AP Calculus BC

2017-2018

Semester 1


Required materials for mathematics classes
Your school-issued laptop, charged!
Two (2) hardbound theme books (i.e. Mead Composition Book) for journaling (no spirals!).  Use one for homework (enter assignments chronologically) and the other for Journal Reflections (this theme book stays in the classroom).
Texas Instruments TI-83+ (or TI-84+) Graphing Calculator + extra AAA cells.  DO NOT expect the school to provide you extra cells!
Basic drawing supplies (ruler that measures in centimeters, compass, protractor).
Several pencils or mechanical pencil with extra lead + a pen for writing JRs.


Click HERE for the Stewart Calculus 7e Website wherein you will find homework hints, "Tools for Enriching Calculus (TEC), additional topics, etc.

Grab THIS document showing the "Essential" and "Recommended" sections of Stewart Calculus 7e.

Linked herein is the topic outline for the course, adapted from the College Board.

  Desmos Graphing Calculator.

Join Wolfram Alpha, which provides step-by-step solutions.

Homework help from Slader.


AP/BC Review Questions (from the end of each chapter).

Greetings
 Date

Entry Task

Activity

Assignment/Homework*

9/6
Find your seat.
Meet 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?
Complete 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.
9/7
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?
Finish the review assessment.
9/8
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.
9/11
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 requiring review.

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?
Complete the suggested problems.
9/12
Work yesterday's suggested problems individually.
Debrief 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.
9/14
Debrief the review problems in your table group. Complete  Review Test 1.1 as if you were actually being tested.  Aim for completion within fifty minutes.  An AP approved calculator may be used.

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).
*Unless otherwise noted, homework is due the next class day.


Chapter 2: Derivatives (selected sections).
 Date

Entry Task

Activity

Assignment/Homework*

9/15
Check to see if the limit exists then compute the left- and right-hand limits
  • Limit as x --> 5 of (x - 5)/|x - 5|
  • Limit as x --> 5 of (x - 5)2/|x - 5|
Debrief ET.

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).
Pgs. 123-124 #13, 23, 33.
9/18
Let f(x) = sec(x)/(1 + tan2(x)).  Perform f '(x) and simplify.
The Chain Rule.  Do the Unbroken Chain activity.

JR:  Under what circumstances would The Chain Rule be used?  Explain in simple language and give an example.
Pg. 154 #7, 23, 25, 47, 51.
9/19
Determine dy/dx for x3 + y3 = 6xy.
Implicit Differentiation.  Review at your 4-top Pgs. 157-161.

JR:  What is the meaning of the result to the ET?
Pgs. 161-162 #7, 23, 25.
9/21
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 cm/min?
Rates of Change in the Natural and Social Sciences and Related Rates.

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.
9/22
Summarize the concept of "related rates" paying particular attention to the terms "related" and "rates" as they apply to calculus.
More work on Related Rates.

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. 181 #23.
9/25
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.
Address 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?
2.  Airplane fuel tanks are located in the wings, which lead to inaccurate fuel quantity measurement using the depth of fuel in the tank.  Consider the cross section (slice the wing front-to-back, look at it from the side) of a wing where part of the tank appears rectangular (twenty-four inches from the wing spar to where it starts curving in the front, ten inches high) and the front of the tank (the leading edge of the wing, therefore curved) can be modeled by an equilateral triangle (all sides = ten inches).  The fuel tank is thirty inches along the wing spar.  Note: a gallon occupies 231 cubic inches.
  • Create an equation that will provide the volume of the tank in gallons versus the depth of the fuel in the tank, d.
  • Compute the volume of the fuel tank in both cubic inches and gallons.
  • Assume the engine burns eight gallons per hour.  Compute the depth of the fuel in the tank after one hour of flight time.
  • At what rate is the depth of the fuel changing at one hour of flight time.
JR:  Assume the voltage, V, across a resistor is modeled by V(t) = sin(t)/(1 + t).  Compute the rate of change of the voltage at 2.0 seconds.
Chapter 2 AP AB/BC Review Questions, Pg. AP2-1 (following page 196).
9/26
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 note card.

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, 38.
*Unless otherwise noted, homework is due the next class day.


Chapter 3: Applications of Differentiation
 Date

Entry Task

Activity

Assignment/Homework*

9/28
Find a positive number such that the sum of the number and twice its reciprocal is as small as possible. Discuss the activities
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.
Pg. 257 #9, 19, 29.
9/29
List the procedure for determining the extrema of a given function.
Complete last year's Test 3.1 (hard copies provided).  Work alone.

JR:  Pg. 277 #45.
Chapter 3 AP AB/BC Review Questions, pg. AP3-1 and AP3-2 (following page 282).
10/2
Score Test 3.1.
Optimization workshop
  • Locate or invent a worthy optimization problem.
  • Write the problem for all to see (whiteboard, projection screen, ...).
  • Work the problems with minimal, if possible, assistance.
  • Participate in problem debrief/demonstration.

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.
10/3
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.
Pg. 277 #47, 48.
*Unless otherwise noted, homework is due the next class day. 



Chapter 4: Integrals
 Date

Entry Task

Activity

Assignment/Homework*

10/5
How is The Chain Rule related to "The substitution Rule?" Give an example.
Clearing The Hill activity.  Loki's Dilemma activity (handouts).

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.
10/6
See problem #13 on page 339.  Explain why "u-substitution" is not necessary to evaluate the integral.
Review 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.
Finish the review and prepare for the quiz.
10/9
Work alone on pg. 318 #17.
Process and quirks of FTC.
  • What it says, fundamentally.
  • How it works, fundamentally.
  • Working with non-simple limits of integration.
  • Why there are two parts.
JR:  List the conditions under which the FTC can be applied.
Watch the You Tube FTC video.  Do Pg. 339 #37, 38; Pg. 340 #53.
10/11
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
  • FTC.
  • U-substitution in integrals.
  • Related Rates.
JR:  List the topics from Chapter 4 you believe you should review more deeply.
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!!!!!).
*Unless otherwise noted, homework is due the next class day.



Chapter 5: Applications of Integration
 Date

Entry Task

Activity

Assignment/Homework*

10/12
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 activity.

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.
Pg. 349 #1, 3, 5, 11; Pg. 360 #1, 5, 7, 9.
10/16
What does the "dx" represent in the volume integrals?
Cylindrical 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 demonstrate.

An Exotic Bagel activity.

JR:  Why does the factor 2π appear with “cylindrical shells” while π appears with “washers?”
Pg. 367 #5, 9, 15.

10/17
Define force, work, and power.
Work (Section 5.4).  Skim and discuss Pgs. 368-371 in your table groups.  The Weighty Chain activity.

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, 3, 7.
10/19
Set 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 the y-axis.
Review of volumes, problem demonstrations.

Do the following problems in your homework notebook:
  • Pg. 360 #11, 14, 15
  • Pg. 367 #21, 37.
  • Pg. 372 #15.
JR:  Give examples of where "discs" is most appropriate to determine a volume and when "cylindrical shells" is most appropriate.
Pg. 372 #19, 21.
10/20
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.
10/23
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.
*Unless otherwise noted, homework is due the next class day.


Chapter 6: Inverse Functions
 Date

Entry Task

Activity

Assignment/Homework*

10/24
Let f(x) = (x - 4)1/3.  Compute
  • f -1(-2).
  • f -1(0).
Let f(x) = 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 Y2 column?

Given f(1) = 2; f(2) = 3; and, f(3) = 1.  What are
  • f(f(1))
  • f -1(f(1))
  • f(f -1(1))

JR:
  Let f(x) = (x - 4)1/3.  Compute (f -1) ' (-2).
Pg. 390 #1, 3, 5, 24.
10/26
Read 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). 
Logarithms.

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.
10/27
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.
10/30
What does a = arcsine(b) mean?  Give an example and include a diagram.
Inverse Trigonometric Functions.

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.
Pg. 460 #23, 29, 31;
10/31
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.
11/2
If f(x) = 2x - 1, compute f-1(3).
Concept Check, 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 trigonometric functions.

JR:  Is tan-1(x) =  sin-1(x) / cos-1(x) ? Why/why not?  Give evidence!
Complete all 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.
11/6
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
  • Use properties of logarithms to expand an expression.
  • Perform derivatives and integrals involving inverse trig functions, either within the function or as a result
  • Determine the equation of a line tangent to a curve at a specified point on an inverse trig function..
  • Evaluate limits of rational functions containing logarithms.
  • Applications of L'Hospital's Rule.
JR:  For which concepts in Chapter 6 are you still uncertain?
Prepare for tomorrow's test.
11/7
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
  • Given facts for a function f(x) that is the inverse of a function g(x), compute the derivative of g(x) at a value.
  • Compute derivatives of various exponential functions, some of which will require logarithmic differentiation.
  • Given a natural exponential function, calculate a point where the tangent line has a given slope.
  • Solve a problem involving exponential growth.
  • Applying the FTC to an exponential function.
  • Compute a limit using L'Hospital's Rule.
JR:  Explain what you know about "partial fractions" and give an example.
Let u and v be functions in x.  Differentiate the product uv versus the variable x.
*Unless otherwise noted, homework is due the next class day.


Chapter 7: Techniques of Integration
 Date

Entry Task

Activity

Assignment/Homework*

11/9
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.
Pg. 492 #3, 11, 17.
11/13
How do you know when to use integration by parts rather than u-substitution?
Pgs. 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!!!).
Finish the problems started in class.

Copy the integration formulae on Pg. 487 and make flash cards of these.  Practice them frequently.
11/14
Compute the integral of t^(1/2) ln(t) dt.
Trigonometric 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).
Pgs. 500-501 #3, 9, 32, 41.

11/16
Trigonometric Substitution.

Look Before You Compute activity.

JR:  Examine the accompanying table.  Explain how each expression is derived from its accompanying identity.
Pg. 507 #5, 7, 9, 11.

Challenge question: Explain how to (realistically) cut a 14" (diameter) pizza into three equal areas using two parallel cuts.
11/17
Perform the integral for x2exdx using integration by parts (or go as far as you can).  Make two attempts with the following substitutions
  • Let u = ex.
  • Let u = x2.
Discussion and problem demonstration for Integration By Parts and Trigonometric Integrals.

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.

11/20
Simplify the 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, 23.
11/21 Perform the integral ∫5/(x–5)2dx.
Addressing repeated linear terms in creating partial fractions (see the Dr. Math Forum, scroll down to "Date: 10/15/2001 at 21:01:46").

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.
11/27
Sit together in groups of four such that there is a clear delineation between groups.
Play 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.
Pg. 523 #1, 5, 9.
11/28
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.
11/30
Let f(x) 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.
  • Left endpoint.
  • Right endpoint.
  • Midpoint.
  • Trapezoid.
Approximate Integration

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, 7.
12/1
Define the following integrals
  • Definite.
  • Indefinite.
  • Improper.
Improper Integrals.

Deceptive Visions activity.

JR:  Explain in simple terms how to determine if an improper integral converges.
Pg. 551 #1, 5, 7, 13.
12/4
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, 31.
12/5

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 tests.
12/7
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
  • Integration by parts
  • Trigonometric integrals.
  • Trigonometric substitutions.
  • Integration using partial fractions.
  • Using Trigonometric identities.
  • Improper integrals.
JR:  For which concepts in Chapter 7 are you still uncertain?
Prepare for tomorrow's test.
12/8
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 7.2.  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
  • Approximate integration. 
  • Approximation will be to four decimal places.
  • The number of intervals, n = 10.
JR:  Quote Simpson's Rule and explain why it is a better method for approximating integration than the other rules.
Develop the Arc Length Formula (see box #2, Pg. 563) from the Pythagorean Theorem.
*Unless otherwise noted, homework is due the next class day.


Chapter 8: Further Applications of Integration
 Date

Entry Task

Activity

Assignment/Homework*

12/11
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.
Pg. 567 #7, 11, 19.
12/12
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?
Pg. 574 #7,11,13.
12/14
Define the following and include appropriate units.
  • Hydrostatic Pressure.
  • Moment.
  • Center of Mass.
  • Centroid.
  • Demand.
  • Consumer Surplus.
  • Cardiac output.
Applications to Physics and Engineering (Section 8.3).  Applications of Economics & Biology (Section 8.4).

Under Pressure activity.

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.
12/15
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."
Execute your study plan with particular emphasis on the few days just before break ends.
1/2
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 8.1. Free response: No note card, no calculator.  Expect questions on
  • Arc length
  • Area of surface of revolution
  • Application (one of: physics, engineering, economics, or biology)

JR:
  Which of the applications in this chapter is most interesting to you.  Explain why.
Solve the differential equation dy/dx = xy (for y).  Simplify!
*Unless otherwise noted, homework is due the next class day. 


Chapter 9: Differential Equations
 Date

Entry Task

Activity

Assignment/Homework*

1/4
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.
Pgs. 608-609 #1, 7, 10, 14.
1/5
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 Direction Field 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!!!
Pgs. 616-617 #9, 11, 13.

Here is a list of several Web-based Direction Field plotters.
1/8
Let 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's Method.

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?
Pg. 617 #21, 22.


1/9
What characteristics of a differential equation allow it to be separable? Separable Equations.

Are They Separable activity.

JR:  Is the differential equation  dy/dx = ln(xy) separable?  Why/why not
Pgs. 624 #3, 15, 21, 23.
1/11
Newton's 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 models.

The Potato Cooling problem from the 2017 AP Calculus exam.

JR:  Why is dy/dx = x2y2 separable and dy/dx = x2 + y2 NOT separable?
Pg. 625 #45, 46.

See THIS website for the differential equation solution to an electrical circuit with a capacitor. 
1/12
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.
Finish 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.
1/18
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 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
  • Solving separable differential equations.
  • Solving differential equations using a slope field.
  • Creating a logistic model.
  • Creating differential equations that model real-world scenarios.
JR:  How did this the test go for you?  Which concepts in this chapter do you believe you should know better?
Prepare 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!
1/19
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 9.2: Differential Equations.  Free response: Note: you may an AP approved calculator (no note card).  Expect questions on
  • Euler's Method.  You MUST show all the calculus, algebra, and arithmetic for each step and write an explicit answer.
JR:  What were the hardest parts of this chapter for you?  What were the (relatively) easiest?
Review previous 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).


*Unless otherwise noted, homework is due the next class day.


Click here for the Reference Pages from Stewart Calculus (7th Edition).

Create and print your own graph paper at THIS Website.

Below are various documents on the operation of the class

Course syllabus

Also note the additional items

    Email: richard.edgerton@highlineschools.org