AP Calculus BC2016 – 2017 Syllabus

Table of ContentsCourse Overview, Requirements, and Expectations 1Grading 2Tests and Quizzes 2Homework 3AP Review 3Course Outline 4Signature Page 8

AP Calculus BC Syllabus

Instructor: Mr. Kasten, Justin D.Email: jkasten2@bfcsmail.comWebsite: bfhskasten2.weebly.com

Course Overview:

This is a college-level calculus course designed to meet the Advanced Placement curricular requirements for Calculus BC (equivalent to one year of college calculus). The major topics of this course are limits, derivatives, integrals, the Fundamental Theorem of Calculus, and series. We will investigate and analyze course topics using equations, graphs, tables, and words, with a particular emphasis on a conceptual understanding of calculus. Applications, in particular to solid geometry and physics, will be studied where appropriate.

Technology Requirement:

You will need to bring a handheld graphing calculator to class every day. Your instructor will use a TI-84, but any AP-approved calculator is acceptable. Your instructor has calculators for check-out, if desired. The list of AP-approved calculators can be found at https://apstudent.collegeboard.org/apcourse/ap-calculus-bc/calculator-policy.

Classroom Expectations:

Every student is an important member of this class. You are expected to actively participate, stay engaged, and discuss ideas. Your instructor will frequently call on students to participate in discussions, expecting that students will make many mistakes as they engage the material. Mistakes are a necessary component of learning - they allow us to identify areas of opportunity, strengthen our mathematical foundations, and maximize our performance.

The classroom is arranged in teams of four so that you may collaborate and share ideas freely. Most classes will include some group work, ranging from a warm-up (often using a past AP exam question) to an exploration. Students will often be asked to present their work to the class.

Everyone is encouraged to form a study group for additional support outside of class. Your instructor will be available before (even before zero period) and after school, as posted, for extra help. Additionally, your instructor is available via email for any questions that can’t be answered in-person. Calculus is a challenging, rigorous course. We can do this together.

Textbook:

Briggs, Cochran, Gillett. Calculus. AP ed. Boston: Pearson, 2014.

Additional Resources:

The class webpage contains PDF files of all notes and handouts. There are also links to online applications, study guides, videos, and AP review resources.

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Quarter Grade Composition:

Component % of GradeTests 40Participation 30Homework/Classwork 20Quizzes 10

Total 100

The overall semester grade will be comprised of the following:

40% from quarter 1 40% from quarter 2 20% from midterm / final exam

Grading Scale:

Benjamin Franklin High School utilizes the following grading scale:

90 - 100% A 80 - 89% B 70 - 79% C 60 - 69% D 0 – 59% F

Tests and Quizzes:

End-of-Year Exam: During the first week of classes, students will be given the end-of-year exam to assess their current level of ability. The end-of-year exam will be given again in May to measure student growth.

Chapter Tests: Each chapter will be followed by a test that covers all material of the chapter. Chapter tests may be retaken within two weeks of receiving your graded test. If you choose to retake a chapter test, you must inform your instructor as soon as possible. You will be given an alternate version of the chapter test, and your retake score will be averaged with your original test score to determine your chapter test grade. IMPORTANT: Scoring lower on a chapter test retake will lower your grade.

Quizzes: Your instructor will give short quizzes (typically covering two or three AP exam problems from previous years) without prior notice. Students may work in small groups but may not use any reference materials (textbook, previous homework, etc.). Students will be successful on quizzes as long as they are attending class, participating, and keeping up with the homework.

Midterm and Final Exam: A midterm exam, worth 20% of your semester grade, will be given December 22nd, and will cover chapters 2 through 6. A final exam, worth 20% of your semester grade, will be given May 24th, and will cover chapters 7 through 11. These dates are subject to change.

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Homework:

Students should expect to have calculus homework to work on every night. AP Calculus BC covers the same level and amount of material that college students cover in two semesters of single variable calculus. A student who fails to submit their homework on time will find it difficult to catch up.

Homework may be scanned and submitted via the homework submission form at www.bfhskasten2.weebly.com. Alternatively, students may submit assignments in class at any time before the deadline. Every homework assignment is due by 11:59 PM on its deadline unless otherwise stated.

AP Review:

Review for the AP exam is ongoing throughout the year. Past multiple choice and free response problems are regularly used for warm-ups and quizzes. Problem sets centered on particular themes and written in AP style are used at least twice in each chapter. Students work through at least two complete practice AP exams with optional weekend practice sessions (dates and times to be determined).

During the two week review period, students will work through practice problems from an AP practice book. As a class, we will analyze sample student responses and cover all nuances of the AP Calculus BC exam.

After students take their final practice AP exam, they will analyze their individual scores to identify areas of opportunity. This will allow students to optimize their studies – focusing on the specific topics and question types that will yield them the largest score increases.

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Course Outline

First Semester: Chapters 2 through 7Second Semester: Chapters 8 through 11

Limits and Continuity (Chapter 2)

2.1 The Idea of Limits 2.2 Definitions of Limits 2.3 Techniques for Computing Limits 2.4 Infinite Limits 2.5 Limits at Infinity 2.6 Continuity 2.7 Precise Definitions of Limits

Sample Activities:

Points of (dis)continuity: Students explore limits at discontinuities in four ways: first, using the table feature on their calculators with decreasing increments; second, using algebraic techniques to simplify the expressions given as formulas; third, using the graph trace feature on their calculators; and fourth, using descriptions of functions written in words to create graphs that match the verbal descriptions. Student work for the activity includes a written summary, using complete sentences, of their findings that compares and contrasts jump, removable, and asymptotic discontinuities. They give an oral presentation describing how the different representations reveal the discontinuities in different ways.

Limit War: This game uses a set of 24 cards with limits expressed analytically. For each round, players find the limits on their card and announce the answer to their group. The player with the highest result (i.e., infinity, finite, values in descending order, negative infinity, and does not exist) wins the cards. Decks are created so that each student receives an equal number of cards with special values in a three-person game. The student with the most cards at the end of the allotted time wins. The game is an introductory week activity with calculators allowed and is repeated as a test review with no calculators. In the cut-throat version, players can steal if their opponent incorrectly evaluates their limit.

Derivatives (Chapter 3)

3.1 Introducing the Derivative 3.2 Working with Derivatives 3.3 Rules of Differentiation 3.4 The Product and Quotient Rules 3.5 Derivatives of Trigonometric Functions 3.6 Derivatives as Rates of Change 3.7 The Chain Rule 3.8 Implicit Differentiation 3.9 Derivatives of Logarithmic and Exponential Functions 3.10 Derivatives of Inverse Trigonometric Functions 3.11 Related Rates

Sample Activities:

Communicating Derivatives: When studying the meaning of derivatives in context, such as position/velocity/acceleration problems described in words, students are asked to interpret the concept

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using the specific numbers - both in writing on homework (using complete sentences) and orally for new questions in class.

Derivative at a Point: Students use a worksheet to explore the derivative function using the limit definition of the derivative at a point. For several points, students compute the difference quotient algebraically using the definition and use a table and/or graph on their calculators to evaluate the limits and interpret their results in terms of the definition to decide fi the derivative does or does not exist at each point. The original function and its derivative values are plotted on two graphs with the same horizontal axes. Students trade papers and validate each other’s answers.

Applications of Derivatives (Chapter 4)

4.1 Maxima and Minima 4.2 What Derivatives Tell Us 4.3 Graphing Functions 4.4 Optimization Problems 4.5 Linear Approximation and Differentials 4.6 Mean Value Theorem 4.7 L’Hôpital’s Rule 4.8 Newton’s Method

Sample Activities:

Search for f : Using the graph of the derivative f ' (x), students determine key features of f (x) (such as increasing/decreasing intervals, local extrema, points of inflection, and concavity intervals) and create a graph of f (x). They exchange f (x) graphs with another student and make a graph of f ' (x) to match their new f (x). The students meet together to check f ' (x) graphs and resolve any differences.

Mean Value Theorem Worksheet: In a worksheet on the Mean Value Theorem, students draw secants and parallel tangent lines on a variety of graphs, considering properties such as continuity and differentiability. More problems are given that ask students about hypotheses and conclusions. In particular, students are asked about the conclusion even when the hypotheses do not hold. Then, students move to finding (if possible) the appropriate value of c described by the MVT, both by hand and with their graphing calculators. Finally, students are asked to explain the conclusions of the MVT in writing for multiple scenarios in context.

Integration (Chapter 5)

5.1 Antiderivatives 5.2 Approximating Areas under Curves 5.3 Definite Integrals 5.4 Fundamental Theorem of Calculus 5.5 Properties of Integrals and Average Value 5.6 Substitution Rule 5.7 Numerical Integration

Sample Activities:

Riemann Sums to Definite Integrals: Students write an expression for an approximation of the area between the horizontal axis and the graph of f (x) for a particular function given as a formula on a specified interval as a left, right, and midpoint Riemann sum using n subdivisions. They then use a Desmos graph with slider to explore sums. The file superimposes rectangular areas on the graph of f (x),

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showing the sum value. The software allows for left, right, and midpoint sums. The slider increases the number of partitions to explore precision. Finally, students write limits of their Riemann sums as n goes to infinity, then identify each as a definite integral, and use the Fundamental Theorem of Calculus to evaluate the integral.

Applications of Integration (Chapter 6)

6.1 Velocity and Net Change 6.2 Regions between Curves 6.3 Volume by Slicing 6.4 Volume by Shells 6.5 Length of Curves 6.6 Physical Applications

Sample Activities:

Honeycomb Volume: Students find the volume of paper party decorations. Students trace the decoration, measure the outline to construct a data table, and use regression to determine the curve(s) of best fit for their object. They write integrals to represent the volume of their objects as solids of revolutions and compute the volumes on their calculators. Students will assess the reasonableness of answers and compare results with other students.

Solids of Revolution: Students will bring a small object from home that is a solid of revolution. Students measure their objects to create a graph whose rotation about the x-axis would produce their object. Then, they use regression and multiple integrals to compute their object’s theoretical volume. Students then use displacement to determine the actual volume and calculate their percentage of error.

Integration Techniques (Chapter 7)

7.1 Basic Approaches 7.2 Integration by Parts 7.3 Partial Fractions 7.4 Improper Integrals 7.5 Trigonometric Substitutions

Sample Activities:

Gaussians: Students graph the Gaussian f ( x )=e−a x2

for three values of a, then compute the area under

each curve ( given ∫−∞

∞

e−ax2

dx=√ πa ). Students complete the square to evaluate∫−∞

∞

e−(ax¿¿2+bx+ c) dx¿,

where a > 0, b, and c are real numbers.

Differential Equations (Chapter 8)

8.1 Basic Ideas 8.2 Slope Fields and Euler’s Method 8.3 Separable Differential Equations 8.4 Exponential Models

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Sample Activities:

Slope Field Card Sort: Sets of cards include 10 differential equations represented symbolically, as a slope field, and by a verbal description. Students match the cards to bring together all three representations.

Sequences and Infinite Series (Chapter 9)

An Overview Sequences Infinite Series The Divergence and Integral Tests The Ratio, Root, and Comparison Tests Alternating Series

Power Series (Chapter 10)

10.1 Approximating Functions with Polynomials 10.2 Properties of Power Series 10.3 Taylor Series 10.4 Working with Taylor Series

Sample Activities:

Modifying Maclaurin: Each student is given a common Maclaurin series, f (x), on a notecard with the first four nonzero terms and general term listed. They are asked to modify the series [for example, f (4 x ) or x2 f (x)], write the new series on the same-colored notecard, and add a new modification for the next group to perform. After four modifications are completed, the modified series is given to a new group with the challenge to decode the modification. The notecards are put in order to check.

Polar, Parametric, and Vector Curves (Chapter 11)

11.1 Parametric Equations 11.2 Calculus with Parametric Equations 11.3 Polar Coordinates 11.4 Calculus in Polar Coordinates 11.5 Vectors in The Plane 11.6 Calculus of Vector-Valued Functions 11.7 Two-Dimensional Motion

Sample Activities:

Polar Art: Students use two to four polar curves to create a design in Desmos, print their graph, and color it like a stained glass window. Students solve for the amount of each color of glass (area) and the amount of copper wire (boundary length) needed to produce their window.

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AP Calculus BC Syllabus, 2016 - 2017

Mr. Kasten

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