AP Calculus (ab) Syllabus
Stephens
HHS
2014/15
Thank you for selecting AP Calculus as one of your courses for next year.
You are holding the course description and syllabus for the 2014-2015 class.
Should you have had summer homework? NO!
Do we hit the ground running? YES!
AP Calculus builds on our study of functions. We look at the manner in which they change
slope and shape, and we examine the ways in which they sketch out area. We emphasize the
interplay between algebraic, numerical, and graphical interpretations and the connections
between these different ways of ‘seeing’ mathematics. Our class is based on the guidelines
created by the college board and presented in its course description booklet. I’ll attach a copy to
our class web page.
Past honors work is not a pre-requisite, but students must be prepared to do a great deal of work
– mental and written – in and out of the classroom.
Calculators:
We rely heavily on calculators to explore hypotheses and estimate values. Think of the
calculator as a mini lab space for our explorations. Thanks to a generous Hastings Education
Foundation grant, the school is able to loan you a computer-algebraic-system (CAS) calculator to
everyone in class. That said, we also need to internalize the connections and processes that we
study, so we also need to be comfortable working without one. We use the calculator on some –
but not all assessments, just as we use them for some, but not all discussions.
Getting Help:
Calculus is a difficult subject and students should expect to need help now and then throughout
the year. Form study groups early and use them often. They are a great way to establish the
basics for each topic and to practice with and test each other.
You can always talk with me after school (until about 3:15). Unfortunately, my chairmanship
duties occasionally make me hard. Don’t panic. Use my email addresses to ask questions:
stephensg@hohschools.org and stephensg@verizon.net please, please treat these carefully. I
use the school one during the day and the home email at night. As a last resort, just pick up the
phone and call me: 478 8667. I can sometimes set you straight in a minute or two, if not, we can
arrange a good time to meet the next day.
Grades
This is not really a class about daily homework. If I give you some, it’s because the work
supports something we’re doing. Problem sets are assigned every other week and you work on
them at home. These ask for annotated solutions to free response questions. Quizzes are
generally 20 – 40 points and often quite difficult. Tests are generally bigger and all the points
add up over the quarter. Your grade is simply points received divided by points offered plus a
smidgeon for classwork. Everything is on the portal; just stop in if you have questions.
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AP Calculus: Stephens
2014/15
Resources
Our textbook:
Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kenned. Calculus: Graphical,
Numerical, Algebraic (Media Update). Needham, MA: Pearson Prentice Hall.
Supplementary material (many of our activities are drawn from these):
Paul A. Foerster. Calculus Explorations. Key Curriculum Press: 1998
Ellen Kamischke. A Watched Cup Never Cools. Key Curriculum Press: 1999
Underwood Dudley, ed. Readings for Calculus. Volume 5, Resources for Calculus. MAA:
1993.
Review and Practice:
David Lederman. Multiple-Choice & Free-Response Questions in Preparation for the AP
Calculus (AB) Examination. 8th ed. Brooklyn: DNS Marketing Systems.
Released FRQs collections 1979-88, 89-97, and the Free Response Questions and scoring guides
from 1998 – present.
Multiple Choice practice problems culled from the acorn booklets of the last several years.
Online: Use the power of the internet!
We’ll use a web-based graphing program, Desmos (www.desmos.com) to fuss with graphs, as
well as GeoGebra, an open source dynamic geometry program that runs installed and streamed.
On the school network, we’ll use a different geometry program, Geometer’s Sketchpad.
I’ll also share things with you through our schoolwires class page and through our google
accounts.
Khan academy has pretty good videos, and there are a few websites I really like, such as Paul’s
Online Calculus from Paul Dawkins and SOS Math. I’ll send you online now and then to see
different approaches and to explore.
Did I leave anything out? Questions? Stop by and ask or shoot me an email.
The pages that follow detail the flow of topics over the year. You’ll want to add in stuff like
breaks and holidays.
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AP Calculus: Stephens
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Pre-Season: We have two days. After we take a peek at the syllabus, I’m going to try to
convince you that math is all about connections rather than procedures. We’ll start with a story
on the first day and look at some experimental data on the second.
Week One: The limit.
Remember limits from precalc?
 We’ll review the concept of limit (or introduce it fresh)
 Explore asymptotes and continuity using a calculator. Develop the concept of the limit
using graphical and numerical approaches.
 Review the Intermediate Value Theorem from PreCalc Honors as a way of reviewing
reasoning and deduction.
Problem Set assigned: Lab Activity #2 from A Watched Cup Never Cools, Ellen Kamischke
(1999: Key Curriculum Press). This lab asks students to explore a limit dynamically in
sketchpad and then find it analytically.
Take-Home Quiz on “Previous Math”
Week Two: Local Linearity
The function ‘machine’ helps us determine the position of a point on the graph of a function.
How does the concept of local linearity help us understand what that point is ‘doing?’
 Use the calculators to zoom in on a specific point on a variety of functions.
 Use the calculators to manage the tiny ∆y’s and ∆x’s.
 Discuss, “what is the difference between where a function is and what it is doing?’
 Formalize the concept of difference quotients: LDQ, RDQ, SDQ.
History Essay/ Class discussion: The role of Newton’s infintesimals.
Quiz on Linearity. Lab Activity #2 due.
Week Three: Working with data.
Back to the experimental data from day 2:
 When is the ball rolling fastest? What happens at the transition ramp to floor? At what
times is the ROC (exactly) 1 foot per second? Is there a new function that describes this
data best?
 Compare the calculator’s estimate of nDeriv as a function to the new function you create
for your data. How are they similar/different?
 Discuss the meaning of the slope or your new slope function for tides.
Problem Set assigned: Practice FRQs related to estimating rates.
Quiz on estimating rates and extrapolating past and future values.
Week Four: Formal definition of the derivative.
What is the difference between determining what a function is doing at a point (last week) and
creating a new function that does this for all points?
 Concept: transition from estimates at a point to a formal function for all points.
 Algebra: practice in manipulating the limit statement to simplify ∆x.
 Re-connect: does the new (derivative) function produce the same results as our ROC
estimates? As our calculator’s nDeriv approximations?
Test: Theorems and Derivatives
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AP Calculus: Stephens
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Week Five: Learn some derivative rules
Students begin to amass a library of derivative rules.
 Applications: Tangents and Normals
 Investigation: Four different teams are assigned a trig function and use the calculator to
estimate slope and create a scatterplot of rates of change. The scatterplot leads
inexorably to a rule for the derivative.
 The groups share and explain their results.
 Newton’s method for finding roots.
Quiz: Developing vs. Memorizing Rules, Applications
Problem Set assigned: FRQs related to Normals and Tangents.
Week Six: Min/Max work
Calculus as an analytic tool: determining extrema.
 Critical points;Relative vs. Global (Absolute vs. Relative) extrema.
 1st derivative test
 derivatives of derivatives.
Quiz: Analyzing extrema
Week Seven: Describing curvature
The role of derivates in analyzing concavity.
 2nd derivative and concavity
 2nd derivative test
 Points of Inflection
 curve sketching: students sketch f(x) from f’(x) and defend their ideas.
Test: 1st and 2nd derivatives.
Problem Set assigned: FRQs related to Extrema and Points of Inflection
Week Eight: Product/Quotient Rules
Students develop routines for managing more complicated functions.
 Product Rule
 Quotient Rule
Problem Set: FRQs related to Optimization
Week Nine: Chain Rule
Exploring dependent, or composed, functions.
 Review of PreCalc composition
 Differentiating composed functions
 Examples from Science and Business to help illustrate how functions and derivatives
illustrate real-world phenomena.
Essay: Who should learn calculus? Students read “Mathematics as a Social Filter,” Davis and
Hersh in Readings for Calculus.
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AP Calculus: Stephens
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Week Ten: Optimization
The calculator suggests solutions . . . calculus nails them down!
 Construct functions to represent problems
 Apply 1st or 2nd derivative tests.
 Emphasize the use of the calculator to i) display functions (making optimal points
obvious) and to ii) sample nearby points.
 Practice creating sound arguments.
Quarter Exam
Problem Set: FRQs related to optimization and theorems.
Week Eleven: A week of theorems
A renewed comparison of the instantaneous to the average
 Intermediate Value Theorem
 Rolle’s Theorem
 Mean Value Theorem (MVT)
Quiz: Theorems
Problem Set (essay): Barbeau’s idea of “good problems.”
Week Twelve: Implicit differentiation
How do we push our understanding to the realm of non-functions?
 Review of conics
 Implicit differentiation
 Classic curves from History
Paper: Detail the history and graph of one of the list of curves. Apply your knowledge of
calculus to questions related to rate.
Week Thirteen: Related Rates
It’s often more meaningful to discuss relationships among rates.
 Review of geometric formulae
 Related rates, including appropriate units.
 Problem solving strategies
Problem Set: FRQs related to Implicit differentiation and Rate
Week Fourteen: Describing a function with derivatives
Where pictures are more descriptive than algebra!
 Slope Fields as a graphical aid to understanding functions.
 Role of an initial condition
 Euler’s Method as a way of re-connecting to linear approximations.
Paper and Pencil simulations
Computer generated fields (program for TI-83, directions for the 89)
Historical Essay: Research and write about one topic from list: Roots, Tangents, Extrema.
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AP Calculus: Stephens
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Week Fifteen: (good-bye, 2007)
Working backwards
 Initial conditions/ Initial value problems
 Tie in to distance/velocity/acceleration.
 Multiple Anti-derivatives.
Problem Set: FRQ from the study of derivatives
Quiz: Timed FRQs
Week Sixteen: Review (hello, 2008)
Help students understand the mix of theory, memorization, and conceptual analysis
 Midterm review and practice
Week Seventeen: Midterm Exam week.
 Review and exam. 2 hours. Open campus.
 The midterm has both short answer and written explanations.
Week Eighteen: Integrals as area
Expand the geometric concept of area.
 Riemann Sums and the summation of area
 RAM approximations.
 Calculator programs and summations, especially …
 The sum( and seq( functions to illustrate the definite integral as a limit of the Riemann
sum.
Car Project: Research your favorite car. How many feet to 65 mph?
Week Nineteen: Antiderivatives
The accumulation of rates as a ‘new’ function
 Fundamental Theorem of Calculus, part I


x
0
f (t )dt : investigate as an “area” function.
 Use the fnInt function of the calculator to describe accumulation.
Short Problem Set: antiderivatives
Week Twenty: Positive vs. Negative areas
Net vs. Total area.
 Discuss the meaning of ‘signed’ area.
 Lab Project: Collect accumulation data in small groups; regress to find a representative
function.
Quiz: the interpretation of area.
Week Twenty-One: Exploring the Definite Integral
Students create their own fundamental rule.
 Fundamental theorem of Calculus, part II
 Formally recognize and memorize Integral rules
Problem Set: FRQs related to the Interpretation of Area
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AP Calculus: Stephens
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Week Twenty-Two: areas contained between two curves
Can we write rules for the distance between two curves?
 Examples from Economics: instantaneous vs. cumulative breakeven.
 Interpreting equations against the y-axis.
 Average Value theorem (MVT for Integrals)
Week Twenty-Three: 1st order Separable Differential Equations
A new twist on our old problems of growth and decay.
 Recovering the equation of a circle.
 Interpreting exponential growth equations.
Problem Set: FRQs related to contained area and differential equations.
Week Twenty-Four: Substitution
Not everything matches the rules we memorize: undoing the chain rule
 Concept of Substitution.
 Contrast substituting back with substituting for the limits of integration as well.
Week Twenty-Five: Integration by parts
Dealing with harder and harder problems.
 Outline of a proof of ∫u dv = uv - ∫v du
 Integration by Parts
 Tabular Integration
Problem Set: FRQs related to managing integration.
Week Twenty-Six: Cross sectional solids
Let’s apply our idea of accumulation to the cross-sections of a solid.
 Lab: Estimate the volume of a curved wooden block.
 Practice problems in cross-sections
 Students build a ‘solid’ out of layers of cardboard.
3rd quarter Exam
Week Twenty-Seven: Disks and Washers
Slices lead to volume.
 Block Period Lab: Find the volume (via cross-sections) of a soda bottle in groups.
 Establish concept of disks contributing to a Riemann sum.
 Formalize idea of disks and washer.
Take-Home problem: Link a piece-wise function to all calculators: Ask for MRAM
approximations of the volume of the resulting solid of revolution.
Problem Set: FRQs related to volume
Week Twenty-Eight: Shells as Volume
Another interpretation of the ‘slice.’
 Practice problems with the x and the y axis. Use the disks model for demo.
 Introduce Shells as an alternative to the algebraic manipulation of functions.
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AP Calculus: Stephens
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Week Twenty-Nine: Transition to the Exam
How is the study of calculus different from exam preparation?
 In class problem-set FRQ from 2006, 2007
 Share/Pair Activity: Students create a FRQ solution guide to four problems in class;
divide into teams and grade sample “student solutions”
 Multiple Choice Jeopardy Review.
Problem Set: Practice FRQ exam (six questions; one week)
Evening Session: Constructing Free Response Answers.
Week Thirty: Review Activities. Last Week of AP Calculus!
So . . . what have we learned this year?
 Answers, Questions, and review
 Problem Set due
Quiz: Multiple Choice with calculator: 40 minute
Quiz: Multiple Choice without calculator + 2 FRQs: 80 minutes
Evening Class: Discussion of last-minute strategies and study guides.
Week Thirty-One: AP Exam Week
 AP Exam
Week Thirty-Two: AP Exam Week
 Textbooks back
 Course Evaluations
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