AP Calculus AB – Course Overview The course develops calculus in an intuitive, conceptual manner and prepares students for the AP calculus exam. Laboratory experiments and hands-on activities are integrated throughout the curriculum as key concepts for learning and conceptual understanding. Students are expected to work collaboratively in study teams; explain; justify and present ideas; and demonstrate persistence when asked to develop difficult concepts for themselves. This calculus course will incorporate technology on a daily basis as a tool for learning. All chapters have labs and other major investigations which will rely on student’s proficiency with the graphing calculator. The course has three major activities during which the students use a CBL or CBR to collect data and analyze rates of change. Key problems also rely on the regression feature of the calculator. Since the AP exam requires students to be proficient in using a graphing calculator, this course assumes that students have access to one in class and at home. Our school uses the College Preparatory Mathematics (CPM) textbook series from 6th grade through AP calculus. This textbook series emphasizes the connection between graphical, numerical, analytical, and verbal representations of the math they are studying. Chapters 1-2: Introduction - 6 weeks Pre-Calculus topics: Functions, holes and asymptotes, limits, continuity, velocity, using rectangles to approximate the area under a curve. Chapters 3-8: Core Calculus AB - 20 weeks Calculus topics: derivatives, integrals, the Fundamental Theorem of Calculus, differential equations, optimization and other applications. Review: 4 weeks Writing

Written explanations and justifications are an integral part of the CPM curriculum. Our AP-Calculus students will have four years of experience with written responses and will be expected to continue to develop and refine this essential skill. All problems listed in the writing section (by chapter) require students to write answers in complete sentences. Students create and keep toolkits to summarize, in their own words, key concepts, definitions, and formulas. These toolkits are primarily comprised of the Math Notes Box sections in each chapter. Bold numbers are the page numbers that contain a Math Note Box. Problems that are underlined are investigations. Investigations are often open-ended and students are encouraged to experiment and explore. Goals of the investigation are clearly stated before students begin. Teachers are responsible for observing, listening and questioning. If key ideas are missing, the teacher is responsible for pointing students in the right direction. Students then continue the investigation after the discussion. This is a time consuming process, but students will remember the results if they go through the investigation. After the investigation is complete, students report their findings in a presentation that requires both written and verbal communication. Students are required to submit an investigative report that includes both analysis and written explanations and justifications for the work submitted. All other problems listed in the writing section (by chapter) require students to explain, justify, describe, and summarize.

Teaching Strategies • Team work • Individual work • Labs • Chapter closure activities and quizzes to verify understanding • Technology

Technology

• TI-83 and TI-84 Graphing calculators - daily • CBL and CBR • Internet activities to show solids and animate concepts • Geometer’s Sketchpad

Student Evaluation

• Summative (Individual Test): 45% • Formative (Learning Targets/Bench Mark/Standards Quiz): 35% • Team Assessments: 10% • Final Exam: 10%

Teacher Resources Textbook: Dietiker, Sallee, Kysh, Hoey. College Preparatory Mathematics Calculus. Second Edition, Sacramento, CA: CPM Educational Program, 2003, 2010. Other Resources

• David Lederman. Multiple-choice & Free Response Questions in Preparation for the AP Calculus(AB) Examination. Eighth edition. New York. D & S Marketing Systems, Inc. Copyright 2003.

• Web site: http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2178.html

o Special Focus Materials o Curriculum Modules o Differential Equations o Related Rates o History of Calculus

• Previously released College Board Exams for AP Calculus.

After the AP Exam

• Study of some BC topics, including Newton’s Method for Approximating Roots, l’Hopital’s Rule, Improper Integrals, Integration by Parts.

• Student projects using calculus topics studied in the course. • Review for the course final exam.

Course Planner Chapter Topics Timeline Chapter 1 – A Beginning Look at Calculus

• Review Topics from previous courses such as piecewise functions,

compositions, inverses, even & odd functions, domain & range, horizontal and vertical asymptotes.

• Develop the concepts of slope and slope functions • Study how particular functions change by examining finite

differences. • Examine both the velocity and distance graph of an object in

motion to find average velocity and acceleration.

Technology Used: CBL with motion detector. Slope Walk. This experiment will challenge students to recreate a given distance/time graph on a graphing calculator by walking toward and away from the motion detector. Students will experience the relationship of their speed to the slope of the graph. Students will then translate the slope of the function into “slope statements” and extend the idea to each of the parent graphs. The Slope Walk will also be used to help students discover relationships between position, velocity, and acceleration. Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 7, 10, 20, 24, 25, 37, 55, 70, 72, 84, 95, 101, 105, 119, 131, 132, 184, 195, 197

Investigation Problem #’s: • 92, 96, 97, 98, 99, 110, 111, 112,

Toolkit Entries Page #’s: • 5, 9, 10, 15, 18, 22, 27, 31, 42, 52, 63

3 weeks, 1 day

Chapter 2 – Rates, Sums, Limits, and Continuity

• Approximate the area under the curve using Riemann sums and

summation notation. • Predict function behavior with limits. • Use limits to define continuity and see how continuity provides

the basis for the Intermediate Value Theorem • Develop a method to approximate the velocity of an object at an

instant. • Discuss the local linearity of well-behaved functions • Analyze proofs of limits of sin(h)/h and (1-cos(h))/h as h goes to

0.

Technology Used: Students will use the sum(seq(Y1, x, a, b, 1) to estimate area under a curve with a TI-84 graphing calculator after first computing the results by hand. The calculator will not only confirm their results but offer an expedited method of finding the area so that the result can be applied to the rest of the problem. Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 7, 8, 20, 21, 36, 37, 38, 81, 105, 107, 125, 128, 138, 143 Investigation Problem #’s:

• 102,103, 112, 113, 132, 135 Toolkit Entries Page #’s:

• 75, 76, 82, 87, 94, 96, 110, 111

2 weeks, 2 days

Chapter Topics Timeline Chapter 3 – Slope and Curve Analysis

• Find slope functions for most Parent Graphs both graphically and

analytically. • Derive and use the formal definition of a derivative as the limit of

the slope of a secant line. • Find derivatives of sine, cosine, and formalize the Power Rule. • Discover what 1st and 2nd indicate about a function’s shape,

including where it is increasing, decreasing and its concavity. • Sketch f’(x) and f’’(x) from f(x). • Connect derivatives and second derivatives with velocity and

acceleration. • Antidifferentiate • Investigate and categorize functions that are not differentiable

everywhere.

Technology Used: Students will use NDeriv or dy/dx functions on their graphing calculators to find the slope of the tangent line. They will use this calculation to help find the equation of a tangent line and confirm their results by graphing the tangent line and original function on the calculator. Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 7, 8, 20, 21, 36, 37, 38, 81, 105, 107, 125, 128, 138, 143 Investigation Problem #’s:

• 7, 52, 53, 54, 55, 56, 70, 96, 99, 111, 112, 144, 145 Toolkit Entries Page #’s:

• 124, 126, 131, 148, 155, 161, 163, 164,

3 weeks

Chapter 4 – The Fundamental Theorem of Calculus

• Set up and evaluate an integral to find the exact area under a curve • Crete area functions to find the area under a curve between a fixed

number and a variable endpoint. • Investigate the properties of definite integrals. • Discover the Fundamental Theorem of Calculus and use it to

evaluate a definite integral. • Calculate the area of a region between curves

Technology Used: Students will use their graphing calculators to evaluate integrals graphically. That way they will connect the shaded area under the curve with the numeric value.

Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 21, 43, 46, 57, 58, 61, 79, 91, 93, 98, 99, 100, 123, 124, 135, 150, 156

Investigation Problem #’s: • 1, 40, 41, 4265, 66, 67, 95, 96, 97, 106, 107, 116

Toolkit Entries Page #’s: • 177, 190, 196, 223

3 weeks

Chapter Topics Timeline Chapter 5 – Optimization and Derivative Tools

• Devise functions to model quantities such as velocity,

acceleration, volume, profit, cost, time, and area. • Distinguish between maximum and minimum values using the

first and second derivatives. • Clarify and categorize the different types of extreme values of a

function. • Develop more derivative techniques: Product Rule, Quotient Rule,

and Chain Rule. • Find the Derivative of sec(x), csc(x), tan(x), and cot(x).

Technology Used: Students will make predictions about the derivatives of given functions, then use their graphing calculators to test their conjectures by drawing the derivative and comparing it to their guess. They can graph their conjecture using a thin line and then graph the slope function for the original using a thicker line. If the thicker line graphs over the thin line, they can assume they have a match. Students can graph the original function in Y1, then use nDerive(Y1, X, x-value) to graph the derivative of Y1 in Y2. They can put their answers to the problems in Y3 to see that it matches the actual derivative.

Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 7, 10, 20, 30, 32, 39, 54, 57, 70, 95, 98, 107, 117, 131, 159 Investigation Problem #’s:

• 1, 2, 12, 13, 48, 59, 60, 78, 110, 111, 128, 129 Toolkit Entries Page #’s:

• 237, 240, 247, 251, 257

2 weeks, 4 days

Chapter 6 – More Derivative Rules

• Revisit exponential functions and study their derivatives. • Explore the mathematical concept “e,” which arises naturally in

many contexts. • Use implicit differentiation as a tool to differentiate relations that

are not solved explicitly for y. • Find the derivatives of all parent graphs and their inverses. • Develop a process to find the mean value of a function, as well as

an average rate of change for a function. • Evaluate improper integrals.

Technology Used: Students use the graphing calculator to generate slopes of tangents at convenient points, and then use a table to find a rule for the slope function.

Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 11, 25, 26, 29, 34, 35, 39, 54, 61, 72, 74, 75, 77, 87, 91, 143, 163, 171

Investigation Problem #’s: • 79, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124

Toolkit Entries Page #’s: • 286, 291, 297, 321, 322, 327, 330

2 weeks, 4 days

Chapter Topics Timeline Chapter 7 – Related Rates and Integration Tools

• Describe the relationship between rates of change for different

scenarios. • Solve for a rate of change given another related rate of change. • Learn the u-substitution method of integration, which will enable

you to easily integrate a wider variety of functions. • Learn how to use integration to solve special equations involving

derivatives, called differential equations. • Graph differential equations and their solutions using slope fields

and Euler’s Method. • Integrate using integration by parts and partial fraction integration.

Technology Used: CBL with temperature probe. Students collect temperature data in the Soda Lab to study and quantify how the rates are changing. Students will then use the data collected to write a differential equation. Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 9, 18, 24, 25, 52, 55, 77, 101, 111, 121, 122, 125, 126, 132, 134, 186, 188, 199, 207, 208

Investigation Problem #’s: • 1, 4, 5, 25, 47, 48, 117, 118, 119, 142, 153, 155

Toolkit Entries Page #’s: • 346, 355, 364, 376

3 weeks

Chapter 8 – Volume

• Slice a solid of revolution to find its volume. • Use cylindrical shells to find volumes. • Slice solids using cross-sections to find volumes. • Determine the length of an arc.

Technology Used: Students will find volumes of solids by hand then confirm results by using the integral function on their graphing calculators.

Writing: Explain, Justify, Describe, and Summarize Problem #’s:

• 41, 52, 59, 62, 68, 79, 88, 100, 109, 140, 143, 144 Investigation Problem #’s:

• 1, 2, 3, 65, 98, 108 Toolkit Entries Page #’s:

• 407, 415, 439

2 weeks 3 days

Review

• AP Calculus AB topics will be completed by the end of March.

Students will then practice taking free response and multiple choice tests by completing and correcting released exams. Review topics will include:

• Motion • Velocity • Acceleration • Optimization • Related rates • Slope fields • Exponential growth • Differential equations • Solids of revolution

We will also take time to discuss complete answers, justification, and rounding to three decimal places.

4 weeks