AP Calculus BC Course Overview AP Calculus BC is designed to follow the topic outline in the AP Calculus Course Description provided by the College Board. The primary objective of this course is to provide a fundamental understanding of Differentiable and Integral Calculus and Polynomial Approximation and Series. The course teaches the concepts of calculus by drawing upon graphical, algebraic, numeric and verbal representations of functions, including parametric, polar and vector functions. Communication of calculus is highly stressed throughout the course. On a daily basis, students are expected to verbally communicate methods and solutions in class discussions and in small group situations. Students are also expected to explain solutions in written sentences on both homework problems and on tests. Course Outline and Pacing Guide The following is a guide to topics and time spent for lecture, instructional labs and group work. Time of review and assessment is also included. The course is taught in a traditional 7-period day, where class times are 51 minutes 4 days/week and 41 minutes 1 day/week. Although class time is divided among lecture, lab and group work, over 50% of the time, students work in small groups with the teacher acting as facilitator. The pacing goal is to complete the AP curriculum with at least 20 instructional days remaining before the AP exam. This time is spent reviewing and preparing for the AP exam.
AP Calculus grades are based upon the following: Tests ~50% In-class Labs and Group work ~30% Homework ~20% Student Evaluation Tests are divided into two sections, a Calculator and a Non-Calculator section. Each of these sections is approximately half multiple-choice and half free-response. Free response questions are multi-part, require students to explain solutions in written sentences and are graded on a rubric that ranges from 5 to 9 points. Homework is collected on a weekly basis. Students are encouraged to work out homework and in-class problems with the assistance of others, but each student must submit his/her own solutions. Many solutions require written explanations to solutions. Technology Resources Students must use a College Board approved graphing calculator. Students are required to use calculators to find: Zeros of a function Points of intersection of functions Derivative of a function at a given point The value of a definite integral
Accompanying Textbook Calculus: Graphical, Numerical, Algebraic (3 rd Edition) Finney, Demana, Waits, Kennedy Book Correlation Topics ~Time Related Assignments (Problem numbers in bold print are categorized Writing to Learn problems in the text and require a written solution.) Chapter 1: Prerequisites for Calculus 1.3, 1.4, 1.5, 1.6 Review Domain and Range Viewing and Interpreting Graphs Exponential & Logarithmic Inverses Trigonometric Odd and Even Inverse Trigonometric 4 days p. 26: 3, 6, 9, 12, 13-18, 23, 30, 39, 41-46 p. 34: 1-4, 5, 7, 9, 10, 14-16, 19, 37, 39, 40 p. 44: 1-6, 17-20, 33-42, 52-57 p. 52: 1, 3-8, 17-22, 25-30, 39-44
2.1 Rates of Change And Limit 2.2 Limits Involving Infinity 2.3 Continuity 2.4 Rates of Change and Tangent Lines Rates of Change Average speed Definition of Limit Chapter 2: Limits and Continuity 2 days p. 66: 1-4, 8-44 even, 50, 54, 67-70 Limit as x a One- and Two-side limits Squeeze Theorem Limits as x End Behavior Models Continuity at a Point Continuous Intermediate Value theorem Average Rates of Change Average vs. Instantaneous Rates of Change Tangent to a Curve Slope of a Curve Difference Quotient Normal to a Curve sin In-Class Group: #75 (Proving lim 1 0 using geometry and Squeeze Theorem) 1 day s p. 76: 3, 6, 9-12, 33-38, 54, 55, 68 2 days p.84: 3-36 mult. of 3, 41-44, 47, 49, 51, 56-59 p.92: 2-10 even, 11-16, 18, 29-32, 37-40 2 days
3.1 Derivative of a Function 3.2 Differentiability 3.3 Rules for Differentiation 3.4 Velocity and Other Rates of Change Chapter 3: Derivatives Def. Of Derivative Notation Relating Graphs of f and f Graphing Derivative from Data How f (a) Might Fail to Exist Local Linearity Numeric Derivative on a Calculator Differentiability Implies Continuity IVT for Derivatives Power Rule Constant Multiple Rule Sums and Differences Products and Quotients Second and Higher Order Derivatives Instantaneous Rates of Change Motion along a Line Position, Velocity and Displacement 5 days p. 105: 1, 2, 6, 9, 12, 13-17, 21, 22, 23, 26, 35 p. 114: 2-10 even, 11-15, 31, 33, 38, 44, 45 p. 124: 3-48 mult. of 3, 55-57 Lab: Given simultaneous graphs of f, f and f ; identify each. 2 days p. 135: 1, 2, 5, 9-12, 19, 22, 32, 36
3.5 Derivatives of Trigonometric 3.6 Chain Rule 3.7 Implicit Differentiation Derivatives of Sine and Cosine developed visually Derivatives of Sine and Cosine functions Developed graphically Using Graphing Calculator s Ability to Graph f and f simultaneously Derivatives of tan, sec, cot and csc Developed Using Rules of Differentiation Review of Composite Chain Rule Repeated Use of Chain Rule Apply Chain Rule to Tabular Derivatives of Parametrically Defined Curves Implicitly Defined Tangent and Normal Lines Higher Order Derivatives Using Implicit Differentiation p. 146: 2-10 even, 23-28, 37, 46-48, 51 Lab: Given graph of a period of sine (cosine) function, use tangent line slopes to visually produce its derivative. 2 days p. 153: 3-39 mult. of 3, 56, 68, 73, 74 p. 153: 41-48 2 days p. 162: 3-39 mult. of 3, 46, 49, 50, 55, 56 Lab: Explicit diff of conic sections compared to implicit method
3.8 Derivatives of Inverse Trig 3.9 Derivatives of Exponential and Logarithmic d 1 sin x Developed dx Geometrically d 1 tan x Developed dx Geometrically Remaining Inverse Trig Function Derivatives Justification of 1 1 f ( x) 1 f f ( x) Geometrically x Derivative of e Derivative of ln x x Derivative of a Derivative of log x Logarithmic Differentiation a p. 170: 3-27 mult. of 3, 28-31, 40 3-27 mult. of 3, 30-50 even, 51, 52, 53, 56
4.1 Extreme Values of 4.2 Mean Value Theorem 4.3 Connecting f and f with the Graph of f 4.4 Modeling and Optimization Chapter 4: Applications of Derivatives Locating Critical Points Absolute Extreme Values Local Extreme Values Finding Extreme Values MVT Physical Interpretations Increasing and Decreasing First derivative Test Second Derivative Test Concavity Points of Inflection p. 193: 5-13, 18, 23, 24, 26, 37, 39, 40, 43-47, 50, 51 p. 202: 1-13, 20, 21, 28, 39-42, 46, 51, 52 p. 215: 4, 5, 9, 12, 16, 22, 23, 24, 28, 37, 39-42, 43-46, 49 2 days Real-world Optimization problems 2 days p.226: 2, 4, 5, 7, 10, 11, 12, 16, 20, 21, 30, 36, 39, 41, 42, 43, 44, 47, 48 4.5 Linearization Linear Approximation Estimating Change 4.6 Related rates Related Rate Problems Solution Strategies Lab: Optimizing fuel consumption as a function of its velocity 1 day p. 242: 1-4, 11, 12, 31, 33, 45 4 days p. 251: 1, 2, 11, 13, 14, 16, 17, 19, 21, 27, 28, 31, 33, 42
5.1 Estimating with Finite Sums Chapter 5: The Definite Integral Distance Traveled Given Velocity 2 days p.270: 2, 5, 6, 15, 17, 19, 23, 30 Rectangular Approximation Method Given Discrete Data Lab: Compare endpoint/midpoint Left-Hand, Right-Hand and methods to produce better estimates of Midpoint Sums area under a curve. p. 282: 8-28 even, 29-36, 41-54, 57 5.2 Definite Integrals Riemann Sums Terminology and Notation of Integration Definite Integral and Area Integrals on a Calculator through graph screen and by using num integration function Discontinuous Integrable 5.3 Definite Integrals and Antiderivatives 5.4 Fundamental theorem of Calculus Properties of Definite Integrals Average Value of a Function Geometrically and Algebraically MVT For Definite Integrals Connecting Differentiable and Integral Calculus FTC, Part I Graphing x a f ( t) dt FTC, Part II Area Connection Graphically Analyzing Antiderivatives 5.5 Trapezoidal Rule Trapezoidal Approximations Comparing Geometric Approximations 4 days p. 290: 1, 3, 7, 8, 11-36, 39-41, 47, 48 2 days 4 days p. 302: 3-18, 27-60 mult. of 3, 64 Lab: Volume of a right circular cone 2 days p. 312: 2-12 even, 19, 20, 22, 34, 36
6.1 Slope Fields and Euler s Method 6.2 Antidifferentiation by Substitution 6.3 Antidifferentiation by Parts 6.4 Exponential Growth and Decay Chapter 6: Differential Equations and Mathematical Modeling Defining and Solving Differential Equations Constructing Slope Fields by Hand Matching Slope Fields with Differential Equations Use Euler s Method to find/graph solutions to initial value problems Evaluating an Indefinite Integral Producing Antiderivatives from Derivatives of Basic Antiderivatives by Substitution of Variables Change of Limits in U-Substitution Integration by Parts derived from Product Rule for Differentiation Tabular Integration Use Parts to integrate Inverse Trig and Log functions Separable Differential Equations Solving Differential Equation for a Given Initial Condition Exponential Growth and Decay Review Newton s Law of Cooling and Other Applications 6.5 Logistic Growth Antidifferentiation with Partial Fractions Logistic Curve and Differential Equation Interpreting Logistic Growth Models p. 327: 3-24 mult. of 3, 25-40, 49, 50, 51, 52 Lab: reading slope fields 2 days p. 337: 2-12 even, 21-34, 40, 41, 53-66, 73, 74, 77, 79 p. 346: 9-15, 18-32 even, 33, 38, 40 2 days p. 357: 1-10, 22, 23, 25, 28, 31, 32, 40-44, 51, 52 p. 369: 3-11, 23-26, 30, 31-34, 43, 44
7.1 Integral as Net Change Chapter 7: Applications of Definite Integrals Velocity to find displacement 2 days p. 389: 3-36 multiples of 3 Velocity to find total distance traveled Solve Problems involving Linear Motion, Consumption over time, work and net change from data 2 days p. 395: 2-16 even, 18-45 multiples of 3 7.2 Areas in the Plane Area between curves Area enclosed by intersecting curves Boundaries of Changing Integrating with respect to y 7.3 Volume as an Integral Volume of known cross-section Disks and Washers Cavalieri s Theorem 7.4 Lengths of Curves Arc Length of an explicitly defined function p. 406: 3-42 multiples of 3, 44, 66-68 1 day p. 416: 3-21 multiples of 3, 25, 27, 35
Chapter 8: Sequences, L Hopital s Rule, and Improper Integrals 8.1 Sequences Defining and Reviewing 1 day p. 441: 3, 6, 12-18 even, 31-39, 57, 58 Sequences Arithmetic, Geometric and Other Sequences Limit of a Sequence 8.2 L Hopital s Rule Indeterminate Forms 1 day p. 450: 3-51 multiples of 3, 58, 70 L Hopital s Rule 8.3 Relative Rates of Comparing Growth Rates of 1 day p. 457: 2-24 even, 39 Growth using L Hopital s Rule 8.4 Improper Integrals Evaluating Integrals over infinite intervals Infinite Discontinuity at an Interior p. 467: 3-39 multiples of 3, 50-53, 57 1 Point dx Infinite Discontinuity at an Investigation: Generalizing p and 0 x Endpoint Test fro Convergence and dx p Divergence (Comparison Test) 1 x
Chapter 9: Infinite Series 9.1 Power Series Geometric Series 2 days p. 481: 3-60 multiples of 3, 62, 68 Identifying Convergent and Divergent Series Representing by Series Finding a Power Series by Differentiation and by Integration Generate Power Series for tan -1 x 9.2 Taylor Series Constructing a Series by 5 days p. 492: 3-42 multiples of 3 Designing a Polynomial to Specifications Series for sin x and cos x Maclaurin Series 9.3Taylor Polynomials Taylor s Theorem and Remainder 2 days p. 1-10, 19, 22, 28, 30, 34 Proving Convergence 9.4 Radius of Convergence Direct Comparison Absolute Convergence Ratio Test Telescoping Series p. 511: 3, 4, 6-14, 26-28, 30-42 even, 47 9.5 Testing Convergence at Endpoints Endpoint Convergence Integral Test Harmonic Series and p-series Limit Comparison Test Absolute and Conditional Convergence Finding Intervals of Convergence 4 days p. 523: 6-42, multiples of 3, 62
10.1 Parametric 10.2 Vectors in the Plane 10.3 Polar Polar Curves Slopes of Polar Curves Chapter 10: Parametric, Vector and Polar Parametric Curves p. 535: 3-33 multiples of 3 Slope and Concavity Arc Length of Parametrized Curves Magnitude and Direction 2days p. 546: 30-50 even Velocity, Acceleration and Speed Displacement and Distance Traveled p. 557: 13-20, 39-52 Areas enclosed by Polar Curves 2 days