class, we will review the video/major topics and practice essential problems. We will have daily warm-ups and students will be called on for answers, steps, etc. Materials Students are expected to have Chromebooks and access to Web Assign which we will use for some of the homework. Students are allowed to use iPads for notes, assignments, etc. In addition, a graphing calculator is required per the college board guidelines for this course. (Refer to the college board website for a list of approved graphing calculators.) If a student cannot obtain a graphing calculator, the library has some available to check out. Students are expected to bring their Chromebooks/iPad & graphing calculator daily and have pencil/paper for any written assignments. Students can access notes through their Chromebook/iPad or print out, but must be prepared daily. Assignments All assigned problems are to be complete per assignment requirement. For Web Assign homework, you must earn a minimum of 80% correct to receive credit on the assignment. You will be given 3 attempts in Web Assign to obtain the correct answer. Any additional assignments will be submitted via Google Classroom or turned in per instructions for the assignment. All steps must be shown on submitted assignments as illustrated in class. NO WORK, NO CREDIT! Late assignments are subject to partial-credit for 1 day late only. If you have an excused absence, you are given an additional day for each day you were absent to turn in the homework, per school guidelines. Tests/Quizzes Tests and quizzes count for the majority of your grade. Since tests are always announced in advance, if you have an excused absence on the test day you are still responsible to take the test the day you return. In addition, if you know you will be absent, you need to schedule a time with me to take the test. An unexcused absence results in a 0 on the test! You are not to discuss test content with other periods. Any discussion of test content will be seen as cheating. Retakes/test corrections are not allowed. Generally, quizzes are also announced in advance. The same absence policy applies for quizzes as the one stated above for tests. Quizzes may also be unannounced. TESTS WILL BE RECOLLECTED AFTER EACH CHAPTER, failure to return test may result in loss of points. Grades *Note: Point value or point totals may be changed. The grade distribution is as follows (without rounding): Approx 120 POINTS PER UNIT 100-90% A 89-80 B 79-70 C 69-60 D etc…. F Hawk Pride · All rules including tardy/truancy policies found in the student handbook will be enforced in the classroom. · All EDHS and district policies/rules will be enforced. · Students are to arrive on time with the required materials and be ready when class begins. · Students will respect the rights of other students when they have permission to talk or answer a question. Raise your hand for attention. · Talking (unless given permission or asking/answering a question) during the presentation of a lesson will not be tolerated…ie when I talk you don’t! • Academic Honesty/Integrity Policy will be in effect, refer to p.26 of student handbook. · Cell phones are not allowed during class time and cannot be used as calculators. Cell phones cannot be out during any testing time and you will receive a 0 if you have one out! Parents: Please monitor homework and grades. If you have any questions, please feel free to e-mail me at RNASR@PYLUSD.ORG or call me at the school. This and more information can be found at NASRS.NET/EDHS |