AP Calculus BC. Course content and suggested texts and reference materials align with the College Board framework for AP Calculus BC.

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1 AP Calculus BC Course Overview Topic Description AP Calculus BC Course Details In AP Calculus BC, students study functions, limits, derivatives, integrals, and infinite series This document details the topics and subtopics that f under each unit and chapter. Teacher Role Prerequisites Course Length Two semesters Throughout the course students write and work with functions represented by written descriptions, mathematical rules, graphs and tabular data. Throughout the course, students develop and exercise skills using the graphing calculator to solve problems, experiment, interpret results, and support their conclusions. Students learn the meaning of the derivative and apply it to a variety of problems, while developing a deeper understanding of the meaning of the solutions to those problems. Students study integrals and learn the relationship between the derivative and the definite integral, using written work and graphing technology to explore and interpret this relationship. Students learn how calculus is used to model real-world phenomena by using functions, differential equations, integrals, and graphing technology to solve problems, support their solutions, and interpret their findings. Students communicate about mathematics through written work and discussion forums with peers that are monitored by the teacher. Asynchronous and synchronous discussion activities throughout the course provide multiple opportunities for students to interact with each other and share ideas about math problems and problem-solving strategies. Discussions also include opportunities for students to work in sm groups where they collaborate on specific assignments. The detailed syllabus outline below indicates where these discussions occur and what the topics are. In this course, the teacher hosts and facilitates weekly synchronous sessions with students who are enrolled. In these regularly scheduled sessions, students communicate with each other and the teacher about course content and assignments. These synchronous sessions ow for timely verbal dialogue about AP Calculus BC content and course assignments. As needed, the teacher guides students through appropriate explanations of assigned problems and solution sets. Helpful guidelines for these sessions are provided to the teacher. Course content and suggested texts and reference materials align with the College Board framework for AP Calculus BC. All students enrolled in this course are assigned to a "section" with a qualified teacher who is responsible for ensuring student success and addressing student questions, problems, and concerns. In addition, each student must have a mentor available at their school or at home to support the student and make sure assignments are completed in a timely manner. Algebra II, Geometry, Pre-Calculus with Trigonometry AP Calculus BC, page 1

2 Course Materials This online course offers instructional content that incorporates required topics in a balanced and comprehensive sequence. Online digital instruction includes text, figures, graphic elements, carefully structured problem sets, exploration guides, and graphing calculator instructions to convey and highlight important information and provide students with specific applications of concepts they are studying. The required virtual content for this course is as follows: Thomas, Paul et al. (editors). AP Calculus BC, K12 digital edition. Herndon, VA: In addition, students should have this required (printed) textbook: Larson, Ron, and Bruce H. Edwards. Calculus of a Single Variable, AP Edition (9th ed.), Belmont, CA: Brooks/Cole, Cengage Learning, [ISBN: ] The following additional (optional) textbooks may be used to supplement the material presented in this course: Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic (3rd ed.), Boston: Pearson Addison Wesley, [ISBN: ] Stewart, James. Single Variable Calculus (7th ed.), Belmont, CA: Brooks/Cole, Cengage Learning, [ISBN: ] The student or the school must purchase a TI-84 Plus calculator (or similar calculator approved by the College Board) for the AP Calculus BC exam. Specific references for use of these texts appear at the end of this document, beginning on page 19. AP Calculus BC, page 2

3 The following details describe different types of instructional activities in this course. Activity Type Learn Video Lectures Explore and Using a Graphing Calculator Description Primary instructional content presented online to teach new concepts through multimedia and interactivity. Paper and pencil activities are included in Explore activities. Graphing calculator activities are also included (these appear in bold in the course outline below). In these Explore activities, students are guided through the key steps for using calculators to explore, experiment, analyze and interpret findings, and support their conclusions For example, in the graphing calculator activity titled Taking More Intervals, students learn how to use the SEQ function to find a Riemann sum, and then use this functionality to test conjectures about the effect that using different interval sizes has on resulting area approximations. Discussion Students discuss topics in threaded discussion boards (these discussions appear in bold in the course outline below). Teachers monitor and participate in these discussions, and students receive credit for appropriate participation. Some discussions include group activities that require student-to-student communication about calculus strategies and concepts. For example, in the Discussion titled Hands-on Solids, students create and compare solids with cross-sections that are circles, squares, or rectangles. Within peer discussion groups, students observe, interact, and compare crosssections and then explain their methods for calculating volume to each other. Practice Try It Problem Set Assessment Quiz Review Lesson Unit Test Students answer online, computer-scored (ungraded) questions to help them synthesize what they have learned in a lesson. This helps them think about the content before using it in a problem set. Every lesson with Video Lectures includes six to ten Try It questions. Every lesson with Video Lectures has a Problem Set, so students can work offline to practice what they have learned. One Problem Set is provided for each lesson as a PDF. Each lesson also includes recommended assignments for each of the three recommended textbooks. Most lessons include a quiz, which is a computer-graded assessment. Review Lessons cover the material presented in a unit or over a semester. Calculator skills are also reviewed in Review Lessons to help prepare students to use them on tests and exams. A unit test is an assessment of the material covered in a given unit. Each test is modeled after the AP Exam. Students complete certain portions of these tests using graphing calculators but are prohibited from using them on other parts of these tests. Each test includes a computer-graded, multiple-choice section and a free-response section that is teacher graded using a detailed rubric. AP Calculus BC, page 3

4 Semester Exam A comprehensive Semester Exam is administered at the end of the semester. Students are required to use graphing calculators to solve problems, experiment, interpret results, support their conclusions, and verify hand-written work. The semester exam is modeled after the AP Exam, so students complete certain parts of the exam using graphing calculators but are prohibited from using them on other parts of the exam. AP Calculus BC, page 4

5 Course Syllabus SEMESTER ONE Unit 1: The Basics (17 Days) [C2] Students prepare to study calculus by reviewing some basic pre-calculus concepts from algebra and trigonometry. They learn what calculus is, why it was invented, and what it is used for. Pre-Calculus Review Introduction to Calculus o Video Lectures: The Study of Change, History of Calculus, Calculus Today, The Study of Calculus o Discussion: Introduction Using a Graphing Calculator o Graphing Calculator: Finding Zeros of Functions Combining Functions o Video Lectures: Sums, Differences, Products, Quotients Composite and Inverse Functions o Video Lectures: Composite Functions, Composite Domains, Inverse Functions, Domains of Inverse Functions o Graphing Calculator: Exploring Functions Graphicy and Numericy Graphical Symmetry o Video Lectures: Symmetry, Even and Odd Functions, Inverse Is Reflection of Original Patterns in Graphs o Video Lectures: Function Families, Rules, Absolute Value o Graphing Calculator: Shifting and Exploring Function Graphs Unit Review Unit Test By the end of this unit, students will be able to: C2 - The course teaches topics associated with Functions, Graph, and Limits; Derivatives; Integrals; and Polynomial Approximations and series as delineated in the Calculus Topic Outline in the AP Calculus Course Description. Write a simple, general definition of calculus. Identify at least three situations where modeling with calculus is appropriate. Explain why calculus was first invented, and name at least one mathematician who was involved in developing calculus. Identify whether a given relationship represents a function and whether that function is oneto-one (the relationship may be given verby, graphicy, or algebraicy). Write functions to represent situations where there is a relationship between two variables. Determine domains (graphicy and algebraicy) for given functions. Determine ranges (graphicy and algebraicy) for given functions. Find asymptotes of rational functions. Determine sums, differences, products, and quotients of functions that are given algebraicy. Determine domains for sums, differences, products, and quotients of functions. Given two functions, identify the graph that is formed by combining two functions by adding, subtracting, multiplying, or dividing the functions. Determine composite and inverse functions that are given algebraicy (including domain restrictions). Graph functions using a calculator. Solve equations numericy using a calculator, and analyticy using algebra. Write functions to represent various geometric and real world situations. AP Calculus BC, page 5

6 Recognize symmetry in a variety of graphs and pictures, and identify the type of symmetry. Identify even, odd, and inverse functions, both from their graphs and from their equations. Write a function from a verbal description or a diagram of a situation involving symmetry. Recognize a new function (algebraicy and graphicy) as an altered form of a familiar function. Use the rules for shifting and distorting to quickly sketch the graph of one function from the graph of another. Match a function with a parameter to a given family of functions. Graph a family of functions when given a function that includes a parameter. Write an equation with parameters to represent a given family of functions. Unit 2: Limits and Continuity (16 Days) [C3] This unit addresses Topic I: Functions, Graphs, and Limits of the College Board s Calculus BC topic outline. Students learn two important concepts that underlie of calculus: limits and continuity. Limits help students understand differentiation (the slope of a curve) and integration (the area inside a curved shape). Continuity is an important property of functions. Introduction o Video Lectures: Limits, Unequal Limits, Ways to Find Limits Finding Limits Analyticy o Video Lectures: Identities, Factoring and Rationalizing, Trigonometric Asymptotes as Limits o Video Lectures: Asymptotes Revisited, Horizontal Asymptotes, Vertical Asymptotes, Drawing a Graph with Asymptote Information, Relative Magnitudes for Limits o Video Lectures: Comparing Algebraic Functions, Comparing Exponential Functions, Comparing Exponential Functions to Algebraic Polynomials and Power Functions o Discussion: Analyzing Examples of Infinities When Limits Do and Don t Exist o Video Lectures: Vertical Asymptotes, Left-and-Right Hand Limits Don t Match, Oscillating Limits Continuity o Video Lectures: What Is Continuity?, Discontinuity Types: Jump, Discontinuity Types: Infinite, Discontinuity Types: Removable, All Together Intermediate and Extreme Value Theorems o Video Lectures: The Intermediate Value Theorem, The Extreme Value Theorem o Discussion: Limits and the Predator/Prey Model Unit Review Unit Test By the end of this unit, students will be able to: C3 - The course provides students with the opportunity to work with functions represented in a variety of ways graphicy, numericy, analyticy, and verby and emphasizes the connections among these representations. Identify when a limit exists. Estimate a limit (approaching from the left and/or right) from a table of data. Estimate a limit (approaching from the left and/or right) from a graph. Estimate a limit (approaching from the left and/or right) numericy, using a calculator (including one-sided limits). Determine a limit using algebraic methods. Determine limits for more complicated expressions, where algebraic manipulation is required, for example, rationalizing, factoring, expanding, finding common denominators, or some combination of four. AP Calculus BC, page 6

7 Apply identities for limits. Calculate limits involving trigonometric functions, using algebraic manipulation when required. Estimate limits at infinity, using numerical or graphical techniques. Solve limits involving infinity, using algebraic manipulation. Use limits to find and describe asymptotes. Reconstruct the graph of a function when given limits that describe the function Determine limits by comparing to known functions. Compare relative magnitudes of functions, including algebraic and exponential functions. Solve problems by comparing relative rates of growth. Find when a limit does not exist and identify why the limit does not exist (for example, RHL/LHL differ, vertical asymptote, or oscillations). Analyze situations that can be described in terms of limits of functions State the definition of continuity at a point. Determine if a function is continuous at a certain point, using the limit definition. Determine the type of discontinuity that may exist, graphicy and analyticy. Describe discontinuities in terms of limits. Solve for parameters in equations that represent continuous functions. State (informy) the Intermediate Value Theorem and the Extreme Value Theorem, including their hypotheses. State why each hypothesis is needed in the Intermediate Value Theorem and the Extreme Value Theorem. Use the Intermediate Value Theorem and the Extreme Value Theorem to predict some of the behavior of a continuous function over a closed interval. Unit 3: The Derivative (25 Days) [C5] This unit addresses Topic II: Derivatives of the College Board s Calculus BC topic outline. Students learn how to calculate a derivative, the slope of a curve at a specific point. They learn techniques for finding derivatives of algebraic functions (such as y = x 2 ) and trigonometric functions (such as y = sin x). Students also interpret the derivative as a rate of change and move fluidly between multiple representations including graphs, tables, and equations. Introduction: Slope and Change o Video Lectures: Slope, Instantaneous Rate of Change Derivative at a Point o Video Lectures: Slope of Curve, Differentiable, Calculating the Derivative o Graphing Calculator: Computing the Derivative of a Function Numericy The Derivative o Video Lectures: Finding and Using the Derivative Function, Units, Slope, Notation The Power Rule o Video Lectures: The Derivative as a Function, The Power Rule, Trigonometric Derivatives o Discussion: Discovering Rules for Derivatives Sums, Differences, Products, and Quotients o Video Lectures: Sums, Products, Quotients, Applying the Quotient Rule Graphs of Functions and Derivatives o Video Lectures: Zeros, Extreme Values, Steepness, Graphical Differentiation, Non Differentiable Continuity and Differentiability o Video Lectures: Review, Discontinuous, Continuous, Differentiable AP Calculus BC, page 7

8 Rolle s and Mean Value Theorems o Video Lectures: Rolle s and Mean Value Higher-Order Derivatives o Graphing Calculator: Higher-Order Derivatives Concavity o Video Lectures: The Second Derivative, Inflection Points Chain Rule o Video Lectures: Units, Chain Rule, Applying the Chain Rule, Derivatives of Complicated Functions Implicit Differentiation o Video Lectures: Implicit Equations and Their Derivatives, Derivative of an Ellipse, Derivative of a Circle and a Hyperbola, Tough Analytical Derivatives, Analytical Unit Review Unit Test By the end of this unit, students will be able to: Calculate average rates of change in various situations where one quantity changes in relation to another quantity. Estimate instantaneous rates of change using data and graphs. Define instantaneous rate of change as a limit of an average rate of change. Calculate an instantaneous rate of change using the limit definition of the derivative. C5 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. Use the concept of the limit to explain how the slope of a tangent line is related to the slopes of secant lines. Find derivatives of functions using the definition of the derivative. Estimate the graph of a derivative function from the graph of its original function. Estimate the graph of the original function when given the graph of the derivative. Read three notations for derivatives (differential or fractional notation, prime notation, and dot notation) and state the situations when each form is commonly used. Determine the derivative of basic power functions and polynomials. Determine derivatives of functions defined as a sum of other functions. Determine derivatives of functions defined as a product of other functions. Determine derivatives of functions defined as a quotient of other functions. Determine the derivative of each of the six basic trigonometric functions: sin, cos, tan, csc, sec, cot. Determine derivatives that may require a combination of the sum, product, and quotient rules for functions that are algebraic, trigonometric, or combinations of both. Use the derivative to solve problems where calculating the slope of a function will help you to determine the solution. Predict features about the derivative graph using the graph of a function. Predict features about the graph of the original function using the derivative. Determine when a function is differentiable. Explain the relationship between differentiability and continuity. Determine the derivatives of piecewise functions. State (informy) Rolle's Theorem and the Mean Value Theorem, including their hypotheses. Use Rolle's Theorem and the Mean Value Theorem to relate average rate of change to instantaneous rate of change for a differentiable function over a closed interval. Solve problems that look new on the surface, but that can be analyzed and solved using the Mean Value Theorem and the concept of differentiability. Determine higher-order derivatives of functions. AP Calculus BC, page 8

9 Identify places where a graph is concave up or down. Use the second derivative to find the inflection points in a graph. Use higher-order derivatives to solve applied problems and analyze rates of change. Write the Chain Rule from memory (using dy/dx notation and prime notation). Use the Chain Rule in combination with the other derivative rules to find derivatives of functions. Identify whether an equation is given implicitly or explicitly. Determine the derivative for implicitly defined curves and relationships. Calculate the slope of the tangent line at points on an implicitly defined curve. Solve complicated problems that look new on the surface, but that can be analyzed and solved using implicit differentiation as taught in this lesson. Unit 4: Rates of Change (17 Days) [C4] This unit focuses on Second Derivatives and Applications of Derivatives within Topic II: Derivatives of the College Board s Calculus BC topic outline. Students learn how to use calculus to model and analyze changing aspects of our world. In addition to the AB topics in this unit, BC students analyze polar and vector-valued functions. Introduction o Exploration: Maximums Extrema o Video Lectures: Extrema, First Derivative Test, Sketching with the Second Derivative, Second Derivative Test Optimization o Video Lectures: Minimizing, Maximizing, Sketching with the Second Derivative, Travel Time, Travel Time 2 o Discussion: Applications of Optimization Tangent and Normal Lines o Video Lectures: The Tangent Line to a Curve, Normal Line, Finding Lines o Discussion: Linear Approximations of sin x Tangents to Polar Curves o Video Lectures: Polar Form of the Derivative, Tangents to Polar Curves, Horizontal and Vertical Tangents to Polar Curves Tangent Line Approximation o Video Lectures: Local Linearity, Approximation, Calculator, Rates and Derivatives o Video Lectures: Rates of Change as Derivatives, Economics, Translating o Discussion: Uses of Rates in Real-World Applications Related Rates o Video Lectures: Related Rates are Applications of the Chain Rule, Related Rates Story Problems Technique, Commonly Needed Formulas and Rules Rectilinear Motion o Video Lectures: Rectilinear, Speed & Velocity o Graphing Calculator: Velocity and Acceleration Motion with Vector Functions o Video Lectures: Magnitude and Direction, Decomposing into Components, Velocity and Acceleration Vectors Unit Review Unit Test C4 The course teaches students how to communicate mathematics and explain solutions to problems both verby and in written sentences. AP Calculus BC, page 9

10 By the end of this unit, students will be able to: Know the meanings of relative and absolute extrema. Identify when a relative maximum or minimum will occur in a function (find critical points). Use a first derivative number line test to identify relative extrema (analyze critical points). Use the second derivative test to identify relative extrema (analyze critical points). Use a second derivative number line test to find inflection points. Analyze curves using a combination of the first and second derivative number line tests. Identify the variables in optimization situations. Solve various types of optimization problems (including those dealing with volume, area, time, and distance). Apply the optimization technique to situations that you have not seen, using units of measure that you have not seen. Write the equation of the tangent line to a curve at a point (using implicit differentiation when necessary). Write the equation of the normal line to a curve at a point (using implicit differentiation when necessary). Write the equation of the normal or tangent line, given the curve and a point not on the curve. Identify areas of local linearity (and absences of local linearity) on a graph; explain the concept of local linearity. Use the tangent line approximation to find approximate values of functions. Use local linearity and tangent line approximation to solve problems associated with unique situations. Translate verbal descriptions involving rates of change into statements written in mathematical symbols. Translate mathematical equations involving rates of change into verbal descriptions. Recognize a related-rates problem. Identify the rates of change in a related-rates problem. Write equations (not necessarily functions) that tie together the variables that are related to each other in a related-rates problem. Determine the rates of change, both known and unknown using implicit differentiation with respect to time. Solve related-rates problems. Apply your knowledge to situations that look new on the surface, but that can be analyzed and solved using the techniques taught in this lesson. Explain the distinction between speed and velocity. Calculate speed, velocity, and acceleration functions from position functions (including algebraic functions, trigonometric functions, and combinations of the two). Analyze rectilinear motion situations using position, speed, velocity, and acceleration functions that you have determined. For example, find distances traveled, find maximum and minimum speeds reached, and graph velocities vs. speeds for the whole function. Find the slope of a tangent line to a polar graph Find the points at which a polar graph has horizontal or vertical tangents. Find the velocity and acceleration vectors for a position function given in vector form. Unit 5: The Integral, Part 1 (16 Days) [C3] C3 - The course provides students with the opportunity to work with functions represented in a variety of ways graphicy, numericy, analyticy, and verby and emphasizes the connections among these representations. This unit focuses on Topic III: Integrals in the College Board s Calculus BC topic outline. Students learn numerical approximations to definite integrals, interpretations and properties of definite integrals, the Fundamental Theorem of Calculus, and techniques of anti-differentiation. They learn how to find areas of curved shapes. AP Calculus BC, page 10

11 Introduction o Graphing Calculator: Analyzing Velocity and Distance for a Car Trip Riemann Sums o Video Lectures: Area, Approximating Area, Inscribed and Circumscribed Rectangles, Improving the Estimate, Riemann Sums Area Approximations o Video Lectures: Trapezoid Rule, From a Function with a Formula, From a Function Graph, From Numerical Data, Error The Definite Integral o Video Lectures: Many Intervals, Definite Integral, Evaluating Definite Integrals, Approximating Numericy, Limit of Sums o Graphing Calculator: Taking More Intervals Properties of Integrals o Video Lectures: Signed Area, Properties, Using Rules Graphing Calculator Integration o Graphing Calculator: Using fnint() Applications of Accumulated Change o Video Lectures: Accumulation, Average Value, Velocity Curves, Exercises, Accumulated Change Antiderivatives o Video Lectures: Going Backwards, Antiderivatives, Some Rules, Differential Equations o Going Between Position, Velocity, and Acceleration Composite Functions o Video Lectures: Chain Rule, Differential Form, Substitution, Another Substitution Example, Practice, Guess & Check, Guess & Check II Unit Review Unit Test C3 - The course provides students with the opportunity to work with functions represented in a variety of ways graphicy, numericy, analyticy, and verby and emphasizes the connections among these representations. SEMESTER TWO Unit 1: The Integral, Part 2 (10 Days) [C3] This unit focuses on Topic III: Integrals in the College Board s Calculus BC topic outline. Students learn the Fundamental Theorem of Calculus, and techniques of anti-differentiation. They learn how to find areas of curved shapes. The Fundamental Theorems of Calculus o Video Lectures: Area Functions, The First Fundamental Theorem, The Second Fundamental Theorem, Units, Names Definite Integrals of Composite Functions o Video Lectures: Fundamental Theorems, Definite Integrals, Area, Upper Limits, Strange Substitutions, When to Substitute Analyzing Functions and Integrals [C3] o Video Lectures: Leibniz s Rule, Leibniz s Rule II, Area Functions, Analyzing Functions, One More Analyzing Functions Example Unit Review Unit Test By the end of this unit, students will be able to: Use summation notation to describe the sum of a series. AP Calculus BC, page 11

12 Calculate exact areas under curves geometricy for circles, trapezoids, triangles, and rectangles. Approximate area under a curve using midpoint, left endpoint, and right endpoint approximations. Write Riemann sums to represent approximations using summation notation. Identify ways in which Riemann sum approximations can be improved. Use the trapezoid rule to approximate the area under a curve. Use numerical methods to approximate the area under a curve, no matter whether the data is given in a formula, graph, or table. Identify the typical error that's present for of the approximation methods. Calculate definite integrals geometricy for circles, trapezoids, triangles, and rectangles. Identify the area under a curve (and the definite integral) as a limit of a Riemann sum. Describe the difference between the area under a curve and a definite integral. Approximate definite integrals using the same methods used for approximating area under a curve. Relate the algebraic properties of the definite integral to the geometric properties of area. Use the properties of the definite integral to solve problems related to area. Identify the definite integral as an accumulator of values. Calculate the average value of a function for a given domain. Calculate the change in position of an object from its velocity curve, using the definite integral. Calculate the net change in a quantity from the area under a rate of change function. Identify the antiderivative of a function as a family of functions. Using the rules for differentiating basic functions, find antiderivatives of basic functions. Identify and solve simple differential equations. Identify when an antiderivative involves a composite function. Take derivatives of functions, using differential notation. Find antiderivatives for composite functions, using substitution. Find antiderivatives for composite functions, without using substitution. Write both Fundamental Theorems of Calculus from memory. Use the First Fundamental Theorem to find derivatives of functions that are defined as integrals. Use the Second Fundamental Theorem to evaluate definite integrals. Use substitution to change the form of a definite integral. Adjust the limits of integration when using substitution to solve definite integrals. Use substitution to identify equivalent definite integrals. Solve definite integrals involving composite functions, with or without using substitution. Analyze functions defined by definite integrals. Use the chain rule and the First Fundamental Theorem of Calculus to analyze functions defined by definite integrals with functions in the limits (for example, Leibniz's rule). Unit 2: Applications of the Integral (13 Days) This unit focuses on Topic III: Integrals in the College Board s Calculus BC topic outline. Students learn to use integrals and antiderivatives to solve problems. In addition to the AB topics, BC students learn to calculate arc length for a smooth curve. Introduction and Area Between Curves o Video Lectures: Accumulation, Two Curves, Multiple Curves, Cutting Area Horizonty More Areas and Averages o Video Lectures: Area Problems, No Formula?, Working Backwards Volumes of Revolution o Video Lectures: Principles, A Calculus View of Volume, Solids of Revolution o Discussion: Hands-on Solids and Volumes AP Calculus BC, page 12

13 Cross Sections o Video Lectures: Cross Sections, Other Shapes for Cross-Sections, Finding Dimensions of Solids Arc Length o Video Lectures: Determine the Arc Length Formula, Arc Length Example y = f(x), Arc Length Example x=f(y) More Rectilinear Motion o Video Lectures: Total vs. Net, Velocity vs. Speed, Putting It All Together, Other Accumulated Changes Other Applications of the Definite Integral o Video Lectures: Geometry, Surface Area, Applications from Physics, Nifty Application, Connections Unit Review Unit Test By the end of this unit, students will be able to: Use the definite integral to calculate the area between two curves (without a calculator). Calculate the area of regions bounded by multiple curves and/or axis lines. Calculate areas by accumulation along the y-axis. Use the definite integral to find the average value of a function. Use numerical integration to estimate the average value of a function given as a table of data. Given an area and a function, find the correct domain for a definite integral to yield that given area. Use the definite integral to find volumes by accumulating cross-sectional area. Use the definite integral to find the volume of a solid of revolution. Calculate volumes of solids of revolution created by rotating curves about lines that are not the x-axis or the y-axis. Calculate volumes of solids that are created with well-defined bases and cross-sectional shapes. Given the shape of a solid (described with a function or a set of functions), determine the limits of integration needed to create a specific volume. Calculate net and total distances traveled by an object. Calculate average speeds and average velocities. Solve problems that incorporate the concepts of motion (speed, velocity, distance, and acceleration) from both semesters. Calculate net and total changes from rates of change presented numericy, analyticy, or graphicy. Use definite integrals to solve problems in new applications where any quantity accumulates. Solve problems where a changing quantity is accumulated over a specified domain. (For example, calculating the total work when there is variable force acting over a specified distance.) Find the arc length of a smooth curve Unit 3: Inverse and Transcendental Functions (23 Days) This unit focuses on Topic II: Derivatives and Topic III: Integrals in the College Board s Calculus BC topic outline. Students learn to calculate and use derivatives, antiderivatives, and integrals of exponential functions (such as y = 3 x where the input variable is an exponent), logarithmic functions (the inverses of exponential functions), and trigonometric functions (such as y = secant x). In addition to the AB topics, BC students learn how to use L Hôpital s Rule and the methods of partial fractions and integration by parts. Also, students learn how to find improper integrals, and derivatives and integrals of parametric functions. AP Calculus BC, page 13

14 Introduction and Derivatives of Inverses o Video Lectures: Inverse Functions, Derivatives of Inverse Functions, The Graphical View, Inverse Trig Functions Inverse Trigonometric Functions o Video Lectures: Domain Restrictions, Derivatives of Arctan and Arccos, Complicated Examples, Using Derivatives Logarithmic and Exponential Review o Video Lectures: Exponential Growth and Decay Functions, Logarithms, Slope, Applications o Discussion: Chenges with Logarithms o Graphing Calculator: Derivatives of Exponential Functions Transcendentals and 1/x o Graphing Calculator: Explore transcendentals and 1/x Derivatives of Logarithms and Exponentials o Video Lectures: Definition, Laws, Logarithmic Differentiation, Exponential Function, Other Bases L Hôpital s Rule o Video Lectures: Indeterminate Quotients and L Hospital s Rule, Indeterminate Products, Indeterminate Differences, Indeterminate Powers Analysis of Transcendental Curves o Video Lectures: Curve Analysis, Tangent and Normal Lines, Optimization, Rates of Change, Related Rates Integrating Transcendental Functions o Video Lectures: Recap Rules, Practice, Strategies, Applications Partial Fractions o Video Lectures: Partial Fractions I, Partial Fractions II Integration by Parts o Video Lectures: Formula and Over Approach, Repeated Use of Integration by Parts, Utilizing Constant Multiples of Original Integral, Definite Integrals with Integration by Parts Improper Integrals o Video Lectures: Improper Integrals with Infinite Limits of Integration, Improper Integrals with Infinite Discontinuities, Volume of an Infinite Solid Applications of Transcendental Integrals o Video Lectures: Area and Averages, Volume, Motion, Accumulations Derivatives of Parametric Functions o Video Lectures: Sketching Parametric Curves, Differentiating a Parametric Curve, Finding the Slope of a Tangent Line to a Parametric Curve, Finding Horizontal and Vertical Tangents to a Parametric Curve Integrating Parametric and Polar Functions o Video Lectures: Length of Parametric and Polar Curves, Area in Polar Coordinates, Surface Area with a Parametric Curve Unit Review Unit Test By the end of this unit, students will be able to: Find an inverse function from a given algebraic or trigonometric function. Find the derivative of an inverse algebraic or trigonometric function, using implicit differentiation. Exploit the graphical symmetry of inverse functions to analyze functions. Identify the domain restrictions of the inverse trigonometric functions. Memorize the derivatives of the inverse trigonometric functions and practice using them. Find derivatives for combination and composite functions involving inverse trigonometric functions. Use inverse trigonometric functions to model situations. AP Calculus BC, page 14

15 Use the derivatives of inverse trigonometric functions to solve problems. Identify whether a function is algebraic, exponential, or logarithmic (these functions may be given as a graph, a formula, or a table of numbers). Use the laws of exponents and logarithms to manipulate expressions involving exponential and logarithmic functions. Solve problems, using the fact that logarithmic functions and exponential functions are inverses of each other. Write equations that model simple exponential growth and decay situations. Find the derivative of a logarithmic function (with any base). Find the derivative of an exponential function (with any base). Combine these rules (for finding derivatives of logarithmic and exponential functions) with the product, quotient, and Chain Rule to find the derivatives of complicated functions involving logs or exponential functions. Use logarithmic differentiation to find the derivative of a complicated product or quotient. Use the derivative to analyze curves for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Use the derivative to optimize situations for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Solve problems about rates of change (including rectilinear motion) for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Solve related rates problems for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Find antiderivatives involving transcendental functions. Use substitution (if necessary) to find more complicated antiderivatives and definite integrals involving transcendental functions. Solve problems related to area for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Solve problems related to average values for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Solve problems related to volume for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Solve problems related to motion for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Use the definite integral to accumulate various quantities for functions covered in this class (algebraic, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these). Describe the indeterminate form for a limit. Use a table or graph to estimate or verify a limit that has an indeterminate form. Use L Hopital s Rule to evaluate a limit. Decompose a rational expression into partial fractions. Use partial fractions to integrate rational functions. Use integration by parts to find an integral. Evaluate improper integrals. Determine whether or not an improper integral converges or diverges. Sketch a parametric curve. Find the slope of a tangent line to a parametric curve. Find the first and second derivatives of a parametric function in terms of the parameter. Find the length of a curve defined in parametric form Find area under a parametric curve Find the area enclosed by a polar curve or the intersection of polar curves AP Calculus BC, page 15

16 Unit 4: Separable Differential Equations and Slope Fields (11 Days) [C4] This unit focuses on Topic II: Derivatives of the College Board s Calculus BC topic outline, specificy, on Equations Involving Derivatives. Students investigate differential equations and solve the equations using a technique ced separating the variables. In addition to the topics covered in AB, BC students also learn to use Euler s method to estimate the solution of differential equations and use logistic equations to model growth. Slope Fields o Video Lectures: What is a Differential Equation?, Slope Fields, Conic Sections, Solving Some Simple Differential Equations, Separating Isn t Always the Answer Differential Equations as Models o Video Lectures: A Field Guide to Differential Equations, English to Math, Separating the Variables, Solving Separable Differential Equations Euler s Method o Video Lectures: Over Approach, Approximating with Euler s Method, Automating the Process Exponential Growth and Decay o Video Lectures: A Family of Exponential Functions, Modeling Exponential Growth, Modeling Exponential Decay, Modified Growth and Decay Logistic Growth o Video Lectures: The Logistic Growth Equation, Modeling Logistic Growth More Applications of Differential Equations [C4] o Video Lectures: Law of Cooling, Fing Bodies, Mixing Problems, Logistic Growth, Connections Unit Review Unit Test By the end of this unit, students will be able to: Identify the order of a differential equation. Identify slope fields associated with given differential equations. Identify differential equations associated with given slope fields. Separate the variables in first-order differential equations. Solve first-order separable differential equations. Translate differential equations from words into math. Translate differential equations from math into words. Solve differential equations given verby. Solve the differential equation dy/dt = ky. Model situations using the solution to dy/dt = ky. Solve separable differential equations that are similar in form to dy/dt = ky. Set up differential equations to model situations. Solve separable differential equations that model situations. Use Euler s Method to approximate the solution to a differential equation. Solve logistic differential equations. Solve logistic growth problems involving populations. Unit 5: Sequences and Series (13 Days) C4 The course teaches students how to communicate mathematics and explain solutions to problems both verby and in written sentences. This unit focuses on Topic IV: Polynomial Approximations and Series of the College Board s Calculus BC topic outline, specificy, on Series of Constants and Taylor Series. AP Calculus BC, page 16

17 Sequences o Series o Video Lectures: Sequences as Functions, Limit Laws and Squeeze Theorem, Bounded Monotonic Sequences Video Lectures: Series and Sigma Notation, Partial Sums and Convergence, Telescoping Series, Geometric Series and Formula Convergence Tests o Video Lectures: Integral Test, P-Series, Alternating Series Test More Convergence Tests o Video Lectures: Direct Comparison Test, Limit Comparison Test Radius of Convergence o Video Lectures: Absolute Convergence, Ratio Test, Test for Divergence, Interval of Convergence Functions Defined by Power Series o Video Lectures: Building a Library of Functions, Differentiating to Obtain Series Representations, Integrating to Obtain Series Representations, Taylor and Maclaurin Series o Video Lectures: Taylor Polynomials, Taylor Series, Maclaurin Series Taylor s Theorem and Lagrange Error o Video Lectures: Error with Series, Taylor s Theorem, Lagrange Form Unit Review Unit Test By the end of this unit, students will be able to: List the terms in a sequence that is defined explicitly or recursively Write an explicit or recursive rule for a sequence Determine whether a sequence converges or diverges, and if it converges, find its limit Draw a graph of a sequence Use term-by-term differentiation or integration to determine whether a series converges Solve problems involving geometric series Use properties of series to solve problems. C4 The course teaches students how to communicate mathematics and explain solutions to problems both verby and in written sentences. Use the Integral Test, p-series Test, Comparison Test, or Limit Comparison Test to determine whether a series converges or diverges. Develop and apply strategies for testing a series for convergence or divergence. Use the Alternating Series Test, Ratio Test, or Root Test to determine whether a series converges or diverges. Determine whether a series is conditiony or absolutely convergent. Develop and apply strategies for testing a series for convergence or divergence. Determine the center of a power series. Determine the radius or interval of convergence for a power series. Determine endpoint convergence for a power series. Use power series to represent functions Differentiate or integrate power series to create other power series Derive power series using known power series and series operations Write the Maclaurin series for a function Construct a Taylor polynomial approximation (at x = 0) for a function. Use Taylor s Theorem to estimate the magnitude of the error for a given polynomial approximation. AP Calculus BC, page 17

18 Unit 6: AP Exam Review and Final Exam (6 Days) Students review what they have learned and become more familiar with AP-type questions in preparation for the AP Exam. Students are also provided with access to previously released AP Exams for practice. Exam Strategies o Video Lectures: T-Minus, One Day, Calculators, Multiple Choice, Free Response, Do s and Don ts Review of Topics Practice Exams o Video Lectures: How an AP Exam Score is Calculated, Rubrics, Strategies, Guesses About What Will Be on the Exam Final Exam Unit 7: Calculus Project (15 Days) If there is sufficient time after the AP Exam, teachers may assign a special project. Project Days: Projects provide an opportunity for students to apply calculus tools and concepts to real-world problems. AP Calculus BC, page 18

19 Semester One Unit 1: The Basics Topic Stewart Finney Larson Pre-Calculus Review Practice: Diagnostic Tests pp. xxiv xxv, #1 10 ; p. xxvi, #1 5 ; p. xxvii, #1 7 ; p. xxviii, #1 9 Practice: pp , #1 43 odd, odd Practice: Review Exercises pp , #1 49 odd; Problem Solving pp , #1 15 odd Introduction to Calculus Read: pp. 1 8 Practice: p. 8, #1 9 Function Basics Read: pp Practice: pp.19 22, #1 13, 23, odd, 63 Combining Functions Read: pp Practice: p. 43, #29 30 Read: Calculus at Work on pp.181, 319, 376, 430, 529 Read: pp , Examples 1 3 Practice: p. 19, #1 19 odd, odd Practice: p. 21, #71; p. 28, #47 Read: pp Practice: p. 47, #1 11 Read: pp Practice: pp , #1 8, odd Read: pp Practice: pp , #9 12, 97; p. 38, #45 Composite and Inverse Functions Read: pp ; pp Practice: p. 43, #31 51 odd; p. 390, #1 31 odd Read: pp , Examples 7 8; pp , Examples 1 2 Practice: p. 20, #51 53 ; p. 44, # 1 23 odd Read: p. 25; pp Practice: p. 28, #59 65 ; p. 349, #1 35 odd Graphical Symmetry Read: pp Practice: pp , #69 79 odd Patterns in Graphs Read: pp Practice: p. 42, #1 23 odd Read: pp , Example 4 Practice: p. 19, #21 30 Read: p. 17, Example 7 Practice: p. 20, #49 50 Read: pp. 2 6; p. 26 Practice: p. 8, #29 57 odd; p. 29, #69 75 Read: p. 23 Practice: p. 28, #49 57 Unit 2: Limits and Continuity Topic Stewart Finney Larson AP Calculus BC, page 19

20 Introduction Read: pp Practice: pp , #1 12 Read: pp , Examples 1 2 Practice: p. 66, #1 4 Read: pp Practice: pp , #2 22 even Finding Limits Analyticy Read: pp Practice: pp , #1 9, odd, 47; Chenge: 58 Read: pp , Examples 3 5 Practice: pp , #5 28 ; Chenge: Read: pp Practice: pp , #1 37 odd, even, odd Asymptotes as Limits Read: pp ; pp Read: pp , Examples 1 5 Read: pp ; pp Practice: p. 61, #29 37 ; pp , #1 6, 7 29 odd, odd Practice: p. 76, #1 7 odd, odd Practice: pp , #1 12, odd, even; Chenge: 69; p. 205, #1 12 Relative Magnitudes for Limits Practice: pp , #10, 12, 26, 34, 36 Read: pp , Examples 6 8 Practice: p. 76, #35 40, odd Read: p. 201 Practice: p. 205, #13 18 When Limits Do and Don t Exist Read: pp Practice: pp , #13 26 ; Chenge: 43 Read: pp , Examples 6 8 Practice: pp , #29 37 odd, ; Chenge: 58 Read: pp Practice: pp , #23 32 ; Chenge: 33 Continuity Read: pp Practice: pp. 90, #1 9 odd, odd Read: pp Practice: pp , #1 16, odd Read: pp Practice: pp , #1 6, 7 13 odd, odd, odd, 52; Chenge: 98 Intermediate and Extreme Value Theorems Read: pp ; pp Practice: p. 92, #51 58 ; pp , #1 10, odd Read: p. 83; pp Practice: p. 85, #45, 46, 51; pp , #1 10 Read: pp ; p. 164 Practice: pp , #83 94 ; p. 164, #a, b AP Calculus BC, page 20

21 Unit 3: The Derivative Topic Stewart Finney Larson Introduction: Slope and Change Read: pp Practice: pp , #1 15 odd, even, 42, 44 Read: pp Practice: p. 92, #1 6, 8; Chenge: 33 Read: pp Practice: p. 103, #1 4 Derivative at a Point Read: pp Practice: pp , #17 31 odd, 47, 51 The Derivative Read: pp Practice: pp , #1 21 odd, odd, even Power Rule Read: pp , ; pp Practice: pp , #1 5 odd, 9 13 odd, 21, odd, 66; Chenge: 76-77; pp. 146, # 1 2, 26 Read: pp Practice: pp , #7 15 odd, 19, 25, 27 Read: pp Practice: pp , #1 11 odd, 21, 24, 29 Read: pp , ; pp ; pp Practice: pp , #1 11 odd, 25, 30, 32; Chenge: 49; p. 146, #1, 3; p. 162, #31 34 Read: pp Practice: p. 104, #5 10 Read: pp Practice: pp , #11 21 odd, 27, 37, 57; Chenge: 64 Read: pp Practice: pp , #1 2, 4 30 even, 31, 38, odd, 55, 59, 63; Chenge: Sum, Differences, Products, Quotients Read: pp ; pp Read: pp ; pp Read: pp ; pp Practice: pp , #2, 8, 12, 18, 22, 26, 36, 50, 68; Chenge: 80; pp , #3 15 odd, 26, 28, 31, 34 Practice: pp , #13 23 odd, 27, 31, 38, 44; Chenge: 50; pp. 146, #5 9 odd, 27 Practice: p. 115, #40 54 even; p. 126, #2 18 even, odd, even Graphs of Functions and Derivatives Read: pp Practice: pp , #2 14 even Read: pp Practice: p. 105, #13 16, 22, 24, Practice: pp , #39 42, Continuity and Differentiability Read: pp Practice: pp , #35 40, Read: pp Practice: p. 114, #1 16, 35; p. 147, #37 Read: pp Practice: p. 106, #89 98, Rolle s and Mean Value Read: pp Read: pp Read: pp AP Calculus BC, page 21

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