# Calculus (AP Math BC) Spring 2018

## Pointer to:

• Kwikis (just below current homework)

## BC Exam Preparation

Here are pointers to important information about the BC Exam from the College Board:

You should take the time to read each of them.

### Calculus Homework

Homework Sources:

Next AP PrepQuiz: Mon May 7, 2018 (generally every Monday until AP exam)

Due Date Assignment
Mon 1/8
WeBWork: Winter-Break-2017-2018

Tue 1/9
* Reconsider two cylinders each with a radius of $$R$$ intersecting at right angles as shown in this diagram. Last semester your volume elements were “rectanglular disks” (squares) to find the the volume of the intersecting region. This time your method is to use “rectangular shells” as volume elements

* TurnIn: A typed second draft of your solution as a complete Four Section Write-up. See Sample write-up. Be sure to comment on the method and why your answer is reasonable.

* (Optional: You might want to review volume by shells by actively watching MIT Volume of revolution via shells.)

Wed 1/10
* WeBWork Volume_Practice_1
Note: Problem 4 is multiple choice, and you only get two tries to get the correct answer(s)
* Finish the Ladder problem from Kwiki 20055

* Start a journal. Each day spend a few minutes summarizing the concepts we worked on in class and in the homework. You might use the journal to ask yourself questions on concepts. For example,
When finding volumes of revolution when is it better to use shells than to use washers?

Thu 1/11
* Carefully read HWS Integration by Parts Introduction
* Use Integration by Parts on $\int x e^{-x} \, dx$ Be sure to check your answer.
* In your journal write-up a solution to question 4 (Battle of the Limits) on Kwiki 20057

Fri 1/12
* Determining $$u$$ and $$dv$$ takes some practice. Actively watch MIT Integration by parts 1
* Finish Kwiki 20058 — (but don’t look up answers and/or solutions)

* Journals:
1. Transfer your journal from your computer to the new book. Now that you have an empty journal book, bring it up to date by printing your computer entries (or writing new ones) for Monday, Tuesday, and Wednesday, and pasting them into the pages of the book. Each day’s entry should have be on a new page with the day and date in the upper right hand corner. Please only use the front side of each sheet of paper. With 168 sheets you will not run out of paper.
2. Today’s entry: You are welcome to use your computer and paste your entries in the notebook, write into the notebook, and do both. At a minimum, today’s entry must include how to show that the product rule leads to the formula for integration by parts.

Tue 1/16
* WeBWorK bic_spring_misc_01
— Note: Only two tries on the matching question.

* TurnIn: A 3rd Draft of Intersecting Cylinder Volume by “rectangular shells”
Focus on improving your explanation of how you develop your volume element. A couple of people in Block 5 annotated this diagram to show a typical rectangular box. You can do this by printing a copy of the diaygram and using a pencil and straight-edge…

* Remember, your journal should include a dated entry after each class on a new page. Today’s Kwiki should be a good starting point.

Wed 1/17
* Be prepared (as a group) to present any of the volume problems from today’s Kwiki on the board.
In addition to the setups, evaluate the integrals.

Thu 1/18 * Work on Particle, the dog (see question 6 of the Kwiki 20061).
* TurnIn: A brief second draft of a write-up of your game plan and solution sections for part (a) of the Particle, the dog question
— Find an equation for a function, $$P(x)$$ that takes the owner’s $$x$$ value and produces the distance Particle has traveled along the sine curve since starting at the origin. Be sure to explain how you developed $$dP$$, a distance element.
Hint: You will ultimately need to use a calulator to get values for $$P(x)$$ other than $$P(0)$$.
Note: If you do not have an equation for $$P(x)$$, write a question where the answer to that question would help you get closer to a equation for $$P(x)$$. Explain how the answer would help you.

Fri 1/19
* The distance Particle, the Dog travels along a curve is called arc length.
Compute the length of the curve $y=x^{3/2} \, \, \text{ for } \,\, x \in [0,4]$
* Actively watch MIT arclength

Mon 1/22
* Select two of the questions from Kwiki 20063 that is not on your group’s poster. Write up each question, and enter a second draft into your journal. A neatly handwritten version or pasting in a typed printout is fine.

* Journal Update: The first test is coming on Wednesday. Make an entry that includes an annotated list of topics we have studied in this second semester. Other ideas are always welcome.

Tue 1/23
* Carefully Read HWS Work Introduction and pulling up a rope using work elements Penn work element
* Do question 2 on Kwiki 20064

Wed 1/24
* Test
There will not be any questions about work
* Be sure to bring in your journal. Among other things you should have an entries on arc length (see 1/19) and an annotated topics list (see 1/22)

Fri 1/26
* WeBWorK Work-bic-1

Mon 1/29
* Background on partial fractions can be found in HWS Partial Fractions
* WeBWorK partial-fractions-1
* Journal Question: Take another look at the last question on Kwiki 20066. Why did I call the pair of functions, $$C(t)$$ and $$S(t)$$? Why did I select the function from the Wednesday’s test to be $$C$$ rather than $$S$$? What else can you say about this pair of functions as analogs to $$\cos(t)$$ and $$\sin(t)$$? Write a journal entry on this.

Tue 1/30
* Do Revisiting Integrals worksheet. Bring it to class.

* Finish today’s Kwiki 20067
* Update your journal. (Definitions of probability…)

Wed 1/31
* Finish Kwiki 20068-with-Homework. Note that there is a third section that contains some homework questions.

Revisiting Integrals in light of doing the kwiki in class. Explain why $$f(x)$$ needs to be invertible on $$[a,b]$$ both from a geometric and algebraic point of view.
— Question 4 on $\displaystyle \int_0^\infty f(x) \,dx$

Thu 2/1
* WeBWorK Misc_Set_04

Fri 2/2
* Actively watch MIT Improper Integrals video
* WeBWorK Improper-bic

Mon 2/5
* Polar Coordinates:
— Do sec 10.1 #1,3,5,9,12,13,21

* Improper Integrals Resources:
HWC Improper Integral notes
MIT Improper Integrals video

Tue 2/6
* Start WeBWorK Misc_Set_05. Stop after 45 minutes even if you have not correctly answered all 7 questions.

Wed 2/7
* Test
* New (or revisited) topics:
— partial fractions
— Riemann Sums
— probability
— work
— improper integrals

* Journals:
— Make sure you have comments on Revisiting Integrals

Fri 2/9
* Polar derivatives:
— Do 10.2 #1, 2, 3, 7, 8

* Try is Desmos to graph some polar equations. Start with my Sample Desmos Polar Graph

* Improper Integrals: Be prepared to present solutions and comments on questions 1 and 2 kwiki 20071.

Mon 2/12
Parametric Curves:
* WeBWorK parametric-intro-2018
* Explore graphing parametric curves. Here is a Sample Parametric Curve Using Desmos
* Be sure you understand question 1(a) on Kwiki 20073
— See Building a cycloid mathlet

Tue 2/13
* WeBWorK parametric-bic-ii
TurnIn A second draft of a Complete WriteUp (4 labeled sections and title) to question 3. Use Desmos Graphing Calculator to create a graph of your solution.

Wed 2/14
* TurnIn: Status report (a few sentences, perhaps equation, and diagrams…) on Clara - Cowculus
Note: You will be doing a write-up, but not yet.

Thu 2/15
* Keep working on Clara - Cowculus. You will be turning in a full write-up on Tuesday, Feb 20. Details on Thursday.

Tue 2/20
TurnIn: A complete write-up Clara - Cowculus
— The key question is finding the area of the grazing region. This will require you to divide the grazing region into several regions. The area of some regions can be computed by simple geometry. Others require Calculus. You should be able to set-up the area elements and integrals.

— After that, you have two options:
Option 1 — Focus on evaluating the integrals
If you can, use integration techniques to get antiderivatives. If you can’t crack an antiderivative, you can use use the Integral Calculator. You should be able to get an exact value for the area of the grazing region.
or
Option 2 — Focus on making a Desmos Graphing Calculator dynamic diagram:
Take a look at my (pre-Desmos) grazing animation at the top See what you can do with Desmos Graphing Calculator. My Moving Cycloid (Desmos version) is a working example of a dynamic diagram which contains some comments on how I put it together.

* Update your journal to help prepare for Wedneday’s test.

Wed 2/21
Test
TurnIn: Journals.

Fri 2/23
* Read Sequences and Series Overview
* Sequences:
— Read Intro to Sequences 11-1
— Be able to explain: “If a sequence is bounded and monotonic then it converges”
— Be able to explain: Example 11.1.15.
* Series:
— Read Intro to Series 11-2
— Do section 11-2 #1-5
* WeBWorK seq_series_01-bic

Mon 2/26
* Actively watch MIT Integral test
* Reread Integral Test and do:
— section 11-3 #5,6,9

Tue 2/27
* WeBWorK seq_series_02-bic

* Update your journal. Include an your understanding of why the Integral Test works. See question 1 from todays Kwiki 20077

Wed 2/28
* Whitman 11-4 Alternating Series read and do:
— section 11-4 #1-5 (all)

* Explore An Alternating Series:
— You can build up points using Demos Start Alternating Series v1
— The first 100 points are plotted at Demos Alternating Harmonic Demo

Thu 3/1
Taylor polynomials are polynomials that can approximate functions where the value of function and the values of all of the derivatives are known for some point $$(a, f(a))$$. For now let’s assume $$a$$ is zero. In this case $$p_1(x)=1+f'(0)x$$, our good friend, the tangent in Taylor form.
* Develop an $$11^{th}$$ degree Taylor polynomial for $$f(x)=e^x$$. see: Pictures of the Class White Boards
* Develop an $$11^{th}$$ degree Taylor polynomial for for $$f(x)=\sin(x)$$.

* Try out your polynomials on Desmos:
Desmos — first three Taylor polynomials of f(x)=e^x
Desmos — Starting Taylor polynomials of f(x)=sin(x)

Note: I know that tonight’s homework is not easy to understand. Do the best you can, but don’t go looking on the internet for answers or explanations of Taylor polynomials. Just do the best you can. Working with one or two other people to talk about the idea of a Taylor polynomial could be useful.

Fri 3/2
* Actively watch 3Blue1Brown Taylor Series video
* Without looking at your notes, (re)do all of the questions on today’s Kwiki 20080.
* There will be a short quiz in class based on the kwiki.

Mon 3/5
* Read and Do Whitman 11-6 and 11-7
— Absolute Convergence: section 11-6 #1-7 (odd)
— Ratio Test: section 11-7 #1-4 (all)
* Actively watch MIT Intro Ratio Test

* Some students have found actively watching Khan– Intro Taylor and Maclaurin Polynomials – Part 1 to be useful.
* Update your journal with a summary of your current understanding of Taylor polynomial and Taylor series. This summary should not be a simple report card. Make it something that will be helpful for rereading in April and May when you are reviewing the material.

Tue 3/6
* Actively Rewatch these sections of the 3Blue1Brown Taylor Series video:
3Blue1Brown Taylor Polynomials – Summary Watch thru about 14:30
3Blue1Brown Taylor Series – Radius of Convergence
Whitman 11-6 Absolute Convergence #1-7 (odd)
Whitman 11-8 Power Series #1,2,3
* Be prepared to present any series from question 3 of Kwiki 20082
* Note: Next test has been moved to the week of March 14.

Wed 3/7
* Spend 20 minutes working on the back of today’s Kwiki 20083
* Actively Watch: MIT Radius of Convergence
Whitman 11-9 Calculus with Power Series #1,2

Thu 3/8
* Actively Watch: MIT Comparison Tests
* WeBWorK Power-Series-bic
* Go over today’s Kwiki 2008

Fri 3/9
* Read Tips on Using Tests of Convergence
* TurnIn: convergence-tests-worksheet

* Reminder: test on Friday, March 16.

Mon 3/12
Catch-Up weekend

* For those with IOUs on either yesterday’s convergence-tests-worksheet, go back and reread Tips on Using Tests of Convergence and turn in the complete worksheet.
* For those who sent me email IOUs on WeBWorK Power-Series-bic, it is should be open for you.

* Do Kwiki 20085 questions 6 and 7 without looking at your notes.
* Do Kwiki 20086 questions 1, 2 and 4 and without looking at your notes.
We will work on question 3 on in class on Monday, but see what you can do by yourself.

Tue 3/13
* Watch Khan– Differential Equation Introduction
* Actively watch Khan– Finding a particular linear solution to a DE
* WeBWorK DiffEQ_01_bic (no reopens)

Wed 3/14
* Review Solutions to WeBWorK DiffEQ_01_bic
* More on slope fields:
Khan– Creating a slope field
Khan– Slope field to visualize solutions
Khan– Matching Differential Equations and slope fileds

Thu 3/15
TurnIn: A unit circle ($$C_1$$) fits snuggly into a 2 by 2 square. Find the radius of the largest circle ($$C_2$$) that fits snuggly in the region of the lower left hand corner of the square outside the unit circle.
* Spend 20 minutes on this problem. (Start by sketching a good diagram.)
* Spend 5 minutes planning an organized write-up. You might not have a complete solution, but you should have things to say about the problem.
* Spend 15 minutes writing (it need not be typed, but you will be turning in a revision on Monday) an organized Four Sections write-up.

Fri 3/16
* Test (nothing on differential equations, but …)

* Update your journal with two paragraph:
— 1) Talk about how much time you spend on the class each night. What is the least amout of time? What is the most you have ever spent? What is your “typical time.” Briefly describe what takes a long time.
— 2) Talk about how you studied for Friday’s test.

* Bring your journal to the test.

Mon 3/19
TurnIn: Complete typed four section write-up for Snuggly Circle Problem. See assignment for Thu 3/15.

Tue 3/20
* Actively watch Khan– Intro Separable DEs
* WeBWorK DiffEQ_02_bic

Wed 3/21
* Turn-in: Bacteria Growth Problem

Note: Homer’s Blood can wait until class on Wednesday.

Thu 3/22
* WeBWorK DiffEQ_06_bic

Fri 3/23
* Actively watch Khan– Intro Eulers Method
* Try using Euler’s Method on Kwiki 20093 (which we did not get to in class today).
* (Some of you need to finish WeBWorK DiffEQ_06_bic)

* Update your journal on Euler’s Method and our discussion of the symmetric difference quotient from today’s class.

Mon 4/2
* Prep Set 1 Set aside 90 minutes to do the six questions. You can use your TI on the first three questions. Write out your solutions as if you were really taking the test. After that you can use resources (e.g. books, the internet, friends) to get a better understanding of any questions that you did not immediately ace. Feel free to email me with any questions over the break. * Prep Set 1 Quiz Monday

Have a good break.

Tue 4/3
* WeBWorK DiffEQ_05a_bic

Wed 4/4
* Polar area:
— Do 10.3 #12, 21
* Trapezoid Approximations
— Watch Khan– Trapezoid Approximations
— Do Khan– Trapezoid Rule Quiz
* Get started on Calculus on a Parabola. (A full write-up on this is coming.)

* Start bringing your graphing calculator to class.

Thu 4/5
TurnIn: Your second draft of Calculus on a Parabola. Use the standard Problem, Approach, Solution, Final Thoughts, organization.

* Test 5 on Monday

* Update your journal on weighted sums; left, right, trapezoid, and middle box approximations for definite integrals.

Fri 4/6
* You will be making an annotated version of the BC Topics Outline provided by the college board. You should start by copying this GoogleDoc – BC Topics List. Note that you will need to use your SFUSD google account to get access to this file.
Do some work on the annotations, e.g. definitions and examples.

* Start tonight. Your complete annotated outline will be collected on Friday, April 13.

Mon 4/9
* Prep Set 2 quiz on Wednesday or Thursday.

* Test (everything upto and including differential equations). You may use a simple calculator for arithmetic. S-104 has calculators you can use.

Fri 4/13
* TurnIn: Your annotated BC Topics Outline. Handwritten annotations are fine. Make this something that will help you study for the AP exam.

Mon 4/16
* Prep Set 3 quiz.

* Update your journal on Friday’s class.

Wed 4/18
or
Thu 4/19

* Read New York Times article on beauty of math

* Review Complex Numbers Be sure you understand these two diagrams:

If not, take a look at Khan Academy on Complex Numbers

* Note — Calculus Final Exam Days: Wed May 2 and Thu May 3, 2018
* Update your journal on complex numbers.

Fri 4/20
* Actively (Re)watch Khan– Taylor Polynomial Remainders Part 1

* Update your journal with more on complex numbers.

Mon 4/23
* Prep Set 4 quiz.

* Update your journal on Friday’s class.

Tue 4/24
* Actively (Re)watch:
Khan– Taylor Polynomial Remainders Part 1
Khan – Remainder Example

* Note: new tables Wednesday

Wed 4/25
* The MAA Mean Policeman is a fun introduction to the Mean Value Theorem from the 1960’s.
* Watch Khan– Mean Value Theorem

* Update your journal on MVT
* Note: new (student choice) tables Wednesday

Thu 4/26
TurnIn: WeBWorK MVT-intro (Just one problem)
* Take a look at Notes on Taylor Reminders
* If time allows: Watch Five Unusual Proofs. Put comments on the video in your journal

Fri 4/27
* Fill out AP Exams Survey to let me know which classes you will miss for AP exams.

* Take another look at Notes on Taylor Reminders. Thanks to several students who spotted an arithmetic error in examples. I’ve annotated the first appearance of the error. (The error is carried forward…)
* Prep Set 5 was handed out in class. PrepQuiz on Monday.

* Update your journal on Taylor remainder.

Mon 4/30
Take PrepQuiz 5
Tue 5/1
* Study for (Wednesday and Thursday)’s Final
* Follow pointers to College Board’s BC Exam Preparation at the top of this webpage.
* Put fresh batteries into your TI.

Wed 5/2
* Final Part I (no TI Multiple Choice)

Thu 5/3
* Final Part II (TI allowed Multiple Choice)

Mon 5/7
* Prep Set 6 quiz.
* Please check that my Missing Calculus for AP Tests List is correct for you.

* Note Room Change for Monday, May 7
— Block 3 goes to 133
— Block 5 goes to 135

* Update your journal on Friday’s class.

Thu 5/10
* If you missed class, spend 30 minutes working on the two FRQs in today’s Kwiki 20105

Fri 5/11
* If you missed class, spend 30 minutes working on the the FRQs in today’s Kwiki 20106
* Bring your graphing calculator to class tomorrow. We will do something more interesting than FRQs.

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## Kwikis

### Set 1S

• Kwiki 20055 sphere by shells (again), ladder problem
• Kwiki 20056 thinking about $$u$$-substitution, intersecting cylinders “shells”, ladder problem
• Kwiki 20057 thinking about $$\displaystyle \int_0^1 \sqrt{1-x^2} \, dx$$, (First Battle of the Limits)
• Kwiki 20058 integration by parts
• Kwiki 20059 integration by parts, another battle of the limits, still thinking about $$\displaystyle \int_0^1 \sqrt{1-x^2} \, dx$$
• Kwiki 20060 Napkin Ring Problem, misc volume problems
• Kwiki 20061 For group presentations: Napkin Ring Problem, misc volume problems; $$\displaystyle \int \sin^2(x) \, dx$$ using integration by parts, start Particle, the Dog
• Kwiki 20062 $$\displaystyle \int \cos^2(x) \, dx$$ using integration by parts, more Particle, the Dog
• Kwiki 20063 Seven potential poster problems
• Kwiki 20064 Using calculus and function attributes to sketch a graph, into work.

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