Course Syllabus

AP Calculus BC

Title of Course

Teacher Name: Elaine Drugan Contact Times: 1:30 – 2:25 pm

E-Mail: Telephone Number: 832-484-5214

TEKS for Course
Nine Goals:
-work with functions represented in a variety of ways
-understand the meaning of the derivative
-understand the meaning of the definite integral
-understand the relationship between the derivative and the definite integral
-communicate mathematics
-model a written description of a physical situation with a function, a differential equation, or an integral
-use technology to help solve problems, experiment, interpret results, and verify conclusions
-determine the reasonableness of solutions
-develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment
Grading Determination
Major grade 70%; Daily grade 20%; homework 10%
Late Work Procedure
Major grade: 10% per day late; Minor grade: ½ credit if late
Reteach/ Retest Procedure
Before and after school study sessions; maximum benefit from test corrections will be ½ credit back on wrong answers; maximum benefit from retake assignment will be a grade of 70; if a fourth major grade is offered in a grading period, it may be used to replace the lowest grade of the same description
Note: There are no provisions for the grade repair program in this AP course.
Projects
No more than 1 per grading period
Outline/Calendar
Syllabus: AP Calculus BC
Unit I: (5 days) Pre-Calculus Review including Polar Co-ordinates
Student will demonstrate understanding of pre-calculus concepts; student will practice using polar co-ordinates: Intercepts, symmetry, point-slope equations, function terms, transformations, classification of functions, composite functions, even/odd, trigonometry, unit circle, study of polar co-ordinates. Student responses are both written and verbal and are supported by use of a graphing calculator. Student renews skills in the use of the graphing calculator.
Assessments: series of unit circle quizzes including questions using polar co-ordinates
Unit II: (8 days) Limits and Continuity
Student builds concepts of limits, continuity: Epsilon/Delta definition of limit, properties of limits, techniques for evaluating limits, continuity, one-sided limits, intermediate value theorem, infinite limits, vertical asymptotes. Limits and continuity are explored using graphs and tables.
Assessments: test on limits and their properties; workbook activity on limits (includes writing definition of continuity, drawing sketches where limits both do and do not exist, one-sided limits, epsilon/delta definition of a limit, Foerster workbook, p.231 )
Unit III: (25 days) Differentiability, Definition of Derivative, Review Parametrics Student defines derivative and learns differentiation rules; student makes connection between graph of a function and its derivative; student uses differentiation to solve related rates problems; student reviews parametric equations: Tangent line problem, differentiability and continuity, power rule, product rule, quotient rule, chain rule including parametric problems, trig derivatives, rates of change, higher order derivatives, implicit differentiation, related rates, sketch parametric curve, eliminate parameter.
Assessments: test on definition of derivative, differentiation rules and differentiation techniques; test on related rates; project on related rates (packet of sample AP problems)
Unit IV: (15 days) Application of Derivative
Student uses differentiation to examine extrema; student learns and applies Rolle’s and Mean Value Theorems: Critical numbers, increasing/decreasing functions, first derivative chart, second derivative test, second derivative chart, concavity, points of inflection. Although this content section requires problem solving without the use of a calculator, graphing calculators are still important as a means of experimenting and to support conclusions. It is also extremely necessary in this unit for the student to explain, both verbally and in writing their understanding of the theorems.
Assessments: test on extrema, Rolle’s Theorem, Mean Value Theorem, increasing/decreasing, concavity; workbook activity comparing f, f’, f” (given a graph of a function, sketch the graph of the derivative; use a derivative graph to explain features of the original function, Foerster workbook, p. 67)
Unit V: (17 days) Analyze Graphs of Functions, Differentials
Student studies curve-sketching techniques and optimization problems:
Limits at infinity, horizontal asymptotes, extrema, optimization, Newton’s method for finding zeros of a function, linear approximation, definition and use of differentials, business and economic applications of derivatives, propagated error. Parts of this unit are intended for students to problem solve without a calculator, but confidence of accurate answers can be achieved by using a graphing calculator to check work. Student explains both local and global behavior of functions including those given in parametric, polar, or vector form.
Assessments: free response problems; test on application of derivative to curve sketching and optimization; test on Newton’s method and differentials; practice AP questions
Unit VI: (15 days) Antiderivatives, Riemann Sums, Review
Student defines antiderivative, learns integral notation; student develops concept of Riemann sum to approximate area; student reviews first semester topics: Antiderivatives, integral notation, initial condition, upper and lower bounds, Riemann sum problems, review limits, continuity, derivatives, graphs.
Assessments: practice AP test; test on antiderivatives, area approximation; workbook activity on indefinite integrals and Riemann sums (includes writing definition of indefinite integral, demonstrating the meaning of upper sum vs. lower sum by marking the graph, demonstrating the difference between delta y and dy by marking the graph, using u-substitution, considering initial condition, Foerster workbook, p. 253)
At End of First Semester:
Semester Final: functions, graphs, limits, derivatives, antiderivatives. Test is modeled after the AP test with calculator and non-calculator sections, split between multiple- choice questions and free response questions, requiring detailed written solutions.
Unit VII: (13 days) Fundamental Theorem of Calculus, Integration, Series
Student studies fundamental theorem as connection between derivatives and integrals; student works with integration techniques; student explores limits of sequences, infinite series: Fundamental theorem of calculus, u-substitution, change of limits of integration, trapezoidal and Simpson’s rules, numerical integration, composite functions, definite integrals, average value of a function, series as a sequence of partial sums.
Assessments: test on integral concepts and fundamental theorem, project of practice free response questions; series of quizzes titled Free Response Friday; activity demonstrating fundamental theorem (Larson, p. 294)
Unit VIII: (12 days) Exponential functions, Inverse functions, Differential equations
Student considers derivatives and antiderivatives of transcendental functions; student studies inverse functions, compares ln to e^x; student begins solving simple differential equations: Derivatives and integrals of exponential functions, bases other than e, growth and decay, Newton’s Law of Cooling, inverse functions, inverse trigonometry and restricted domains, Euler’s method to approximate solutions to differential equations, logistic differential equations problems. The graphing calculator is essential in gaining confidence in work with transcendental functions.
Assessments: test on derivatives and integrals of transcendental functions; project on Riemann sums, trapezoidal rule, Simpsons rule (compute area under a curve using each technique and finally using a definite integral). Student responses include an explanation connecting integrals to their numerical approximations.
Unit IX: (20 days) Area, Volume, Integration techniques, Slope fields
Student uses definite integral to find area of a region, volume of a solid, length along a curve; student studies techniques for solving limits, integrals, and differential equations: Area between two curves, solids of revolution, disc method, shell method, solids with known cross sections, arclength, integration by parts, partial fractions, improper integrals, L’Hospital’s Rule, slope fields. Student learns calculator techniques to assist in the computations and is able to explain how the calculator is programmed to arrive at an answer.
Assessments: test on area, volume, arclength, slope fields; test on sample AP concepts to date; project calculating volume of actual physical objects; exploration on slope fields (objective is to solve a differential equation graphically, using its slope field, and make interpretations about various particular solutions, Foerster workbook, p. 109)
Unit X: (13 days) Series, Taylor Polynomials, Power Series
Student learns formulas for known series; student uses graphing calculator to explore convergence as the limit of the sequence of partial sums; student uses graphing calculator to investigate Taylor polynomials as approximations for known functions; student studies power series: Series (geometric, harmonic, p-series, alternating), convergence tests (integral, direct comparison, limit comparison, alternating, ratio, root), Taylor polynomials, Taylor series centered at x=a, Maclaurin series for e^x, sin x, cos x, 1/(1-x), form new series from known series, radius of convergence, interval of convergence, LaGrange error bound.
Assessments: test on Taylor and Maclaurin series, convergence; practice AP problems; workbook activities (e.g. power series for a familiar function: learn what a power series is, and how it can fit closely a particular function, Foerster workbook, p. 157)
Unit XI: (20 days) Parametric equations, Polar equations, Vectors
Student works with polar and parametric equations; student learns special polar graphs; student practices vector problems including distance between points: Compute derivatives and definite integrals of polar and parametric functions, interpret integrals, solve problems involving position, direction, velocity, speed, acceleration, polar area, parametric arclength.
Assessments: practice AP multiple-choice test; practice free response problems; workbook activities (e.g. derivatives of a position vector: given the vector equation for motion in two dimensions, find velocity and acceleration as functions of time, Foerster workbook, p. 145)
Unit XII: (16 days) Semester Review
Student demonstrates mastery of Calculus I concepts: Prepare review project of specific topic to be shared with class, demonstrate techniques for solving problems along with explanations of both theory and application.
Assessment: project for year-end review
At End of Second Semester:
Semester Final: functions, graphs, limits, derivatives, integrals, polynomials approximations, series. Test is modeled after the AP test with calculator and non-calculator sections, split between multiple-choice questions and free response questions, requiring detailed written solutions.
Student Absence Procedures
Student is responsible for scheduling make-up work within same number of days as days missed
Test Days
Tuesday, Friday

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