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Advanced Placement Calculus AB Syllabus Course Description Calculus AB is primarily concerned with developing the student's understanding of the concepts of calculus and providing experience with its methods and applications. The course will consist of a four pronged approach to calculus. Concepts, results, and problems will be expressed graphically, numerically, analytically, and using writing in context. The connections among these representations also are important are constantly related. Broad concepts and applicable methods are emphasized. The focus of the course is neither manipulation nor memorization of a predetermined list of functions, curves, theorems, or problem types. Although students must deal with manipulation and computational competence, they are not the sole purpose of this course. Technology, in the form of a TI-84 calculator, will be used throughout the course to reinforce the relationships among the various representations of functions; f(x) to f ’(x) to f ’’(x), to confirm written work, to encourage experimentation, and to assist in interpreting results relating to slope, maxima/minima, and area using tables, rectangular and trapezoidal approximations, and graphs. Through the use of the calculator, students will become aware of connections between derivatives, integrals, limits, approximations, and applications with modeling. As a result, the course becomes a connected whole rather than a collection of unrelated topics. Text Calculus: Graphical, Numerical, Algebraic by Finney, Demana, Waits, Kennedy; Scott Foresman,Addison Wesley 1999 Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus AB Examination by David Lederman and Lin McMullin, 2004, D&S Marketing Technology All students either have or are supplied with TI-84 calculators. The students have been using these calculators in Algebra II Honors, Pre- calculus Honors, and in AP Calculus AB to interprete graphs, determine limits, identify asymptotes, find maxima and minima points, and to determine where a function is increasing, decreasing concave up or down using graphs and tables. Calculation of Grades Students will be tested approximately every 2 to 4 weeks. Each test will consist of questions using a format that is similar to the AP Exam. These questions will contain multiple choice questions that include methodology questions, interpretation of graphs using calculators and derivatives, numerical tables and their relationship to slope of a curve, and essay questions from past exams that will include justification using written statements to interpret what is happening mathematically. Each test will consist of 4 components. Component 1 will be a timed multiple choice test of approximately 9-10 questions. This component is non-calculator active. Component 2 will be a timed multiple choice test of 5-6 questions. This component is calculator active. Component 3 will consist of 1 essay question (show your work) consisting of multiple parts that must be completed in 15 minutes. This question will be calculator active. Component 4 will be 1 essay question to be completed in 15 minutes. Calculators may not be used on this question. Grading of the test will be completed in the following manner: Components 1 and 2 will be combined. Students will earn credit based upon right, wrong, and blank answers using the formula (# correct - 1/4 # wrong) x 1.2 = points earned. Blank answers will not count against the student. 2. Components 3 and 4 will be graded based upon the student's work. Answers only will only earn the answer point. Components 1 and 2 are worth 18 points and Components 3 and 4 are worth 9 points each for a total of 36 possible points on each test. Grades are assigned in the following manner: 36 points----------100 31-35 points--------95 25-30 points--------93 19-24 points--------90 13-18 points--------83 7-12 points--------74 1-6 points--------65 Students may earn bonus points weekly on 10 minutes quizzes given on Tuesday and Thursday. These points will be totaled at the end of the nine weeks and divided into 5 zones. Zone 1 will earn 0 points. Zone 2 will earn 1 point. Zone 3 will earn 2 points. Zone 4 will earn 3 points. Zone 5 will earn 4 points which will be added to the nine weeks average. These quizzes cannot hurt the student’s grade, but they do consist of questions that are AP formatted with regards to the multiple choice form. Each nine weeks average will be computed as a simple average. Each semester average will be computed as 1st nine weeks times .4 plus 2nd nine weeks times .4 plus the exam grade. This total will then be divided by 5 to obtain the semester average. The grading scale for our District is as follows: A 93-100 B 85-92 C 77-84 D 70-76 F 69 and below Our students do receive additional grade point value for each grade due to AP calculus being considered an honors course. Topics and Timeline for First Semester Section Topic Timeline 1.1 – 1.3 Equations of lines; domain, range, and 1day zeroes of functions; exponential growth and decay functions; parametric forms 1.4-1.6 Continuity; vertical and horizontal 1day Asymptotes (calculator activity) 2.1 Average/instantaneous rate of change; 3 days Definition of limit; Properties of limits; Left/Right-hand limits; Sandwich Theorem (calculator activity-analysis of functions) 2.2 Limits as x approaches form of infinity; 2 days Limits of infinity; End behaviors (calculator activity-table analysis of function Behavior) 2.3 Continuity at a point; Continuous function; 2 days Combinations/Composition of continuous Functions; Intermediate Value Theorem 2.4 Average rate of change and slope; Equations 2 days of tangent lines; Slope of a curve; Normal lines to a curve; Speed 3.1 Definition of derivative; Notation; Connections 4 days between graphs of f and f ’; Graph derivatives from data and tables Test Review 2 days Test # 1 Covers all topics listed above 2 days 3.2 Failure of f ’(x) to exist; Differentiability implies 2 days linearity and continuity; Intermediate Value Theorem for derivatives; Use calculator to find derivatives and graph same 3.3 Rules for differentiation: positive integer 3 days powers; Sums and differences; Products and quotients; Negative integer powers; Higher order derivatives 3.4 Instantaneous rate of change; velocity; motion 3 days along a line; sensitivity to change; Applications 3.5 Derivatives of sine/cosine; Simple harmonic 2 days motion; Jerk; Derivatives of tangent/cotangent and Secant/Cosecant Test Review 2 days Test # 2 Derivatives of above, definition of derivative, 2 days multiple derivatives, domain of a function, velocity/acceleration, limits, reflections, zeroes, equations of tangent/normal lines, parallel/ perpendicular lines, position, speed, jerk 3.6 Derivatives of composition; Repeated use of 5 days chain rule; Slopes of parametric curves; Chain rule with powers of functions 3.7 Implicit functions and their derivatives; Tangents 2 days and normals; implicit derivatives of higher order; rational powers and their derivatives 3.8 Derivatives of inverse trigonometric functions 2 days 3.9 Derivatives of all forms of exponential functions; 3 days derivatives of all forms of logarithmic functions Test Review 2 days Test # 3 Derivatives of any type of function mentioned 2 days above, zeroes of functions, multiple derivatives, laws of logarithms, continuity, asymptotes, equations of tangent/normal lines, derivatives of composition, definition of derivative, odd/even functions, vertical/horizontal tangents, slope of a curve, 4.1 Absolute extreme values; Local extreme values; 2 days finding extreme functional values 4.2 Mean Value Theorem; interpretation; increasing/ 2 days decreasing functions 4.3 First derivative test for local extrema; concavity; 3 days points of inflection; second derivative test for local extrema; functions from derivatives Test Review 2 days Test # 4 Asymptotes, increasing/decreasing functions, 2 days Concavity, maxima/minima, point of inflection, Odd/even functions, behavior of tangent lines, Equations of tangent lines, implicit differentiation, Derivatives, limits, maxima/minima with justification, Concavity/point of inflection with justification, Sketch graph using first derivative information, Use graph of first derivative to find maxima/minima with written justification, use graph of first derivative to find inflection points with written justification 4.4 Modeling and optimization Methodology; Business 5 days and industry; Mathematics; Economics 4.5 Linearization; Newton’s method; Differentials; 5 days L”Hopital’s Rule Estimating change with derivatives; Absolute, relative, and percent of change 4.6 Related rates; solution strategy 5 days Test Review 2 days Test # 5 Normal/tangent lines, related rates, derivatives, 2 days Definition of derivative, limits, maxima/minima, Evaluate derivatives, odd/even and symmetry, Differentiability, zeroes of a function, Mean Value Theorem for derivatives First Semester Exam—covers all topics listed above; Split format: 28 questions-multiple choice without calculator and 17 questions-multiple choice with calculator. Topics and Timeline for Second Semester Section Topic Timeline 5.1 Distance traveled; Rectangular Approximation 2 days Left, right, and midpoint rectangular approx- imation; 5.2 Summation notation; Reimann sums; 2 days Notation of integration; Definite integrals and area; Constant functions; Integrals on calculator; Discontinuous integrable functions 5.3 Properties of definite integrals; Average value 3 days of a function; Mean Value Theorem for Definite Integrals; Connecting Differential and Integral Calculus 5.4 Forms of Fundamental Theorem of Integral 3 days Calculus; Area connection; Applications 5.5 Trapezoid and Simpson’s approximations 2 days 6.1 Initial value, anti-derivatives and indefinite 2 days integrals, properties of indefinite integrals, slope-fields 6.2 Integration by substitution, change of limit, 2 days separation of variables for differential equations Test review 2 days Test # 6 Symbolic differentiation, First Fundamental Th. 2 days of Integral Calculus, definite integrals, properties of integrals, definition of derivative, integrals with absolute value, Fundamental Theorem of Integral Calculus (alternative version), integration with unknown limits, max/min, symmetry, inverses, area between curves, equations of tangent lines, points of inflection with justification 6.3 Integration by parts, repeated use, solve unknown 2 days integrals 6.4 Solve exponential change using separation of 3 days variables 6.5 Exponential and logistics growth models; Partial 2 days fractions 6.6 Euler’s method 2 days Test review 2 days Test # 7 Definite integrals, derivatives, integrals with 2 days unknown limits, derivatives and distance using graphs, tangent or normal lines, functions from graphs, increasing or decreasing functions, concavity, Fundamental Theorem of Integral Calculus, area under curve, definition of derivative and continuity, Mean Value Theorem for derivatives, Maxima and minima with analysis, roots and average value of a function, initial conditions problem, velocity 7.1 Linear motion, net change from data, work 3 days 7.2 Area between curves, boundaries with 3 days changing functions, integration with respect to y 7.3 Volumes of solids: disc, washer, and shell; 6 days known cross-sections and volume 7.4 Length of smooth curves, vertical tangents, 3 days corners, and cusps Test Review 2 days Test # 8 Concavity, indefinite integrals, tangent lines, 2 days approximation of zeroes, limits from graphs, definite integrals, relative extrema, derivatives, initial conditions problems, area using properties of integrals, area between curves, volume of known cross-sectional area, integration by parts, Intermediate Value Theorem, instantaneous and average velocity, total distance traveled, area between curves, length of a curve Review for AP exam Practice problems in exam format. 20-25 days Students will initially work in groups of 2-3. Practice problems will be worked in class in an exam format. Students will consult with each other or the teacher to address specific difficulties. After 10 class days, the students will work individually on practice tests under a time format that conforms to AP guidelines. After each session, questions may be asked about specific problems. Practice AP exam Previous AP multiple-choice exam and last 1 day year’s essay exam will be given following exam time and form requirements Final Review Students will discuss the practice exam or any 5 days specific topics they feel are necessary to help them achieve their desired score on the exam Total days 162 days AP Exam Last days of class Review of essay exam and preparation 14 days for BC calculus