Math 21, Winter 2018 — Schaeffer/Solis**Improper Integrals, Sequences, and Series**

**Getting started!**

**Familiarize yourself with the course policies:**The complete, detailed version and a quick summary of the main points.**If you are having problems enrolling in the course:**Read the information here and fill out the linked webform.**If you want more information on the ACE (Additional Calculus for Engineers) Program:**Read the information here (the application can also be found through that site).*Please note that ACE acceptances are made by Prof. Noe Lozano in the School of Engineering and not by Dr. Schaeffer.***If you are on a Stanford University Athletics team:**Fill out the athletics survey.**If you have an accommodation from the OAE:**Please inform Dr. Schaeffer of the details by email as soon as possible.**Write down the exam dates:**Midterm 1 is on__January 31st__(7–8:30 PM), Midterm 2 is on__February 28th__(7–8:30 PM), and the Final is on__March 19th__(7–10 PM).*If you have a conflict with any of those dates, email Dr. Schaeffer as soon as possible. Please note that pursuant to Math Department policy, there are no alternate exam dates.*- Here are some helpful general tips for success in undergraduate math courses (h/t Adva Wolf)!

**Piazza and Gradescope**

- Piazza is where to go to ask homework questions and anything else about the course material!
- Homework in Math 21 is submitted and graded on Gradescope (submission guide), and your exams will be graded there as well.

This is also where your grades are recorded and where you can view your graded work. - Grade redress form.

**Weekly Schedule**

Monday | Tuesday | Wednesday | Thursday | Friday | |
---|---|---|---|---|---|

George Schaeffer | Lec. 10:30–11:20 (380-Y) Lec. 1:30–2:20 (Herrin T195) O.H. 4:30–5:30 (381-G) |
Lec. 10:30–11:20 (380-Y) Lec. 1:30–2:20 (Herrin T195) O.H. 2:45–4:15 (381-G) |
Lec. 10:30–11:20 (380-Y) Lec. 1:30–2:20 (Herrin T195) |
||

Pablo Solis | Lec. 9:30–10:20 (380-F) Lec. 11:30–12:20 (Hewlett TC 101) O.H. 2:45–4:15 (384-F) |
Lec. 9:30–10:20 (380-F) Lec. 11:30–12:20 (Hewlett TC 101) O.H. 4:30–5:30 (384-F) |
Lec. 9:30–10:20 (380-F) Lec. 11:30–12:20 (Hewlett TC 101) |
||

Ben Lim | Disc. 9:30–10:20 (200-013) Disc. 10:30–11:20 (460-301) Disc. 11:30–12:20 (Herrin T195) |
O.H. 5:30–7:00 (380-U1) | O.H. 3:00–4:30 (380-U1) | ||

Adva Wolf | Disc. 10:30–11:20 (STLC 115) Disc. 11:30–12:20 (STLC 115) Disc. 12:30–1:20 (STLC 115) O.H. 2:00–3:30 (381-L) |
O.H. 1:00–2:30 (160-B40) | |||

Calista Bernard | O.H. 5:30–7:00 (380-R)* | Disc. 1:30–3:20 (380-Y)† | O.H. 5:30–7:00 (380-R)* |

† Calista's discussion sections are reserved for students in the ACE program—you must be enrolled in Math 21A to attend. (You may attend Calista's office hours regardless of your enrollment.)

* Note that in order to enter the first floor of Building 380 after 4:30 PM you may need to use the

**Homework**

- Homework in Math 21 is submitted online using Gradescope.
- Here's a short guide on how to submit assignments.
- You may also view your graded work and all recorded grades on Gradescope.

- Assignments are
__due at 11 AM on Friday__every week except on exam weeks (Weeks 4 and 8).*Please give yourself enough time to submit, and try to upload your submission at least 30 minutes before the deadline just to be sure.* - Late homework is accepted up to one day after the deadline with
__steep__penalties—see the course policies for details. No exceptions are made. - Your lowest score on HWs 1–7 is dropped.
- Homework assignments:
- Homework 0, due Friday, Jan. 12th by 11 AM
__on Gradescope__(not for credit, but highly recommended). Solutions to Homework 0. - Homework 1, due Friday, Jan. 19th by 11 AM
__on Gradescope__. (Solutions) - Homework 2, due Monday, Jan. 29th by 11 AM
__on Gradescope__. (Solutions) - Homework 3, due Monday, Feb. 12th by 11 AM
__on Gradescope__. (Solutions) - Homework 4, due Monday, Feb. 19th by 11 AM
__on Gradescope__. (Solutions) - Homework 5, due Monday, Feb. 26th by 11 AM
__on Gradescope__.

- Homework 0, due Friday, Jan. 12th by 11 AM

**Prerequisites**

- The math department has developed an online precalculus refresher that will give you an opportunity to watch mini-lectures on precalculus topics and do many practice exercises.

We strongly encourage you to spend some time using this resource to brush up on these topics and skills. For more information, click here. - Limits are a very necessary calculus prerequisite for Math 21. We will be taking a lot of limits this quarter!

If you feel that you need to review limits and/or L'Hôpital's Rule, Paul Dawkins' Online Math Notes provide excellent review:- Limits "Cheat Sheet" (Note: You will not need the formal ε–δ definition in Math 21. You should simply know how to evaluate limits
*practically*and have some understanding of them,*conceptually*.) - The Limit
- One-Sided Limits
- Limit Properties/Rules
- Computing Limits
- Infinite Limits
- Limits at Infinity (especially important in Math 21): Part I, Part II
- Indeterminate Forms and L'Hôpital's Rule (also important in Math 21)

- Limits "Cheat Sheet" (Note: You will not need the formal ε–δ definition in Math 21. You should simply know how to evaluate limits
- The complete integration table from the book (some printed versions are missing a page)

**Lectures, Handouts, and Notes**

We (Drs. Schaeffer and Solis) will

However, they should help you if you have an extended absence, fall behind, or think you missed something!

Finally, our lectures are coordinated to cover the same material, but we may have different 'approaches' to it.

Topics covered | Book | Schaeffer (Winter 2018) | Solis (Winter 2018) | |
---|---|---|---|---|

Lecture 1 (1/08) | What is Math 21 about? | — | Notes | Notes |

Lecture 2 (1/10) | The central question(s) of Math 21 Evaluating improper integrals |
7.6 | Notes | Notes |

Lecture 3 (1/12) | Convergence/divergence of improper integrals Limit comparison for improper integrals to infinity |
7.6, 7.7 | Handout (Solutions) | |

Lecture 4 (1/17) | Growth and decay of functions Limit comparison for improper integrals to infinity |
7.6, 7.7 | Notes | Notes |

Lecture 5 (1/19) | Growth and decay of functions Limit comparison for improper integrals to infinity |
7.6, 7.7 | — | Notes |

Lecture 6 (1/22) | Improper integrals where the integrand is unbounded improper integrals with multiple issues |
7.6, 7.7 | Notes | Notes |

Lecture 7 (1/24) | Direct comparison for improper integrals | 7.6, 7.7 | Notes More notes on problem/bystander |
Notes |

Lecture 8 (1/26) | Geometric sums and series | 9.2 | Notes | Notes |

Lecture 9 (1/29) | Review for midterm | — | Notes | Notes |

Lecture 10 (1/31) | Review for midterm | — | — | — |

Lecture 11 (2/02) | Sequences and series | 9.1, 9.3 | Notes | Notes |

Lecture 12 (2/05) | Divergence and integral tests | 9.3 | Notes | Notes |

Lecture 13 (2/07) | Comparison tests | 9.4 | Notes | Notes |

Lecture 14 (2/09) | Ratio test | 9.4 | Notes | Notes |

Lecture 15 (2/12) | Absolute convergence, alternating series test | 9.4 | Notes | Notes |

Lecture 16 (2/14) | Convergence test overview, which test to use when? | 9.4 | Notes 20 series' solutions |
Notes |

Lecture 17 (2/16) | Power series, intervals of convergence | 9.5 | Notes | Notes |

**Midterm 1**

- Midterm 1 is scheduled for 7:00–8:30 PM on Wednesday, January 31st, in Cubberley Auditorium.
- There will be assigned seating in the Main (Cubberley) Exam: Seating Chart
- Review session: 5:30–7:30 on Monday, January 29th, in Hewlett 200, hosted by Adva Wolf.
- Midterm 1 expecations (tells you what is and is not testable material)
- Practice midterms (note that these are exams from previous quarters; material that we haven't yet covered and which you cannot be tested on is highlighted in red)—solutions will be posted later on:
- Practice problems from the book:
- 7.6 (Improper Integrals): 5, 7
^{a}, 9, 11^{b}, 13, 15, 19^{c}, 21^{c}, 25^{c} - 7.7 (Comparison for Improper Integrals): 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25
- Ch. 7 Review: 141, 143, 145, 147, 149
^{c}, 151, 153^{d}, 155, 157^{e} - 9.2 (Geometric Series): 1, 3, 5, 19, 21, 23, 25

^{a}Evaluating this requires L'Hôpital's rule (it is an example in the improper integrals summary notes). This integral is worth memorizing.^{b}Requires integration by parts, so is not exam-inable.^{c}Requires the use of a table entry (which would be provided on an exam).^{d}Remember that tangent = sine / cosine.^{e}On an exam you would be provided the inequality 0 ≤ sin(*x*) ≤*x*when 0 ≤ x ≤ 1. - 7.6 (Improper Integrals): 5, 7
- Copy of the midterm*, and solutions*.

(*Both the copy and the solutions are one of two forms. Other than one typographical error, there were no*significant*differences between the forms.)

**Midterm 2**

- Midterm 2 is scheduled for 7:00–8:30 PM on Wednesday, February 28th, in Cubberley Auditorium.
- There will be assigned seating in the Main (Cubberley) Exam: Seating Chart will be posted 2/27 PM.
- Review session: TBD hosted by Ben Lim.
- Midterm 2 expecations (tells you what is and is not testable material).
- Practice midterms (note that these are exams from previous quarters; material that we haven't yet covered and which you cannot be tested on is highlighted in red)—solutions will be posted later on:
__Note:__All redacted problems in Practice Midterms 1A–C now cover testable material for Midterm 2 (exception: A's Problem 3).

It is likely that at least one of these will appear on your exam, verbatim, so they're worth reviewing.- Practice Midterm A
- Practice Midterm B
- Practice Midterm C

- Practice problems from the book (those before the // are mostly straight-up convergence problems, those afterwards are more involved and/or conceptual in nature—only the
*odd*problems are given; the answers are in the back of the book):- Section 9.1: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 31, 63, 65, 67, 69, 71, 73, 75
- Section 9.2: 1, 3, 5, 19, 21, 23, 25
- Section 9.3: 13‐31 // 53, 55, 57, 61, 63
- Section 9.4: 61–65, 67
^{a}, 69, 71, 73^{b}, 75, 77 // 97–107, 113^{b}, 115, 117 - Section 9.5: 11–17, 27–37 // 45abd, 55, 61, 65
- Ch. 9 Review: 13, 15, 37–53, 57, 63, 65

^{a}tan(1/x) is asymptotic to 1/x.^{b}Remember that absolute convergence implies convergence.

**Final Exam**

**Extra Help**

- The official Math 21 Piazza page
- Tutoring: VPTL individual appointments (free), VPTL drop-in tutoring (free)

**Important Dates**

~~(1/08) First day!~~~~(1/12) Homework 0 due (optional).~~~~(1/15) Martin Luther King Jr. Day (no classes or office hours).~~~~(1/19) Homework 1 due.~~~~(1/26) Add/drop deadline (5 PM).~~~~(1/29) Homework 2 due.~~~~(1/31) Midterm 1 (7:00–8:30 PM, in Cubberley Auditorium).~~~~(2/12) Homework 3 due.~~~~(2/19) Presidents' Day (no classes or office hours).~~~~(2/19) Homework 4 due.~~- (2/26) Homework 5 due.
- (2/28) Midterm 2 (7:00–8:30 PM, in Cubberley Auditorium).
- (3/02) Course withdrawal and change of grading basis deadline (5 PM).
- (3/09) Homework 6 due.
- (3/16) Homework 7 due.
- (3/19) Final Exam (7:00–10:00 PM, location TBA).