Calculus: Early Transcendentals Eighth Edition, by Howard Anton, Irl Bivens, and Stephen Davis

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1 AP Calculus AB Textbook: Calculus: Early Transcendentals Eighth Edition, by Howard Anton, Irl Bivens, and Stephen Davis Course Outline Unit 1: Prerequisites for Calculus (2 weeks) In this unit, students are reviewing Pre-Calculus topics that are essential for AP Calculus. The textbook is not used in this unit. Students are given handouts and worksheets to practice problems for the following topics. Solving Inequalities with Quadratics, Polynomials, and Rational Expressions Equations of Lines Graphing Parent Functions Trigonometric Functions Discontinuities of Functions Domain of Functions Chapter 2: Limits and Continuity (2 weeks) 2.1 Limits (An Intuitive Approach) -Evaluating limits graphically. 2.2 Limits (Computational Techniques) -Evaluating limits analytically. 2.3 Limits at Infinity -Evaluating limits at infinity both graphically and analytically. 2.5 Continuity -Using the definition of continuity to determine if a function is continuous. 2.6 Limits and Continuity of Trigonometric Functions -Evaluating the limits of trigonometric functions both graphically and analytically.

2 Chapter 3: The Derivative (3 weeks) 3.2 The Derivative -Using the definition to find the derivative of a function. -Analyzing the relationship between the graph of the function and the graph of the derivative. -Analyzing the points on a graph where a derivative does not exist. -Analyzing average rate of change versus instantaneous rate of change. 3.3 Techniques of Differentiation -Using the power rule to find the derivative of a function. -Using a variety of notations for the derivative. -Discussing how the first derivative is applied to velocity and the second derivative is applied to acceleration. 3.4 Product Rule and Quotient Rule -Using the product rule and the quotient rule to find the derivative of a function. 3.5 Derivatives of Trigonometric Functions -Using the derivatives of the trigonometric functions. 3.6 Chain rule -Using the chain rule to find the derivative of a function. 3.8 Differentials and Linear Approximations -Writing a derivative in differential form. -Using the tangent line to find approximations. Chapter 4: Logarithmic and Exponential Functions (3 weeks) 4.1 Implicit differentiation. -Solving free response questions related to implicit differentiation. 4.2 Derivatives of the Exponential Functions. -Using the derivatives of the exponential functions.

3 4.3 Derivatives of the Logarithmic Functions and the Inverse Trigonometric Functions. -Using the derivatives of the logarithmic functions. -Using properties of logarithms when finding derivatives. 4.4 L Hopital s Rule -Using L Hopital s Rule to evaluate limits. -Using L Hopital s Rule for indeterminate forms 0/0 or /. -Analyzing a function to insure the use of L Hopital s Rule is appropriate. 3.7 Related Rates -Applying the derivative to find rates of change. Chapter 5: The Derivative in Graphing and Applications (2 weeks) 5.1 Analysis of Functions: Increase, Decrease, and Concavity -Analyzing the first derivative to determine where a function is increasing and decreasing. -Analyzing the second derivative to determine where a function is concave up and concave down. -Determining points of inflection. 5.2 Analysis of Functions: Relative Extrema, Critical Points -Analyzing the first derivative to determine relative extrema for the function. -Determining critical points. -Analyzing the graph of the derivative to determine the relative extrema for the function. -Analyzing the graph of the derivative to determine inflection points for the function and to determine concavity for the function. 5.4 Absolute Maxima and Minima -Analyzing the first derivative on a closed interval to determine the absolute extrema for a function. -Analyzing the first derivative on an open interval to determine the absolute extrema for a function. End of first semester

4 Chapter 5: Continued (2 weeks) 5.5 Applied Maximum and Minimum Problems -Using the derivative to solve applied optimization problems. 5.6 Newton s Method -Using Newton s Method to find the zeros of a function. 5.7 Mean Value Theorem -Using the Mean Value Theorem to find values in an interval that satisfy the conclusion of the theorem. -Applying the Mean Value Theorem to velocity problems. 5.8 Motion Along a Line -Using the first derivative to analyze a velocity function. -Using the second derivative to analyze an acceleration function. -Determining where an object is speeding up or slowing down. -Determining the direction an object is moving. -Determining the total distance an object travels over an interval. Chapter 6: Integration (3-4 weeks) 6.2 The Indefinite Integral -Evaluating indefinite integrals. 6.3 Integration by Substitution -Evaluating indefinite integrals using u-substitution. 6.4 Riemann Sums -Approximating the area under the curve by using Riemann Sums. These approximations include using the right endpoint, the left endpoint, and the midpoint. 6.5 The Definite Integral -Finding the area under the curve by evaluating definite integrals geometrically. 6.6 The Fundamental Theorem of Calculus -Evaluating definite integrals by using the Fundamental Theorem of Calculus. -Applying the Fundamental Theorem of Calculus part 2 to various problems. 6.7 Motion Along a Line Using Integration

5 -Finding the position function by integrating a velocity function. -Finding the velocity function by integrating an acceleration function. -Finding the distance an object travels by integrating the absolute value of the velocity function. 6.8 Evaluating the Definite Integral with Substitution -Evaluating definite integrals using u-substitution. Chapter 7: Applications of the Definite Integral (3 weeks) 7.1 Area Between Curves -Evaluating integrals to find the area between two curves. -Finding the area by integrating with respect to x and integrating with respect to y. 7.2 Volumes by Disks and Washers -Finding the volume of a solid when an enclosed region is revolved over the x-axis or the y-axis. -Finding the volume of a solid when an enclosed region is revolved over a horizontal axis or a vertical axis. 7.2 Volumes by other Cross Sections: Semicircles, Squares, Equilateral Triangles -Finding the volume of a solid with known cross sections. These cross sections include: semicircles, squares, and equilateral triangles. 7.6 Average Value Applications -Finding the average value of a function over a closed interval. -Applying the use of average value in the context of real-world problems. 8.7 Numerical Integration: Trapezoidal Rule -Approximating the area under the curve by using the trapezoid rule. Chapter 9: Differential Equations (1 week) 9.1 Solving Differential Equations by Separation of Variables -Solving a separable differential equation by integration. 9.2 Slope Fields -Sketching a slope field for a differential equation. -Analyzing the slope field to determine the behavior of the function.

6 Student Evaluation Both formative and summative assessments are used throughout the course. The course is run in a block schedule where students attend class every other day for 90 minutes. Formative assessments take place during nearly every class session with summative assessments occurring about every 2-3 weeks. The 18-week grade in each semester is calculated such that 85% of the grade is test scores and quizzes, 15% is homework. The semester grade is calculated such that the 18-week grade accounts for 80% and the semester exam accounts for 20%. In the first semester there are five chapter tests and in the second semester there are three chapter tests. The final exam in the second semester is a variation of a released AP Exam. Beginning with Chapter 4, at the completion of a unit students work through a few freeresponse questions from previous AP Exams. The scoring guidelines for these questions are discussed in class so that students understand what is expected when they present their solutions. The free-response questions allow students the opportunity to work with functions described graphically, numerically, analytically, and verbally. At this point in the course, free-response questions are included in each chapter test. In addition, we require students to correctly use mathematical syntax both in written and oral form in explaining their solutions. The ability to correctly speak the language of mathematics is highly valued both in class discussions and on exams. Graphing Calculators All students are expected to have their own graphing calculator. The TI-84 plus is the recommended calculator. However, students also have access to Desmos and other online graphing tools. A graphing calculator is not allowed on exams in the first semester of the course. However, in the second semester students use calculators on chapter exams in order to hone and demonstrate the necessary skills in preparation for the AP exam. Students learn to use calculators to perform certain operations. These include, but are not limited to; graphing a function on a specified window, finding the zeros of a function, finding intersection points between two functions, evaluating a definite integral, and evaluating a derivative at a given value. The graphing calculator is a tool to help students solve problems and to interpret results. Yet, students must be able to communicate the reasoning behind solutions found with the calculator.

7 Preparing for the AP Exam Typically, there are 3-4 weeks to review for the AP Exam. Due to the alternating block schedule, the time is necessary for an adequate review. During the review time, the students work on multiple-choice problems as well as free-response questions. Students are encouraged to work together and discuss their solutions. With all free-response questions the scoring rubric is discussed so that students understand how to present their work and how to justify answers when necessary. The students also take a mock AP Exam in April.

8 Component 2A. The course provides opportunities for students to reason with definitions and theorems. In section 2.5 (continuity), students use the definition of continuity to determine if functions are continuous at x = c. 2x + 3, x 4 Ex. f(x) = { , x > 4} x In section 5.7 (mean value theorem), students must apply the Mean Value Theorem. Ex. A particle moves along the x-axis in such a way that its velocity at any time t > 0 is given by v(t) = 3t 2 4t 4. The particle s position x(t) has a value of 1, when t = 1. For what value(s) of t, 0 < t < 4, is the particle s instantaneous velocity on the closed interval [0,4]? Component 2E. The course provides opportunities for students to build notational fluency. Throughout Chapter 3 students experience and are required to use a variety of notions for derivatives. Ex. f (x), dy, d2 y, d f(x), dx dx 2 dx f (x), y, and y Regular formative assessments, in small groups as well as individually, provide opportunities for students to demonstrate their growth, For example, students display work from homework or challenge problems, on a classroom white board, allowing for immediate feedback concerning their entire problem-solving process, including proper notations. Thus, both their peers and teacher provide and receive insights on progress toward mastery of notational fluency. Students build their notional fluency in many ways throughout the course. For example, using sample free-response questions where students are given the graph of the derivative, students practice notational fluency via a Sage and Scribe strategy. The Sage describes the behaviors of the function, based on the graph of the first or second derivative. The Sage must articulate details that the Scribe documents. Details may include increasing and decreasing intervals, relative extrema, points of inflection, and concavity intervals. A second problem of similar nature is then worked so that the sage and scribe switch roles. At the conclusion of the activity, two S&S pairings collaborate and evaluate the work and notations of the others. Whole group discussions also take place in order to hone the appropriate notational usage and oral communications.

9 Component 2F. The course provides opportunities for students to communicate mathematical ideas in words, both orally and in writing. Throughout the course, formative assessments during class are part of the regular routine and course expectations. Cooperative learning groups (from partners to groups of 3-4) also are a regular feature of all math courses at RHS. Students collaborate in problem-solving activities and share results with others, both in small and whole group settings. In addition, students present solutions daily via warm-ups and closure activities. Summative assessments, as well as regular homework assignments, require students to communicate ideas in writing. Ex. Given, f (x) = 4x 3 9x, find all the critical points and determine the relative extrema. Justify your answer using complete sentences. Throughout the spring semester, students engage in Think, Ink, Pair Share and Dyad strategies to articulate problem-solving processes on free-response questions. For example, in Chapter 7 students work area and volume problems. Prior to actually completing the problems, they first use the assigned strategy within their group. A Dyad requires one person to orally communicate ideas without interruption for 1-2 minutes. The listener then becomes the speaker and orally communicates his/her ideas without interruption for 1-2 minutes. Following the Dyad, both students compare and contrast ideas and develop questions for the teacher or whole group if there are processes upon which they were unsure or disagreed. Thus students regularly practice oral communications using appropriate Calculus terminology, and maximize their productive problemsolving time since areas of weakness are corrected before embarking on paper and pencil work. For example, students would discuss whether a washer or a disk is the proper set up when asked to find a volume. Students use Silent Dialogue strategies to hone writing skills while problemsolving in Calculus. Using multiple choice questions, students are given a question and four possible solutions. Students work in pairs, to find the correct solution. The first student has two minutes to review the problem and set up the first step needed to start solving. Since this is a silent activity, after documenting the set-up, the first student must write in complete sentences, what he/she chose to do, and why. Next, the second student has two minutes to review the problem, the initial set-up, and explanation. And, then either document the next step, with a complete explanation, or correct error(s) evident in initial set-up. This process continues with each side taking no more than two minutes, to completely document each step and present rationale/explanation in complete sentences. At the end of the allotted time, dialogue pairs share their work with a proximity pair and the four students discuss processes and clarity of communications.

10 Component 3A. Students have access to graphing calculators. Students are required by the district to have graphing calculators for all math courses Algebra 2 and above. Thus, students who are unable to purchase their own device may check out a graphing calculator from the school library (50+ devices available as per grant funding). In addition, the math department has a classroom set of calculators (also via grant funding), and most teachers have 4-6 calculators in their classrooms to provide access to technology for all students. Component 3B. Students have opportunities to use calculators to solve problems. Throughout the fall semester students use calculators to explore the behaviors of various functions. Although students do not use calculators on exams in the fall semester, throughout the course they use devices/technology to enhance, deepen, and verify/validate their learning. Throughout the spring semester students use calculators for homework and exams during Chapters 6 and 7 as they master integration. Students also use graphing calculators to continue verifying/validating their learning, as they prepare for the calculator portion of the AP Exam, using previously released AP questions. Component 3C. Students have opportunities to use a graphing calculator to explore and interpret calculus concepts. Students explore the behaviors of functions such as y = sin (lnx) from (0,a) using online graphing utilities in order to investigate the behaviors near zero. Students also will solve equations such as 0 = x cos x, and 0 = (x 3) 2 using graphing calculators and online graphing utilities. In addition, students use graphing calculators to evaluate integrals such as cos x dx. 1 x 4 Although students are not allowed to use the sketches of graphs from devices as evidence to argue behaviors, students explore and master typical behaviors of functions by using graphing calculators regularly in class, and on homework assignments. Thus, enhancing their abilities to communicate and justify their solutions.

11 Student Activity Motion Along a Line Let s(t)= sin (2t) be the position function of a particle moving along a coordinate line, where s is in meters and t is in seconds. a) Find the velocity function and the acceleration function. b) Complete the table showing the position, velocity, and acceleration to two decimal places at times t= 1, 2, 3, 4, and 5. Time Position Velocity Acceleration t =1 t =2 t =3 t =4 t =5 c) At each of the times in part b, determine the direction (left or right) the particle is moving. d) At each of the times in part b, determine whether the particle is speeding up or slowing down. e) Use the appropriate graphs to determine the time intervals (from t =0 to t =5) on which the particle is speeding up and on which it is slowing down. Use the calculate zero function when analyzing the graphs.

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