# Improper Integrals Homework Assignments

## Calculus II

**Course:** Math 142, Section B (Fall 2017)

**Time & Place:** TR 10:10 – 11:30 am in Hegeman 106

**Instructor:** Jim Belk (belk@bard.edu)

**Office Hours:**

- Thursday 7–9 pm (RKC 100)
- Friday 4–6 pm (RKC 100)

**Course Tutor:** Alexis Zheng (hz1459@bard.edu)

**Math Study Room:** Sun–Thurs, 7–10 pm (RKC 111)

## Announcements

#### Final Exam

The final exam is Thursday, December 21 at the following times:- 10 am – 12 pm: 3rd floor Albee
- 12 pm – 6 pm: Albee 100

**calculator**to the exam. You may also bring an 8.5" × 10" sheet of paper with notes written on the front and back.

#### Extra Office Hours

I will be having the following office hours this week.- Tuesday 7–9 pm (RKC 100)
- Wednesday 3–5 pm (3rd floor Albee)
- Wednesday 7–9 pm (RKC 100)

#### Final Exam Practice Problems

Here are some practice problems for the final exam: Here are the solutions:#### Quiz Solutions

Here are the solutons to the recent quizzes:#### Homework Solutions

Here are the solutons to homeworks 6 and 7:#### Estimating Double Integrals

Here are some practice problems on estimating double integrals numerically: Answers are on the second page.#### Homework 9 (Revised)

The last written homework assignment is due this Friday, December 15. Here is the assignment: Note that there has been a slight revision to part (b) of the last question.#### Double Integrals

Here are some practice problems on double integrals over rectangles:- Section 12.2 # 3, 5, 9, 17, 19, 21

- Section 12.3 # 1, 3, 9, 21, 23, 25

#### Homework 8

The eighth written homework assignment is due this Friday, December 8. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Areas and Volumes

Here are some practice problems on areas and volumes:- Section 6.1 # 3, 5, 9, 17, 23
- Section 6.2 # 1, 3, 5, 7, 13, 25

#### Homework 7

The seventh written homework assignment is due this Friday, December 1. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Quiz Tuesday

There will be a long (2 page) quiz this**Tuesday**, November 21. It will cover integration odds and ends and improper integrals. (See the practice problems below.)

#### Improper Integrals (with answers)

Here are some practice problems on improper integrals Answers are on the second page.#### Integration Odds and Ends

Here are some problems on some of the aspects of integration that we've been discussing recently:- Section 5.2 # 35, 37, 41, 43
- Section 5.4 # 7, 9
- Section 5.5 # 49, 50, 59, 60
- Section 6.1 # 1, 13

#### No Homework This Week

There is no written homework due this week.#### Homework 6

The sixth written homework assignment is due this Friday, November 10. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Advanced Substitution and Integration by Parts

Here are some harder problems on substitution and some problems on integration by parts:**Substitution:**Section 5.5 # 17, 31, 51**Integration by Parts:**Section 5.6 # 1, 3, 7, 11, 19, 25, 27, 29

#### Basic Substitution Problems

Here are the basic substitution problems that you did last week:**Indefinite Integrals:**Section 5.5 # 1, 3, 5, 7, 13, 21**Definite Integrals:**Section 5.5 # 43, 45, 47

#### Midterm Exam Friday

The midterm exam is this**Friday**from 9 am to 6 pm in RKC 101. Remember:

- • The midterm is designed to be about 2 hours long, but there is no time limit.
- • You should bring a
**calculator**to the midterm. - • You can bring a 3" × 5" index card to the midterm with anything you want written on the front and back.

#### Office Hours Tonight

I will be having extra office hours**tonight**(Wednesday) from 7 pm to 9 pm in RKC 100.

#### Midterm Practice Problems

Here are some practice problems for the midterm exam: I've written solutions to these practice problems:#### Homework Solutions

Here are the solutions to the first four homework assignments:#### Quiz Solutions

Here are the solutons to the first five quizzes:#### Paper on Functions without Elementary Antiderivatives

For those students who are interested, here's an expository article from the*College Mathematics Journal*on the mathematics of proving that certain functions have no elementary antiderivatives:

#### Practice Problems: Integrals and Area

Here are some practice problems on integrals and area. This material will be covered on the midterm exam. Answers are on the second page.#### Quiz Thursday

There will be a quiz this**Thursday**, October 19. All of the questions will involve evaluating integrals using the fundamental theorem of calculus (see the exercises below), which is covered in Section 5.3 of the textbook.

#### Practice Problems: Fundamental Theorem of Calculus (with Answers)

Here are some practice problems on the fundamental theorem of calculus: Answers are on the second page.#### Homework 5

The fifth written homework assignment is due this Friday, October 20. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Spreadsheet from Class

Here is the Google spreadsheet we used in class: In case it's helpful, I've also made an Excel version of this spreadsheet:#### Homework 4

The fourth written homework assignment is due this Friday, October 13. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Quiz 5

The fifth quiz is this Thursday, October 12th. It covers critical points and the second derivative test. See the practice problems below.#### Practice Problems: Critical Points

Here are some practice problems on critical points and the second derivative test.- Section 11.7 # 1, 3, 5, 7, 9, 13

#### Quiz 4

The fourth quiz will be this Tuesday, October 3. It will cover higher derivatives, functions of three or more variables, and partial differential equations. See the practice problems below.#### Practice Problems: Higher Derivatives

Here are some practice problems on second and higher partial derivatives:- Section 11.3 # 51, 53, 55, 59, 61, 63

#### Practice Problems: Functions of Three or More Variables

Here are some practice problems on functions of three or more variables:- Section 11.3 # 1, 29, 31, 35, 41, 79
- Section 11.4 # 19

#### Practice Problems: Partial Differential Equations

If you haven't seen differential equations before, you might want to start with a couple of problems on ordinary differential equations: Now here are some practice problems on partial differential equations:- Section 11.3 # 69, 70 (answer)

#### Homework 3

The third written homework assignment is due this Friday, September 29. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Quiz 3

The third quiz was on Tuesday, September 26. It covered partial derivatives. See the practice problems below.#### Practice Problems: Partial Derivatives

Here are some practice problems on partial derivatives:- Section 11.3 # 3, 15, 17, 19, 21, 23, 25, 27, 39
- Section 11.4 # 1, 3, 5, 17

#### Homework 2

The second written homework assignment is due this Friday, September 22. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Quiz 2

The second quiz was on Tuesday, September 19. It covered cross-sections, contour plots, and equations of planes. See the practice problems below.#### Practice Problems: Equations of Planes

Since the book doesn't cover equations of planes very well, I've written some brief notes on the subject as well as some practice problems. The answers to the exercises are on the second page of the PDF.#### Practice Problems: Cross-Sections and Contour Plots

Here are some practice problems on cross-sections:- Section 9.6 # 18 (answer)
- Section 11.1 # 9, 11, 14 (answer), 19, 23, 35, 37, 45

#### Homework 1

The first written homework assignment is due this Friday, September 15. Here is the assignment: You should feel free to work together with other students on the assignment, but everyone must turn in their own copy of the solutions, and you must acknowledge any collaborators. I will be having office hours this Thursday and Friday to help with the homework.#### Cross-Sections Animation

Here is the animation of cross sections that I showed in class yesterday:#### Practice Problems: Multivariable Functions

Here are some practice problems on multivariable functions:- Section 9.6 # 9, 13, 15, 23

#### Reading

Please read sections 9.6 and 11.1, and 11.3 before next Tuesday's class.#### Derivative Practice Problems

Here are some practice problems on basic derivatives. They are from the Chapter 3 Review on pg. 248. (The exercises start at the bottom, after the “Concept Check” and the True-False Quiz.)- Chapter 3 Review Exercises: 1, 3, 5, 7, 9, 11, 17, 25, 27, 31, 41

#### Review Videos

You might find the following videos helpful for reviewing derivatives:#### Welcome!

Welcome to Math 142. I will be using this course webpage to post all announcements and documents, including homework assignments, homework solutions, and practice exams.## Textbook Information

The textbook is*Calculus: Concepts and Contexts, 4th edition*by James Stewart. You will need a copy of the textbook for reading and homework problems, though you do not need to bring it to class. An electronic version of the book can be rented for $24 from the publisher, or you can buy the Kindle version for $105 from Amazon.com. You can also rent a hardcover version from Amazon.com for $90.

## Course Policies

#### Introduction

Math 142, second-semester calclus, will continue to explore the fundamental ideas of differentiation and integration that you learned in Calculus I. Topics to be covered include functions of several variables, partial derivatives, integrals of single-variable functions, techniques of integration, improper integrals, applications of integration, and multiple integrals.#### Prerequisites

This course is designed for students who have completed Calculus I (Math 141), or who have taken a calculus course in high school. We will be assuming basic knowledge of derivatives (including the product rule and chain rule) and integrals.#### Textbook

The text is*Calculus: Concepts and Contexts*, 4th edition, by James Stewart. We will cover most of the material in chapters 5, 6, 11, and 12, as well as a few topics from elsewhere in the text. You are encouraged to read the text

*before*coming to class, and then again while doing the homework.

#### Written Homework

There will be a weekly written homework assignment due each Friday consisting of a few longer problems, with an emphasis on the conceptual side of calculus. You are encouraged to work with other students in solving the homework problems, but you should write your own solutions, and you must acknowledge anyone that you work with. Your solutions should be written clearly and in complete sentences, with enough detail that another student in the class would be able to follow your reasoning.#### Practice Problems and Quizzes

Computation is an important component of mathematics, and is a key part of any calculus course. I will often recommend practice problems from the textbook, and you are strongly advised to try at least some of these problems. To make sure your computational skills are progressing, there will be a short (10–20 min.) quiz each Tuesday consisting of a few relatively straightforward computational problems.#### Exams and Grading

The grade will be based on the weekly homework, quizzes, the midterm exam, and the final exam:Written Homework | 25% |

Quizzes | 25% |

Midterm Exam | 25% |

Final Exam | 25% |

The midterm exam is two hours long, and will be offered on **Friday, October 27**. The final exam is two hours long, and will be on **Thursday, December 21**.

#### Calculator

You will need a calculator for the quizzes and the exams. Any calculator is permitted, and even a scientific calculator app on a smartphone would suffice.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**

**Note:**You may attend any of our our office hours (O.H.) as listed below.

*However, please only attend the lecture and discussion section in which you are enrolled or plan to enroll.*

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

*front*entrance (i.e., the one which faces out towards Serra Mall and The Oval).

**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__. (Solutions) - Homework 6, due Friday, Mar. 9th by 11 AM
__on Gradescope__. - Homework 7, due Friday, Mar. 16th by 11 AM
__on Gradescope__.**YOU WILL WANT TO READ THE DIRECTIONS**

- 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

__sometimes__upload lecture notes throughout the quarter.

*WARNINGS: These notes may not be complete/detailed (since they're for us), may differ from actual lecture material, and are NOT a substitute for attending lecture.*

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.

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 |

Lecture 18 (2/21) | Power series representations | — (~10.3) | Notes | Notes |

Lecture 19 (2/23) | Power series representations | — (~10.3) | Notes | Notes |

Lecture 20 (2/26) | Review for midterm | — | Notes Solutions | Notes |

Lecture 21 (2/28) | Review for midterm | — | — | — |

Lecture 22 (3/02) | Taylor series (general method, sin(x) and cos(x)) | 10.2 | Notes | Notes |

Lecture 23 (3/05) | Taylor series (binomial series (x+1)^{p}), Taylor polynomials | 10.2, 10.1 | Notes | Notes |

Lecture 24 (3/07) | Manipulating of Taylor series Applications of Taylor series | 10.3 | Notes | Notes |

Lecture 25 (3/09) | Applications of Taylor series | 10.3 | Notes | |

Lecture 26 (3/12) | Review for final | — | — | 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}

- 7.6 (Improper Integrals): 5, 7
- 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.

(*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
- Review session: 6:30–8:00 PM on Tuesday in 260-113, 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:
- 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. - Copy of the midterm*, and solutions*.

(*Both the copy and the solutions are one of two forms. There were no significant differences between the forms.)

**Final Exam**

- The final exam is scheduled for 7:00–10:00 PM on Monday, March 19th, in 420-40 and 420-41.
- There will be assigned seating, TBA.
- Final exam expecations (tells you what is and is not testable material).
- Breadth questions.
- Practice finals (note that these are exams from previous quarters; material that we did not cover (and for which you are not responsible) is highlighted in red):

**Extra Help**

**Important Dates**

- (3/09) Homework 6 due.
- (3/16) Homework 7 due.
- (3/19) Final Exam (7:00–10:00 PM, location TBA).

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