AP Calculus AB

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Darlene Casey
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1 AP Calculus AB Course Objectives: To achieve college level calculus knowledge. To receive Advanced Placement credit and be prepared to start Calculus BC (either here or in college). Course Description: This course is designed to prepare students for the AP exam by providing a balanced approach to the teaching and learning of calculus that involves analytical (functional), numerical (table values), graphical, and verbal (written) methods (as outlined in the AP Calculus Course Description). The first semester will focus on the concepts of limits, derivatives, and the applications of derivatives. The second semester will focus on definite and indefinite integrals, their applications, and preparation for the AP exam. The course incorporates a variety of teaching techniques in an effort to limit the amount of lecture. Students are expected (on a daily basis) to work cooperatively to investigate and solve problems, to write about their conclusions, and present their conclusions orally. The use of the graphing calculator is integrated throughout the course and will be used as an investigation tool. Pre-Requisites: Students will have completed two years of algebra, one year of geometry, and a Pre-Calculus course including elementary functions, analytic geometry, and trigonometry. Students must also be willing to do rigorous work at a level equivalent to a first year college course. Calculator Usage: Each student will be provided access to the TI-Inspire graphing calculator as the primary calculator used during class time. [CR3a] Each student may have their own TI- Inspire or alternate calculator for home use. The graphing calculator is used throughout the course to not only solve problems, but also to explore concepts. Students will be given opportunities to use the calculator to experiment, make predictions, support conclusions, and verify and interpret results, including reasonableness of solutions. Calculator usage should facilitate students ability to make connections between graphs of functions and numerical or analytic information. Students should be able to graph functions in an appropriate window, find zeroes of functions, intersections of functions, numerical derivatives, and definite integrals. Textbook: Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus, Graphical, Numerical, Algebraic: AP Edition. Third Edition, Boston: Pearson, Prentice Hall. Homework: Homework is give almost every night. It is essential to success in this class. It is to be completed by the next class (unless otherwise stated). Calculators may be used on any homework assignment. Please do your homework nightly; not doing so will cause you to fall behind and catching up will become a REAL challenge. When doing homework, please make note of any questions that arise, and bring them in for discussion during the next class. Exams: Exams will be multiple choice, free response or combination of the two formats. Multiply choice exams will be graded right or wrong, while free response will be scored for correctness of method (problem solving) and final answer. Explain will always mean justify your answer, which indicates your methodology or reasoning needs to be communicated in written form. Please be complete and concise with your explanation, more points are not awarded for more words. The ability to write an intelligent sentence or paragraph using sound mathematical reasoning is essential for this course. Students will need to be able to describe and/or interrupt what they have done on their calculator, in written form, to receive full credit for some exam questions.

2 Study Groups: It is encouraged that you develop a small study group of classmates as an outside support mechanism. If you miss class, this will give you a venue for determining what you missed and allow you to be prepared for the next class. Plan to meet and study together. Discuss concepts and bounce ideas for solutions or projects off each other. Being able to explain a problem to another student helps reinforce your understanding, as well as helping the other student gain insight. This will also improve your ability to use mathematical reasoning both orally and in the written form. Grade Determination: 60% - Tests & Projects, 40% - Assignments, Labs, Quizzes, & Homework. Academic Dishonesty: Academic dishonesty is considered a serious offense in my class. Students caught cheating will face serious consequences. I encourage collaboration on all assignments but I expect the work you hand in to be your own. Chapter / section AP Calculus AB 1st Semester [16 weeks] Topic Timeline Chapter 1 Pre-Calculus Review 3 weeks * Calculator orientation (procedures and graphing format) 1.1 Lines: slope, equation, parallel and perpendicular 1.2 Functions and their graphs: library of curves, piecewise 1.3 Exponential Functions: Growth and Decay 1.4 Parametric Equations: separating x(t) and y(t),graphing 1.5 Logarithms and other inverse functions 1.6 Trigonometric Functions: review values at (0, π/4, π/3,π/2, π) Chapter 2 Limits and Continuity 3 weeks 2.1 Rates of Change and Limits: Lab - Connecting table values with graphical limits; definition, properties, 1 & 2 sided, average and instantaneous rates of change Activity: Once students understand limits, they are given an oral quiz that is presented to the entire class. Each student is given a separate problem. Students must describe how to find a limit for the given function (which may be presented algebraically, graphically, or numerically) and explain what the limit means in the context of the problem. [CR2f: oral] 2.2 Limits Involving Infinity: end behavior, horizontal asymptotes, x ±, x a Activity: Students are grouped in pairs. One student is given a function and asked to analytically determine its limit. The other student graphs the function on a calculator and determines the limit by inspecting the graph. [CR3b] They then share answers and explain how they arrived at their solutions. For the next problems, students switch so that each takes a turn using the calculator. This is especially useful in visualizing the ways in which various limits do not exist.

3 2.3 Continuity: at a point, closed & open intervals, compositions, Intermediate value theorem Activity: Continuity is discussed, and, in a subsequent lab, students are given a set of functions, some presented as formulas and some as graphs. [CR2d: analytical, graphical] Students discuss the continuity of each function with members of their group. [CR2f: oral] Then as a group, students write a question for another group, asking the other group to create a function satisfying continuities and discontinuities at certain values. They may include certain limits they want to exist and other qualities they want the function to possess. The other group creates this function, presents it to the class, and leads a discussion around their solution. 2.4 Rates of Change and Tangent Lines: slope of a curve, normal, velocity/speed, writing equation of tangent lines Activity: Students revisit rates and change by determining slope of a curve using secant and tangent lines. Student work for the activity includes a written summary, using complete sentences, of their findings that compares and contrasts average rates of change with instantaneous rates of change and the slopes through secant and tangent lines. [CR2f: written] They will then give an oral presentation describing how the different lines represent the different rates of change. [CR2d: graphical, verbal] Calculator functions for taking limits AP question analysis with discussion on multiple methods of solution Chapter 3 Derivatives 5 weeks 3.1 Derivative of a function: definition, notation, relationship of the graphs of y=f(x) and y = f (x), table values Activity: Students are given a lab in which they must calculate derivatives using the definition, tables, and graphs. [CR2d: analytical, numerical, graphical] Real-life examples are also included, and students must clearly interpret the derivative s meaning using the proper units. Students discuss their answers in pairs. 3.2 Differentiability: sharp points, holes, gaps, local linearity, use of calculators to calculate derivatives Activity: Students are given a lab in which they must use their calculators to explore the local linearity of functions to try to determine their differentiability. [CR3c] They then discuss the continuity of the same functions. The lab guides them to the conclusion that differentiability implies continuity; however, continuity does not necessarily imply differentiability. [CR2a]

4 3.3 Rules of differentiation: include higher order, product and quotient rules Activity: The product and quotient rules are proven in a traditional manner after following an activity that asks students to use their calculators to consider a function, h(x), that is approximated by the product of the local linear approximations of two other functions, f and g, in an attempt to discover the formula for the product rule before it s analytic proof. [CR3c] When they determine the product of these approximations and consider its derivative at a point a, they have essentially derived the product rule. Students use linear approximations as a way of comprehending nonlinear functions. 3.4 Velocity and Other Rates of Change 3.5 Derivatives of Trig Functions 3.6 Chain Rule 3.7 Implicit Differentiation: include higher order 3.8 Derivatives of Inverse Trig Functions: domain/range, graphical 3.9 Derivatives of Exponential and Log Functions Calculator functions for derivatives, min, max, zeros AP question analysis with discussion on multiple methods of solution Project on types of derivatives Chapter 4 Applications of Derivatives 5 weeks 4.1 Extreme Values of Functions: absolute & relative max/min 4.2 Mean Value Theorem: increasing/ decreasing functions 4.3 Curve Sketching: 1st & 2nd derivative test, points of inflections, concavity, sketch y from y, sketch y from y, sketch y & y from y 4.4 Modeling and Optimization: all types 4.5 Linearization: differentials, absolute, relative and percentage changes 4.6 Related Rates * Related Rate Project (orally presented to class) Calculator functions for derivatives AP question analysis with discussion on multiple methods of solution Selected group project on Modeling and optimization Total 16 weeks

5 Chapter / section AP Calculus AB 2nd Semester [17 weeks] Topic Timeline Chapter 5 The Definite Integral 5 weeks 5.1 Estimating with finite sums 5.2 Definite Integrals: Riemann Sums, notation, sigma, integrals on a calculator 5.3 Definite Integrals and Anti-derivatives: properties, average value of a function, Mean Value Theorem, connecting Differential and Integral Calculus 5.4 Fundamental theorem of Calculus: 1st & 2nd parts, area connection (graphical) Activity: Students are given a lab of past free-response questions in which they must use the Fundamental Theorem of Calculus. Within these problems, they are often required to calculate a definite integral with their calculators. In addition, they must answer questions about extrema and inflection points of g using calculus if given a function x g ( x) f ( t) dt [CR2e] and a graph of f. a 5.5 Trapezoidal Rule; Rt & Lf hand sums, midpoint sum, error analysis Calculator functions for integrals AP question analysis with discussion on multiple method of solution Project on sample AP questions on integrals Chapter 6 Differential Equations and mathematical Modeling 6.1 Slope Fields and Euler s Method: differential equations 5 weeks Activity: After being introduced to the idea of a slope field, students are given a series of differential equations, asked to draw the slope field, and then describe its solution curve. They then analytically solve the differential equation to reinforce their assertions. [CR2d: connection between graphical and analytical] To emphasize the usefulness of slope fields, students are then given a differential equation they cannot solve and asked about the solution curves. They quickly sketch the slope field to answer probing questions and understand their purpose. 6.2 Anti-differentiation by u-sub: converting to u-values (definite) 6.3 Anti-differentiation by parts: tabular integration, inverse trig functions, log functions Activity: Students are given a lab in which they must calculate definite integrals using the definition, tables, and graphs. [CR2c] Real-life examples are included, and students must clearly interpret the integral s meaning using the proper units. In pairs, students discuss their answers.

6 6.4 Exponential Growth and Decay Calculator functions for Differential equations AP question analysis with discussion on multiple method of solution Chapter 7 Applications of Definite Integrals 4 weeks 7.1 Integral as net change: motion 7.2 Areas in the plane: between two curves, vertically and horizontally set-ups 7.3 Volumes: disk, washer, known cross-sections 8.2 L Hôpital s rule: Indeterminate forms (all types) REVIEW Preparing for the AP Exam: students will review all 3 weeks Total topics and practice the exam format Calculator functions for integrals AP question analysis with discussion on multiple method of solution Group project on application problems 17weeks Topics to cover post AP Exam 7.3 Shell Method 3 days 7.4 Lengths of Curves 2 days 7.5 Work/Fluid Problems 1 week 8.4 Improper Integrals (If time permits)

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only.

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only. Calculus AB Priority Keys Aligned with Nevada Standards MA I MI L S MA represents a Major content area. Any concept labeled MA is something of central importance to the entire class/curriculum; it is a