Ap Calculus Course Outline

AP CALCULUS BC SYLLABUS

Instructor: Mr.Gulsoy Course Overview: The Advanced Placement Calculus Course BC An Advanced Placement course in calculus consists of a full academic year of work in calculus comparable to courses in colleges and universities. It is expected that students who take and AP course in calculus will seek credit or placement, or both, from institutions of higher learning. This is a full-year calculus course designed for students who will take the AP Calculus BC exam in the spring. Calculus is explored graphically, numerically, verbally and analytically. The course covers all AP Calculus AB concepts and the following additional material: Analysis of planar curves given in polar, parametric, and vector form, numerical solution of differential equations using Euler method, L’Hopital’s rule, derivatives of parametric, polar, and vector functions, antiderivatives by substitution of variables including parts, and simple partial fractions, improper integrals, solving logistic differential equations, concept of series including convergence and divergence, series of constants including geometric, harmonic, and alternating series with error bound, tests for convergence and divergence of series such as the integral test and the ratio test, Taylor and Maclaurin series, functions defined by power series, radius and interval of convergence of power series and Lagrange error bound for Taylor polynomials. The use of technology is integrated throughout the course and students are required to have a graphing calculator. The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

Major Text: Calculus of a Single Variable, Finney-Thomas, 1994 second edition

Prerequisites Before studying calculus BC, all students should complete calculus AB and four years of secondary mathematics designed for college bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry and elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions.

Teaching Strategies I explain topics by following ‘the rule of four’: Graphically, numerically, algebraically and verbally. In my presentations, I stress out the importance of seeing the big picture. For that purpose, I explain why we are doing the major applications of calculus and the connections between the concepts. I encourage students to ask “what if” questions, which develop critical thinking skills. Our classroom contains SMART BOARD which facilitates the presentation of the concepts and enables to incorporate online applications. Throughout the year, students will work in small groups and individually to develop graphical, numerical and analytical solutions to the problems. Students will explain solutions to problems both verbally and in written sentences. Students develop analytical arguments in written form to justify their solutions. Throughout each chapter, they will work on assignments which require communicating mathematical ideas in oral and written form.

Graphing Calculators

The calculator helps students develop a visual understanding of the material that they would not otherwise have. They also use graphing calculators to approximate answers found algebraically to see if they are reasonable. My students use TI-83 and TI-84 graphing calculators for the following purposes:  To graph a function in an arbitrary window to explore the properties of functions.  To graph functions in polar and parametric modes to justify their hand- sketched graphs of parametric and polar functions  To explore the roots, the extreme points, the increasing-decreasing intervals and the inflection points of a given function.  To determine the intersection points between the two functions graphed on the same set of axes.  To use the TABLE and the TRACE features to investigate the limit of a function when the x value approaches to a certain value or infinity.  To determine the value of the derivative at a certain point and to numerically calculate the value of a definite integral.

Key class assignments, and tests will be incorporating multiple choice and free response AP applications where the use of graphing calculator is allowed.

AP Calculus BC Course Outline

FIRST SEMESTER UNIT 1: Functions (2 weeks)  Identify graphs of the following functions: constant, linear, quadratic, cubic, square root, cube root, absolute value, greatest integer, exponential and logarithmic and trigonometric  Define, identify and apply concepts of functions with respect to domain, range, intercepts, symmetry, asymptotes, zeros and odd and even functions  Apply the algebra of functions by finding sum, difference, product and quotient, composition and inverse where they exist  Describe transformations of graphs from given equations and change equations to produce desired transformations  Explore parametric relations  Use mathematical models to predict information from data  Use a graphing calculator to produce the graph of a function within an arbitrary viewing window  Use a graphing calculator to find the zeros and maximums and minimums of a function UNIT 2: Limits and Continuity (2 weeks)  Use a graphing calculator to estimate the limit of a function both numerically and graphically 1.0, 1.2  Use the properties of limits to calculate the limit of a sum, difference, product and quotient of functions 1.1  Compare growths of logarithmic, polynomial, and exponential functions  Determine when a limit does not exist 1.0  Determine and evaluate limits involving infinity 1.0  Evaluate one-sided limits 1.0  Use the definition of continuity to determine whether a function is continuous at a point or on a interval 2.0  Verify the Intermediate Value Theorem for a specific function 3.0  Find the points of discontinuity of a function and determine if they are removable or nonremovable 2.0  Understand the concept of infinite limits and use the concept of infinite limits to find the vertical asymptotes of a function 1.0

UNIT 3: Differentiation (4 weeks)  Define and compute the derivative of a function using the limit process 4.0  Apply the concepts of average and instantaneous rates of change of a function 4.2  Define the derivative of a function in a variety of ways including slope of the tangent line, rate of change of a function and instantaneous velocity 4.1  Demonstrate an understanding of the relationship between differentiability and continuity of a function 4.3  Use the differentiation rules to compute derivatives of functions 4.4  Use the chain rule to differentiate composite functions 5.0  Find the derivative of implicitly-defined functions 6.0  Compute higher order derivatives of functions 7.0  Use the derivative to find the slope of a curve and to write equations of tangent and normal lines to a curve 4.1  Use the graphing calculator to compute the derivative of a function at a point 4.0  Use the graphing calculator to compute equations of tangent lines 4.1  Use the graphing calculator to compute the numerical derivative of a function 4.0  Differentiate exponential and logarithmic functions 4.4  Find the derivative of the inverse of a function 4.4  Find the derivatives of inverse trigonometric functions 4.4

UNIT 4: Applications of Differentiation (4 weeks)  Apply the Extreme value Theorem to find the maximum and minimum values of a function on a closed interval 3.0  Use the graphing calculator to find the maximum and minimum values of a function 9.0  State and apply Rolle’s theorem and the Mean value theorem 8.0  Use the first derivative test to find the intervals on which a function is increasing and decreasing to determine relative extrema of a function 9.0,11.0  Use the second derivative test to determine intervals of concavity of a function and to locate inflection points 9.0,11.0  Use the second derivative test to determine relative extrema of a function 9.0,11.0  Use information about intervals of increase and decrease, relative extrema, intervals of concavity and inflection points to sketch the graph of a function 9.0  Use information about the derivative of a function to determine the sketch of the graph of the function and to determine the concavity of the function 9.0  Use derivatives to solve optimization problems 11.0  Solve problems involving related rates 12.0  Apply the derivative to problem solving involving position, velocity and acceleration 4.2  Use Newton’s method to approximate the zeros of a function 10.0

REVIEW FOR SEMESTER EXAM (1 WEEK)

SECOND SEMESTER

UNIT 5: The Definite Integral (3 weeks)  Understand the concept of area under the curve using Riemann sums over equal subdivisions 13.0  Compute Riemann sums analytically and by graphing calculator using left endpoints, right endpoints and midpoints as evaluation points 13.0  Use the limit of a Riemann sum to approximate a definite integral 13.0  Use the Fundamental Theorem of calculus to evaluate definite integrals 15.0  Compute simple antiderivatives and the average value of a function15.0  Use the graphing calculator to calculate definite integrals numerically 21.0  Use the Mean Value theorem for integrals to find the average value of a function on a interval  Use the second Fundamental Theorem of calculus to find derivatives 15.0  Use Simpson’s and trapezoidal rules to approximate a definite integral 21.0  Integrate exponential and logarithmic functions 15.0,17.0  Evaluate integrals that yield inverse trigonometric functions 17.0  Calculate antiderivatives by using techniques of integration such as substitution, integration by parts, trigonometric substitution, partial fractions and completing the square. 17.0 and 19.0  Compute improper integrals (as limits of definite integrals) 22.0

UNIT 6: Differential equations and Mathematical Modeling (2 weeks)  Apply the antiderivative to solving problems such as exponential growth and decay 27.0  Find general and particular solutions to differential equations with separable variables 27.0  Solve initial value problems with separation of variables 27.0  Construct a slope field for a differential equation 27.0  Use a graphing calculator to construct a slope field 27.0  Solving the logistic model and antiderivatives by partial fractions  Solving initial value problems by Euler’s method

UNIT 7: Applications of definite integrals (2 weeks)

 Use definite integrals to find area under the curve 16.0  Use definite integrals to find area between two curves 16.0  Find the volume of a solid of revolution using both the disk and washer method 16.0  Find the volume of solids with known cross sections 16.0  Find the distance traveled by a particle along a line 16.0  Find the area of surface of revolution and arc length 16.0  Model a given physical real-world situation with an appropriate integral and write an appropriate Riemann sum and set up the definite integral 14.0,16.0

UNIT 8: Parametric, Vector, and Polar Functions (2 weeks).

 Calculate the length of parametrically defined curves  Explore vectors and vector-valued functions  Calculus of vector functions  Calculus of polar functions including slope, length, and area

UNIT 9: Sequences (2 weeks)

 Define arithmetic, geometric, harmonic, and alternating harmonic sequences  Define convergence and divergence  Distinguish bounded, monotonic, and oscillating sequences  Explore limit properties of sequences  Apply comparison test to improper integrals

UNIT 10: Series (4 weeks)

The following concepts will be explored:  Definition and notation of series; sequence of partial sums; geometric, harmonic, alternating harmonic series  Infinite geometric series  Terms of series as areas of rectangles  Power series; interval and radius of convergence defined  Taylor series  Maclaurin series for ex, sin x, cos x, and functions  Functions defined by series  Taylor polynomials  Taylor’s theorem with Lagrange form of the remainder  Radius of convergence: nth term test; direct comparison test; absolute and conditional convergence; alternating series test  Interval of convergence and testing endpoints; integral test; p-series; limit comparison test; alternating series test

REVIEW FOR THE FINAL EXAM (1-2 WEEKS)

III. CLASS PROCEDURES

 Students are expected to have the correct textbook, a calculator, loose-leaf paper, graph paper, a folder, homework notebook and a pencil each day.  Be in seat and prepared for class when the bell rings.  You treat each other with respect. Raise your hand when you want to talk.  All assignments must be done in pencil. You need to show all your work to get credit.  When you miss a test, you will take a make-up test the day you return in class. If it is a planned absence, you need to let me know latest the day before the test. I will not give any make-up quizzes.  I maybe adding extra sessions when needed.

IV. ASSIGNMENTS, TESTS, AND QUIZZES

 I assign homework daily, which includes numerical problems, graphs, and writing assignments. Each homework assignment is due to the next class day, unless otherwise specified. I expect to see all your work on the assignments. All work must be shown neatly. The writing assignments need to have detailed analysis. I collect and check the homework assignments daily.  Class-work assignments are part of your grade. I assign as often as possible small- group exercises during the class. Everyone in the group must contribute equally to the exercise and all work must be shown neatly to get credit.  Quizzes and tests: You will be given a number of quizzes and one test per chapter. The quizzes will be open book and the test will be closed book. Quizzes may be announced or not. You need to be prepared for every class. I announce the test at least one week before its date. You will also receive a study guide for each test.  Final exams: You will be given one final exam at the end of the first semester. You will have 2 Practice AP tests at the end of the second semester which will count as final exam each.  Review quizzes: This class prepares you for the AP test next May. You need to review constantly what you learned throughout the year. For that purpose, I will give every month review quizzes which will include multiple-choice and free-response questions and cover all subjects done up to the day of the quiz.  My website: All assignments will be posted at mathrepublic.webs.com. The site will be updated by the second week of the school. Please make sure that check the website if you miss a class or unsure about the assignment V. GRADING:

Chapter Tests: 100 points each

Quizzes: Up to 30 points each

Review Quizzes: Up to 80 points each

Final exam: 150 points

Homework and class-work assignments: 10-20 points for each numerical assignment and up to 40 points for each writing assignment.

Grades are cumulative through the end of each semester and will be based only on demonstrated mastery of concepts and development of skills.

Grade Scale: A+ 100-97 A 96-93 A- 92-90 B+ 89-87 B 86-83 B- 82-80 C+ 79-77 C 76-73 C- 72-70 D+ 69-67 D 66-63 D- 62-60 F 59 % and below

VI. TUTORING HOURS: Tue-Th 3:00-4:00pm You can also see me M-TH during nutrition and the first half of the lunch by appointment.

Instructor Contact: Mr. Gulsoy Phone: 818-894-6417 Ext. 315 E-mail: [email protected]