A Sage helper for integrals You can use the

*Sage*helper to crack difficult integrals.

Circle Area is \(\pi r^2\) using Concentric Circle Slices

graph match - big derivative puzzle

The idea is to look at the properties of amount functions (e.g. \(f(x)\)) and their rate functions (the derivative functions, e.g. \(f'(x)\)).

*A note on the puzzle:*

You can make the puzzle task more interesting by scrolling the browser to hide the true/false hints.Building a rate function Using the slope of tangents

Mapping Diagrams Representing functions using parallel number lines for the domain (e.g. (x) axis) and codomain (e.g. (y) axis).

- Kwikis (just below current homework)

**Homework Sources:**

**Next Calculus Test:** Fri March 16, 2018 (generally every other Wednesday)

Due Date | Assignment |
---|---|

Mon 1/8 | WeBWork: Winter-Break-2017-2018Enjoy your break. |

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. * Update your journal. |

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. * Update your journal. |

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 * Put comments on arc length into your journal. |

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 * Update your journal. |

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 * Update your journal. |

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.* Update your journal (e.g. comments on partial fractions). |

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.* Update your journal with comments about: — 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 * Update your journal. |

Fri 2/2 | * Actively watch MIT Improper Integrals video * WeBWorK Improper-bic * Update your journal. |

Mon 2/5 | * Polar Coordinates: — Read Polar Coordinates — Do sec 10.1 #1,3,5,9,12,13,21 * Improper Integrals Resources:– HWC Improper Integral notes – MIT Improper Integrals video * Update your journal. |

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

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: — Read Polar Slopes — 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.* Update your journal. |

Mon 2/12 | Parametric Curves: * Read 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 * Update your journal with thoughts about parametic curves. |

Tue 2/13 | * Read Parametric Calculus * 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. * Update your journal with thoughts about calculus on parametic curves. |

Wed 2/14 | * TurnIn: Status report (a few sentences, perhaps equation, and diagrams…) on Clara - CowculusNote: You will be doing a write-up, but not yet. * Update your journal about Calculus on parametric curves |

Thu 2/15 | * Keep working on Clara - Cowculus. You will be turning in a full write-up on Tuesday, Feb 20. Details on Thursday. * Update your journal about Remain Sums and Integrals |

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 * Update your journal with comments on sequences and series |

Mon 2/26 | * Read about the Integral Test (listed as Theorem 11.3.3). * Actively watch MIT Integral test * Reread Integral Test and do: — section 11-3 #5,6,9 * Update your journal |

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. Update your journal with comments and observations about this assignment. |

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. Update your journal with comments and observations about this assignment. |

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 * Read and Do — 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 * Update your journal. * 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 * Read and Do: — Whitman 11-9 Calculus with Power Series #1,2 * Update your journal |

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

Fri 3/9 | * Read Tips on Using Tests of Convergence * TurnIn: convergence-tests-worksheet * Update your journal * 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. * Update your journal |

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) * Update your journal with comments on Differential Equations |

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 * Update your journal with comments on Differential Equations |

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. * Update your journal |

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. * Update your journal with comments about the Snuggly Circle Problem and Friday’s test. |

Tue 3/20 | * Actively watch Khan– Intro Separable DEs * WeBWorK DiffEQ_02_bic * Update your journal with comments on separable DEs, |

Wed 3/21 | * Turn-in: Bacteria Growth Problem Note: Homer’s Blood can wait until class on Wednesday. * Update your journal with comments on the Bacteria Growth Problem. |

- Kwiki 20086 beginning to introduce differential equations
- Kwiki 20087 cooling coffee and catch up
- Kwiki 20090 separable differential equations
- Kwiki 20091 Homer’s blood (Can wait until class on Wednesday)

- Kwiki 20076 Converging sequences, Fun with Fibonacci
- Kwiki 20077 Integral test, Converging sequences, Fun with Fibonacci
- Kwiki 20078 Integral test, Converging sequences, Fun with Fibonacci
- Kwiki 20079 Converging sequences, limits, \(u\)-substitution
- Kwiki 20080 Maclaurin Polynomials
- Kwiki 20081 Building Maclaurin series
- Kwiki 20082 Building and Using Maclaurin series
- Kwiki 20083 More Building and Using Maclaurin series
- Kwiki 20084 Still Building and Using Maclaurin series
- Kwiki 20085 Misc
- Kwiki 20086 beginning to introduce differential equations
- Kwiki 20087 cooling coffee and catch up
- Kwiki 20089 \(\pi\) and \(\Pi\) day
- Kwiki 20090 separable differential equations

- Kwiki 20072 Polar coordinates and Cartesian coordinates
- Kwiki 20073 Parametric curves, rolling circle, Desmos
- Kwiki 20074 Improper integrals, parametric line segment
- Kwiki 20075 Geometric series

- Kwiki 20065 Work on \((Work)^2\) Day
- Kwiki 20066 partial fractions, Riemann sums, two pairs of functions
- Kwiki 20067 probability
- Kwiki 20068
- Kwiki 20069
- Kwiki 20070 Improper integrals
- Kwiki 20071 Improper integrals plus

- 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*.