your peers & teachers.
Debrief Summer Homework and compile a list of needed review items.
JR: What are your goals in this class for this year?
Getting to Know You.
Download the course Syllabus. Sign and return the signature sheet.
List in your homework notebook what you believe you know well and what you need to know better.
Bring two hardbound "theme books"--one for journal and one for homework.
||Get ready for the Expectations Quiz. Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.||Clarify
expectations, both from the class syllabus
and the Student Handbook.
JR: What do you believe will be your biggest challenge this year in AP Calculus BC?
the review assessment.
||Begin Diagnostic Tests (Stewart Pgs. xxiv through xxviii).||Continue.
JR: Hypothesize what you believe will be the most common (mathematical) error students will make this year with their calculator.
|Finish the Diagnostic Tests. Write a summary of what you did well and what concepts need further support.|
||List the concepts or topics on which you believe you need work (whether they were on the review assessment or not). Compile a list at your 4-top of the concepts and processes requiring review.||Achieve
a whole-class consensus for the concepts and processes
Create a list of selected questions to complete to overcome the class' perceived needs.
JR: Which topic from AP Calculus AB would be the most difficult for you if it were to appear on the AP Calculus BC test?
the suggested problems.
the problems as a group.
Suggest more problems.
JR: How proficient are you with vectors? List what you can and cannot do.
|Complete the suggested problems.|
||Debrief the review problems in your table group.||
Review Test 1.1
as if you were actually being tested. Aim for
completion within fifty minutes. An AP approved calculator may be
Debrief the review test.
JR: How are limits and continuity related (both similar and different)?
|Chapter 1 AP AB/BC Review Questions, Pgs. AP1-1, 2 (following page 102).|
||Check to see if
the limit exists then compute the left- and right-hand
The Revenge of Orville Redenbacher activity.
The Derivative Function activity (see second page of Revenge.
JR: Must a function be continuous at a point to be differentiable at the point? Why/why not (include an example). If a function is continuous at a point must it be differentiable at the point? Why/why not (include an example).
#13, 23, 33.
f(x) = sec(x)/(1 + tan2(x)).
Perform f '(x) and simplify.
Chain Rule. Do the Unbroken Chain
JR: Under what circumstances would The Chain Rule be used? Explain in simple language and give an example.
154 #7, 23, 25, 47, 51.
dy/dx for x3 + y3
Differentiation. Review at your 4-top Pgs. 157-161.
JR: What is the meaning of the result to the ET?
161-162 #7, 23, 25.
||At what rate is
the volume of a cube increasing when the length of a side
is 10.0 cm and the rate of increase of a side is 2.5
Change in the Natural and Social Sciences and Related
Follow That Particle activity (§ 2.7).
JR: The sides of a rectangle are the lengths a and b. Over a particular time interval a increases at 5.0 mm/s and b decreases at 5.0 mm/s. Under what conditions is the area of the rectangle decreasing?
|Pg 174 #15; Pgs.
180-181 #9, 13, 15.
concept of "related rates" paying particular attention to
the terms "related" and "rates" as they apply to calculus.
||More work on
Nobody Escapes the Cube activity (§ 2.8).
JR: A moderately sized amount of water will form a sphere in microgravity (as when aboard the ISS) due to surface tension and cohesion). The rate at which the volume changes is proportional to the surface area of the sphere. Prove the rate of change of the radius of the sphere is always constant under these conditions.
|Pg. 175 #29; Pg.
||The following is
a graph of f ', the derivative of some function f.
Where is f increasing?
Where does f have a local minimum?
Where does f have a local maximum?
Where is f concave up?
Assuming that f(0) = −1, sketch a possible graph of f.
remaining Chapter 2 issues and work selected problems
(such as the following ones).
1. Two chimpanzees on bikes are separated by 350 meters—Chimp A is due West of Chimp B. Chimp A starts riding North at a rate of 5.00 m/sec and seven minutes later Chimp B starts riding South at 3.00 m/sec. At what rate is the distance separating the two chimps changing twenty-five minutes after Chimp A starts riding?
|Chapter 2 AP AB/BC Review Questions, Pg. AP2-1 (following page 196).|
||Get ready for the quiz! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people||Quiz on Related
Rates. You may use an AP approved calculator and a
JR: Explain a strategy that will result in a passing score on a Related Rates problem even if you do not know the mathematical relationships within. :-)
|Pg. 182 #31, 33,
||Find a positive number such that the sum of the number and twice its reciprocal is as small as possible.||Discuss
The Shape of a Can.
The Waste-Free Box.
Do the Calculus in England activity.
JR: The textbook outlines a method for solving optimization problems (see Pgs. 250-251). At which step would it be best to check your work using a graphing calculator? Include what you would graph on the calculator.
257 #9, 19, 29.
the procedure for determining the extrema of a given
last year's Test 3.1 (hard copies provided). Work
JR: Pg. 277 #45.
|Chapter 3 AP AB/BC Review Questions, pg. AP3-1 and AP3-2 (following page 282).|
JR: A company estimates that the marginal cost (in dollars per item) of producing x items is 2.07 – 0.002x. If the cost of producing one item is $561.00, compute the cost of producing 100 items.
|Complete last year's Test 3.2. Work alone.|
||Get ready for the quiz! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people|| Quiz:
optimization. Note card and calculator OK.
JR: State Fermat's Theorem in plain language.
277 #47, 48.
||How is The
Chain Rule related to "The substitution Rule?" Give
||Clearing The Hill
activity. Loki's Dilemma activity
JR: Explain how to identify what to use for a "u-substitution" and why. Give an example.
|Pgs. 335-336 #3, 13, 21; Pg. 336 #23, 39, 47, 59.|
||See problem #13
on page 339. Explain why "u-substitution" is not
necessary to evaluate the integral.
exercises: Concept Check (Pg. 337) and True-False Quiz
(Pg. 338). Include stem and correct answer for the
T/F Quiz. Show all work!
Chapter 4 AP AB/BC Review Questions, pg. AP4-1 (following page 342). Write processes and answers in your homework notebook. Show all work!
JR: State the Net Change Theorem and explain its application. Include several examples.
review and prepare for the quiz.
||Work alone on
pg. 318 #17.
quirks of FTC.
||Watch the You
Tube FTC video. Do Pg. 339 #37, 38; Pg. 340
||Get ready for the quiz! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.||Quiz 4.1.
Free response: you may use one 3 x 5 note card and an AP
approved calculator. Expect questions on
||What role did Newton and Leibniz play in the "invention" of calculus? You may use your computer to find sources of information (remember to reference sources of information you use!!!!!).|
||Explain the process that will compute the "volume of a function of revolution" independent of its axes of revolution.||Area
between curves. Practice
With Areas activity and Edible Volumes
JR: A part having a circular cross-section is to be made such that it's radius is given by y = ln(x + 2) + 2 (the curve is rotated about the x-axis). A hole is bored into the part having the radius given by y = x/2 + 1. The height of the quasi-cylinder is 5.0 units. Set up the integral to solve for the part's volume.
349 #1, 3, 5, 11; Pg. 360 #1, 5, 7, 9.
does the "dx" represent in the volume integrals?
Shells. At your 4-top, determine three
different ways to slice a bagel to get pieces that are
circular (in some way). Cut a "bagel" to
An Exotic Bagel activity.
JR: Why does the factor 2π appear with “cylindrical shells” while π appears with “washers?”
367 #5, 9, 15.
work, and power.
5.4). Skim and discuss Pgs. 368-371 in your table
groups. The Weighty
JR: Under what circumstances MUST work be computed with an integral? Give an example. Under what circumstances is an integral not necessary to compute work? Give an example.
|Pgs. 371-372 #1,
up the integral to compute the volume of the solid bounded
by the x-axis and y = x(x -
1)2 over [0, 1] if the region is rotated around
of volumes, problem demonstrations.
Do the following problems in your homework notebook:
372 #19, 21.
||Set up and evaluate the integral to solve for the volume of a sphere of radius r using Cylindrical Shells.||Concept
Check, Pg. 377; Exercises Pg. 378 #3,8, 9, 16.
JR: Examine the general form of Average Value of a Function (Pg. 374). Write a plain language version of what this means.
|Chapter 5 AP AB/BC Review Questions, Pg. AP5-1 (following page 382). Show all work, do all problems.|
||Get ready for the quiz! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.||Quiz: volumes
from rotation. No calculator or note card.
JR: For which concepts in Chapter 5 are you still uncertain?
|Read the first section on Inverse Functions (Pgs. 384-389). Write a plain language version of what "inverse function" means.|
= (x - 4)1/3. Compute
= 2x - 4. Create f-1(x).
Enter both into the Function Editor of your calculator and
examine the TABLE (START = 0; deltaTbl = 1). What
does it mean when a value in the X column appears in the
Given f(1) = 2; f(2) = 3; and, f(3) = 1. What are
JR: Let f(x) = (x - 4)1/3. Compute (f -1) ' (-2).
|Pg. 390 #1, 3, 5, 24.|
and note the rules of exponents and derivatives of
exponential functions in Section 6.2 (Pgs. 391-400).
Read and note the rules of logarithms and derivatives of logarithmic functions in Section 6.3 (Pgs. 404-408).
Prove WHY d/dx (ln(x)) = 1/x.
Irrational, Impossible Relations activity.
Logarithmic Differentiation activity.
JR: Explain, extensively and definitively, why the log of a negative number does not exist. Include a diagram.
|Pg. 408 #5, 7, 11. Show all steps, do NOT use a calculator!!! Pg. 418-419 #7, 9, 27.|
||A pain killer is administered every eight hours with the expectation that half of the dose will remain in the patient's blood half the time before the next dose is to be given. Sketch a graph of the concentration of pain killer in the patient's blood over time. How much of the initial dose would be expected in the patient's blood when the next dose is due?||Exponential
growth and decay (Section 6.5, Pgs. 446-451).
The Sedative activity. I case you are curious, the "previous exercise" dealing with Homer's Blood Pressure can be viewed HERE.
A comment on Newton's Law of Cooling. Discuss, together, Example 3 (Pgs. 449-450).
JR: Explain the basics of "Newton's Law of Cooling" (Pg. 449) and include why it essentially works as a "differential equation" rather than an exponential.
|Pgs. 451-452 #3, 9, 13.|
does a = arcsine(b) mean?
Give an example and include a diagram.
Inverse Trickery activity.
Note the table of derivatives on the bottom of page 457.
JR: List the domain and range for sin-1 , cos-1 , and tan-1.
460 #23, 29, 31;
||Given a circle with radius l. Create a formula for the area of the sector subtended by the central angle theta.||Intermediate
forms and l'Hospital's Rule (see Pg. 470).
See The Sector Ratio activity.
JR: Explain Cauchy's Mean Value Theorem in simple language.
|Pg. 477 #7, 9, 17, 19.|
= 2x - 1, compute f-1(3).
Pg. 480-481; True-False Quiz Pg. 481 (copy stem!); Chapter
6 AP AB/BC Review Questions, Pg. AP6-1 (following page
486). Omit all problems dealing with hyperbolic
JR: Is tan-1(x) = sin-1(x) / cos-1(x) ? Why/why not? Give evidence!
problems from the class activity and include Pg. 482 #4,
21, 38, 59.
|11/3||Let y = x^(e7x). Differentiate the function.||In-class review.
JR: Could there be logarithms having a negative number as their base? Why/why not? Give some examples.
|Prepare for the tests.|
||Get ready for the test! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.||Test 6.2.
Multiple Choice: you may use one 3 x 5 note card and an AP
approved calculator. Expect questions on
||Get ready for the test! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.||Test 6.1. Free
response: you may use one 3 x 5 note card and an AP
approved calculator. Expect questions on
and v be functions in x.
Differentiate the product uv versus the variable x.
||Explain in plain language The Product Rule for differentiation. Give an example.||Integration
By Parts. View this
Khan Academy video.
Guess The Method activity.
JR: Explain the formula for integration by parts in plain language.
492 #3, 11, 17.
do you know when to use integration
by parts rather than u-substitution?
492-493 #19, 27, 37, 45.
JR: Explain how to select u and dv in integration by parts. Also explain what to do if the u and dv you selected create an integral you cannot perform (there should be several contingencies!!!).
the problems started in class.
Copy the integration formulae on Pg. 487 and make flash cards of these. Practice them frequently.
the integral of t^(1/2) ln(t) dt.
Integrals (Section 7.2).
Prove the substitutions for sin2 x and cos2 x on the top of Pg. 496.
Discuss each of the examples in Section 7.2 at your 4-top.
An Equality Tester activity.
JR: Copy the strategies for integration in Section 7.2 (Pgs. 497 & 498).
500-501 #3, 9, 32, 41.
Look Before You Compute activity.
JR: Examine the accompanying table. Explain how each expression is derived from its accompanying identity.
507 #5, 7, 9, 11.
Challenge question: Explain how to (realistically) cut a 14" (diameter) pizza into three equal areas using two parallel cuts.
integral for x2exdx
using integration by parts (or go as far as you
can). Make two attempts with the following
problem demonstration for Integration By Parts and
JR: Explain why a good choice for "u" during Integration By Parts would be in the order Logarithmic, Inverse Trig, Algebraic, Trigonometric, Exponential.
|Pg. 493 #29, 33,
35; Pgs. 500-501 #11, 17.
following difference: 1/(x + 3) - 1/(x +
2). Check both functions in your calculator to see
if they produce the same outputs in a table.
||Read through the
background material and examples on Partial Fractions in
Section 7.4 (Pgs. 508-516). Discuss each of the
examples with your table partner.
JR: Under what conditions would creating partial fractions make integration easier? When should you use polynomial division instead? Give examples.
|Pg. 517 #11, 15,
Addressing repeated linear
terms in creating partial fractions (see the Dr.
Math Forum, scroll down to "Date: 10/15/2001 at
Do the Partial Fractions activity. Those facing North do Version 1, those facing South do Version 2.
JR: Refer to Exercise #1 on page 554. Explain why this integral can be performed using simple (e.g. Power Rule) techniques. Compute the integral.
|Pg. 517 #29, 33, 43, 49.|
together in groups of four such that there is a clear
delineation between groups.
Integration Jeopardy with 4-tops.
JR: In an integration problem, how could substituting x = tan (theta) for the term 1 + x2 possibly make things simpler? Explain and give an example.
523 #1, 5, 9.
||Get ready for the quiz! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.||Quiz:
Integration By Parts--expect two problems. No
calculator or note card.
JR: How many of the integrals in our text's Reference Pages have you memorized?
|Pg. 528 #3, 7, 9.|
be a continuous function that cannot be integrated but you
must know the "area under the curve" from a to b.
Write a plain language explanation of each of the
following methods of performing a Riemann Sum using n
intervals. Include a diagram and state which
function types will result in the integration being an
overestimate and which will be an underestimate.
Do Comparison of Methods activity. Discuss each of the approximation rules and their associated error bounds.
Discussion of Simpson's Rule.
JR: The Midpoint Rule and the Trapezoid Rule appear identical. Explain why they are different and include diagrams.
|Pg. 540 #3, 5,
Deceptive Visions activity.
JR: Explain in simple terms how to determine if an improper integral converges.
|Pg. 551 #1, 5,
||Sit together in groups of four such that there is a clear delineation between groups.||Play Improper
Integration Jeopardy with 4-tops.
Alternate: Integration Jungle activity.
JR: Consider the integral from e to infinity of dx/(x ln(x)p). For what values of p will the integral converge? Why? Give examples.
|Pg. 551 #17, 27,
|| In-class review
(follow THIS link for the
example problems presented and discussed in class).
Consider these Review Exercises Pgs. 554-555 #1, 4, 11, 17, 27, 37, 45.
Consider doing the Chapter 7 AP AB/BC Review Questions, Pgs. AP7-1, 2 (following page 560).
JR: Which concepts in this chapter concern you the most?
|Prepare for the
||Get ready for the test! Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.||Test 7.1. Free
response: Note: you may use both sides of ONE 3x5 note
card, no calculator. Expect questions on
Free response: Note: you may use both sides of ONE 3x5
note card and an AP approved calculator. Tables of
integrals will be provided. Expect questions on
||Develop the Arc Length Formula (see box #2, Pg. 563) from the Pythagorean Theorem.|
||Draw y = sin(x) from x = 0 to π/2 on a half-sheet of graph paper as accurately as you can. Measure the arc length in segments of 0.2 radians. Record the data in a table. Add the arc length segments and convert to the units of the sine graph.||Arc
Length. Derive the arc length formula.
How to Define π activity.
JR: Use the arc length integral to perform the ET.
567 #7, 11, 19.
||Compute the volume generated by rotating y = sin(x) from x = 0 to π/2 about the x-axis.||Areas
of A Surface of Revolution.
Gabriel's Horn activity. North side of classroom do Page 495, South 496. Discuss differences!!! :-)
JR: How is finding a surface from a revolution the same as/different from finding an area?
following and include appropriate units.
Physics and Engineering (Section 8.3). Applications
of Economics & Biology (Section 8.4).
Homer’s Blood activity.
JR: Explain what will happen to a probability distribution as the sample size increases. Hint: see Section 8.5.
|Pg. 584 #13;
Pgs. 590-591 #10, 12, 16.
||Chapter 8 AP AB/BC Review Questions; AP8-1, 2 (following page 602). Do them all!||Create a study
plan to address your areas of weakness over the
break. Be specific and include sample problems.
JR: Define normal in the context of "normal distribution."
study plan with particular emphasis on the few days just
before break ends.
||Test 8.1. Free
response: No note card, no calculator. Expect
JR: Which of the applications in this chapter is most interesting to you. Explain why.
differential equation dy/dx = xy (for y).
||What is a "differential equation?" Give an example.||Modeling
with Differential Equations. Read Pgs. 604-607.
How do Solutions Behave: A First Look activity.
Fun with Differential Equations activity.
JR: Suppose y1(t) is a solution to the second differential equation in today's activity such that y1(0) = -3. What can you say about the long-term behavior of y1? Explain your reasoning.
608-609 #1, 7, 10, 14.
||Create (by hand) the direction field (AKA Slope Field) for y' = -x for -4 ≤ x ≤ 4 and -4 ≤ y ≤ 4 by substituting various values for x and y to determine the slope at various coordinates. Pleace a short dash to represent the approximate slope at the coordinates. Refer to Pgs. 610-611 for examples of direction fields.||Trace
Practice sheets (copies will be provided).
Direction Fields activity (copies will be provided).
JR: Explain in simple terms (as if to a ten-year-old) how a population increases if it is non-logistic, that is dP/dt = kP. Write several sentences!!!
616-617 #9, 11, 13.
Here is a list of several Web-based Direction Field plotters.
f(x) = (x + 3)(1/2).
Compute f(1) and f ' (1). Create the
equation of the line tangent to f at x = 1
and use it to estimate f(1.1).
Euler With Care activity.
Consider viewing these resources on Euler's Method
JR: What is special about the direction field of an autonomous differential equation as compared to the direction field of an arbitrary differential equation?
617 #21, 22.
||What characteristics of a differential equation allow it to be separable?||Separable
Are They Separable activity.
JR: Is the differential equation dy/dx = ln(xy) separable? Why/why not
|Pgs. 624 #3, 15, 21, 23.|
Law of Cooling asserts a liquid cools at a rate
proportional to the difference between the temperature of
the liquid and the temperature of the environment.
Let T represent the temperature of a hot cup of coffee, k
the proportionality constant, and E the temperature of the
environment. The rate (e.g. degrees per minute) the
coffee cools is, therefore, given by dT/dt = k(T - E)
where the constant "k" must be negative. Solve the
differential equation for T.
||Watching Water Cool:
collect cooling data from a cup of hot water. Solve
the differential equation to get an exponential
function. Calculate the temperature for several
five-minute intervals. Compare the actual reading
with the mathematical predictions and see how the
differential equation fares against various exponential
The Potato Cooling problem from the 2017 AP Calculus exam.
JR: Why is dy/dx = x2y2 separable and dy/dx = x2 + y2 NOT separable?
625 #45, 46.
See THIS website for the differential equation solution to an electrical circuit with a capacitor.
||Write an integral that requires both a trig substitution and Integration By Parts to solve. Explain why these are required.||Chapter 9 AP AB/BC
Review Questions; AP9-1, 2 (following page 658). Do
them all and show all work! Confirm answers with
table partners and Edge as you work.
JR: Assume a population, P, begins with 100 individuals and grows at the rate 0.1P(1 - P/2000). Solve the initial value problem and use the solution to predict the size of the population when t = 20.
the review questions.
|1/16||Prepare questions to ask for today's review.||In-class review.
JR: Explain the purpose and process of Euler's Method using terminology understood by a person taking Algebra I.
|Prepare for the tests.|
9.1: Differential Equations. Free response: Note:
you may use both sides of ONE 3x5 note card and an AP
approved calculator. Expect questions on
for tomorrow's test.
Recall the syllabus says "no late work will be accepted in the last five school days of a grading period." That means NO LATE WORK after 25 January through the end of the semester!
Differential Equations. Free response: Note: you may
an AP approved calculator (no note card). Expect
homework, quizzes, and tests and bring questions on any
concept you are still uncertain. Final exams will be
administered next week.
Consider signing up and attending the UW Math Day for High School Students (19 March 2018; Cost: $15 per person).
Create and print your own graph paper at THIS Website.
Below are various documents on the
operation of the class
the additional items