**Campus: **Ohio University, Athens Campus

**Department: **Mathematics

**Course Description: **First course in calculus and analytic geometry with applications in the sciences and engineering. Includes basic techniques of differentiation and integration with applications including rates of change, optimization problems, and curve sketching; includes exponential, logarithmic and trigonometric functions. Calculus is the mathematical language used to describe and analyze change. The course emphasizes how this abstract language and its associated techniques provide a unified way of approaching problems originating in disparate areas of science, technology, and society, highlighting how questions arising in different fields are connected to the same fundamental mathematical ideas. No credit for both MATH 2301 and 1350 (always keep 2301).

**Prerequisites: **(B or better in MATH 1350) or (C or better in 1300 or 1322) or (Math placement level 3)

**Meeting Times and Locations:**

- Lecture Section 100 meets Mon Wed, Fri 8:35am – 9:30m in Morton Hall 235.

- Recitation Section 101 meets Tue 8:00am – 8:55am in Morton 318
- Recitation Section 102 meets Tue 9:30am – 10:25am in Ellis 107
- Recitation Section 103 meets Tue 12:30pm – 1:25pm in Morton 122
- Recitation Section 104 meets Tue 2:00pm – 2:555pm in Morton 318

- Lecture Section 110 meets Mon Wed, Fri 10:45am – 11:40am in Morton Hall 237.

- Recitation Section 111 meets Tue 9:30am – 10:25am in Morton 218
- Recitation Section 112 meets Tue 11:00am – 11:55am in Ellis 107
- Recitation Section 113 meets Tue 2:00pm – 2:55pm in Morton 126
- Recitation Section 114 meets Tue 3:30pm – 4:25pm in Morton 122

**Information about the Instructors: **

**Instructor for Lecture Sections 100 and 110:** Mark Barsamian

**Office Location:**Morton 521**Office Hours:**Mon, Wed, Fri 1:00pm – 2:00pm (No appointment necessary)**Office Phone:**740-593-1273**Email:**barsamia@ohio.edu

**Instructor for Recitation Sections 101, 102, 103, 104:** Isaac Agyei

**Office Location:**Morton 532**Office Hours:**Mon, Wed 3:00pm – 4:00pm (No appointment necessary)**Office Phone:**XXX**Email:**ia520320@ohio.edu

**Instructor for Recitation Sections 111, 112, 113, 114:** Kenny So

**Office Location:**Morton XXX**Office Hours:**XXX (No appointment necessary)**Office Phone:**XXX**Email:**ks698620@ohio.edu

**Special Needs: **If you have specific physical, psychiatric, or learning disabilities and require accommodations, please let Mark Barsamian know as soon as possible so that your learning needs may be appropriately met. You should also register with the
Office of Student Accessibility Services
to obtain written documentation and to learn about the resources they have available.

**Final Exam Date: **All Athens Campus Sections of MATH 2301 have a Common Final Exam on Thu Dec 14, 2023, from 2:30pm – 4:30pm in various Morton Hall rooms. (Room assignments will be made later.)

**Syllabus: **This web page replaces the usual paper syllabus. If you need a paper syllabus (now or in the future), unhide the next four portions of hidden content (Textbook Information, Exercises, Grading, Calendar) and then print this web page.

**Textbook and WebAssign Information: **

**Required Online Course Materials: ** Through a program called *Inclusive Access*, the University has negotiated with the publisher a special price for this course's *Required Online Course Materials*. On the first day of class, you will receive access to an an online system called *WebAssign*. The *WebAssign* system includes an *eText* version of the textbook and an *online homework system*. The cost of the Online Course Materials is a discounted Inclusive Access Price of $45 plus 7% Ohio sales tax, for a total of about $48.15. That cost will be automatically billed to your Ohio University Student Account. If you drop the course before the drop deadline (Fri, Sep 8, 2023), your student account will be credited for any amount billed. After you register, you will receive more information about the Inclusive Access program, including an option to "Opt Out" of participation in the program. To "Opt Out" means that your payment for the Online Course Materials is not handled by the Inclusive Access program. If you do that, you can still use the Online Course Materials, but in order to access them, you will be asked to make a credit card payment for the Retail Price of the materials. (Note that the Retail Price is $111 plus 7% ohio tax, for a total of about $118.77. That is significantly higher than the Inclusive Access Price.)

**Optional Print Copy of the Textbook: ** Many students (and instructors) prefer reading printed textbooks rather than eTexts. Students in Ohio University MATH 2301 Sections 100 and 110 can purchase a print copy of the book at the
**College Bookstore**
(at the corner of Court Street and Union Street in Athens) for the discounted price of $33.50 + 7% Ohio sales tax, for a total of around $35.85. This is an extraordinarily low price for a print textbook, and you are strongly encouraged to buy the print copy. Note that your purchase of the print copy will be **in addition to** the *Online Course Materials* that you receive as part of the *Inclusive Access* program, described above. So if you do buy the print copy, your total expenditures will be $48.15 (for the *Online Course Materials* purchased through the *Inclusive Access* program) plus $35.85 (for the print copy of the textbook, purchased at the College Bookstore) for a total of $84. That is still an excellent price for course materials. The print copy is a loose-leaf book; its full description is:

**Title:**Essential Calculus, Early Transcendentals, Second Edition, Loose-Leaf Edition**Author:**James Stewart**Publisher:**Cengage (2012)**ISBN:**9780357005262**Available at:****College Bookstore**at the corner of Court Street and Union Street in Athens

**Exercises: **

(from Stewart Essential Calculus Early Transcendentals 2nd Edition)

Your goal should be to write solutions to all 333 exercises in this list.

Printable PDF of the Exercise List

- 1.3 The Limit of a Function: 1, 5, 7, 10, 11, 12, 13, 15, 23, 33, 39
- 1.4 Calculating Limits: 5, 7, 10, 11, 17, 21, 23, 25, 27, 31, 33, 35, 38, 42, 49, 51, 55
- 1.5 Continuity: 3, 5, 7, 17, 19, 27, 33, 39, 43, 47
- 1.6 Limits Involving Infinity: 1, 5, 7, 9, 10, 13, 19, 21, 25, 29, 33, 35, 40, 41, 45, 49
- 2.1 Derivatives & Rates of Change: 1, 5, 9, 11, 15, 16, 18, 25, 27, 29, 31, 33, 35, 43, 47
- 2.2 The Derivative as a Function: 1, 3, 5, 9, 11, 13, 19, 20, 22, 23, 25, 33, 35, 39
- 2.3 Basic Differentiation Formulas: 1, 7, 9, 11, 13, 19, 27, 29, 31, 33, 35, 37, 39, 45, 50, 57, 69
- 2.4 The Product & Quotient Rules: 3, 5, 7, 13, 16, 17, 19, 21, 26, 27, 31, 34, 37, 41, 51, 55
- 2.5 The Chain Rule: 1, 7, 13, 14, 17, 21, 25, 35, 43, 47, 51, 55, 63, 64
- 2.6 Implicit Differentiation: 5, 7, 9, 11, 13, 19, 21
- 2.7 Related Rates: 4, 5, 11, 13, 15, 20, 23, 25, 27, 28, 31
- 2.8 Linear Approx & Differentials: 1, 5, 6, 11, 13, 17, 19, 21, 23
- 3.1 Exponential Functions: 1, 5, 7, 9, 13, 15, 16, 17, 27, 29, 30
- 3.2 Inverse Functions, Logarithms: 5, 7, 9, 11, 15, 17, 18, 22, 23, 25, 35, 36, 39, 67, 71, 76
- 3.3 Derivs of Log. & Exp. Functs.: 1, 3, 4, 6, 13, 20, 26, 31, 35, 41, 45, 55, 57
- 3.4 Exponential Growth & Decay: 1, 2, 3, 9, 13, 16
- 4.1 Maximum & Minimum Values: 5, 9, 18, 19, 21, 25, 29, 35, 39, 43, 47, 49
- 4.2 The Mean Value Theorem: 1, 3, 5, 7, 9, 11, 13, 15, 17, 23, 25
- 4.3 Derivs. & Shapes of Graphs: 1, 5, 7, 10, 13, 15, 19, 23, 27, 35, 37, 45
- 4.4 Curve Sketching: 1, 9, 11, 13, 15, 19, 31, 34, 39
- 4.5 Optimization Problems: 2, 7, 11, 15, 17, 22, 25, 30, 37, 39, 53, 57
- 4.6 Newton's Method: 4, 7, 9, 11, 13
- 4.7 Antiderivatives: 1, 2, 7, 12, 13, 15, 20, 27, 38, 40, 47, 53, 55
- 5.1 Areas and Distances: 2, 3, 4, 5, 9, 13, 16, 18
- 5.2 The Definite Integral: 1, 3, 9, 11, 15, 25, 30, 33, 35, 39, 40, 44
- 5.3 Evaluating Definite Integrals: 3, 7, 11, 18, 26, 29, 49, 51, 56, 59, 61, 65, 69
- 5.4 The Fund. Thm. of Calculus: 1, 3, 5, 10, 15, 25, 27
- 5.5 The Substitution Rule: 7, 11, 13, 17, 19, 23, 26, 27, 33, 37, 39, 44, 50, 53, 55, 61

**A Suggestion for Studying: **Even though *WebAssign* does not require that you write stuff down, you will learn a lot by focusing on your writing. Furthermore, having good writing skills will really help when working on a written Quiz or Exam. Therefore, you should write down a complete solution to each problem *before* you type the answer into the answer box in *WebAssign*. Focus on the clarity and correctness of your written solution. Keep your written work organized in a notebook. Compare your written solutions to my written solutions in lectures. Find another student, or a tutor, or your Recitation Instructor, or Mark Barsamian, to look over your written work with you.

**Grading: **

During the course, you will accumulate a * Points Total* of up to

**WebAssign:**28 Assignments @ 1 point each = 28 points possible (Extra Credit points)**Recitation:**15 Tuesday Recitation Activities @ 5 points each = 75 points possible**Quizzes:**Best 8 of 9 Quizzes @ 30 points each = 240 points possible**Exams:**Best 2 of 3 Exams @ 220 points each = 440 points possible**Final Exam:**245 points possible

At the end of the semester, your * Points Total* will be divided by \(1000\) to get a percentage, and then converted into your

Observe that the **Total Possible Points** is \(1028\), but your points total is divided by \(1000\) to get the percentage that is used in computing your course grade. This is because the \(28\) points that can be earned by doing **WebAssign Homework** are considered **Extra Credit Points**.

The **90%, 80%, 70%, 60% Grading Scale** is used on all graded items in this course, and is used in computing your * Course Letter Grade*.

- A grade of
**A, A-**means that you mastered all concepts, with no significant gaps.- If \(93\% \leq score \), then
*letter grade*is**A**. - If \(90\% \leq score \lt 93\%\), then
*letter grade*is**A-**.

- If \(93\% \leq score \), then
- A grade of
**B+, B, B-**means that you mastered all essential concepts and many advanced concepts, but have some significant gap.- If \(87\% \leq score \lt 90\%\), then
*letter grade*is**B+**. - If \(83\% \leq score \lt 87\% \), then
*letter grade*is**B**. - If \(80\% \leq score \lt 83\%\), then
*letter grade*is**B-**.

- If \(87\% \leq score \lt 90\%\), then
- A grade of
**C+, C, C-**means that you mastered most essential concepts and some advanced concepts, but have many significant gaps.- If \(77\% \leq score \lt 80\%\), then
*letter grade*is**C+**. - If \(73\% \leq score \lt 77\%\), then
*letter grade*is**C**. - If \(70\% \leq score \lt 73\%\), then
*letter grade*is**C-**.

- If \(77\% \leq score \lt 80\%\), then
- A grade of
**D+, D, D-**means that you mastered some essential concepts.- If \(67\% \leq score \lt 70\%\), then
*letter grade*is**D+**. - If \(63\% \leq score \lt 67\% \), then
*letter grade*is**D**. - If \(60\% \leq score \lt 63\%\), then
*letter grade*is**D-**.

- If \(67\% \leq score \lt 70\%\), then
- A grade of
**F**means that you did not master essential concepts.- If \(0\% \leq score \lt 60\%\), then
*letter grade*is**F**.

- If \(0\% \leq score \lt 60\%\), then

**Attendance:**Attendance is recorded but is not part of your course grade**Written Solutions to Homework Exercises:**There is a list of Homework Exercises on this web page. To succeed in the course, you will need to do lots of them (preferrably*all*of them), writing the solutions on paper. Those written solutions are not graded and are not part of your course grade. (Your scores on the online*WebAssign*homework*will*be part of your course grade.)

Use this to calculate your ** Current Letter Grade** before the Final Exam: Grade Calculation Worksheet

Attendance is required for all class meetings, and your attendance (or absence) will be recorded, but attendance is not used in the calculation of your course grade.

**Missing Class: **If you miss a class for any reason, it is your responsibility to learn the stuff that you missed. You can do this by studying a classmate's notes, or reading the Lecture notes that Mark Barsamian posts online, and by reading the textbook. Your Instructurs will not use office hours to teach topics discussed in class meetings to students who were absent.

**Missing a Quiz or Exam Because of Illness: **If you are too sick to take a quiz or exam, then you must do these three things:

- Send Mark Barsamian an e-mail
**before**the quiz/exam, telling him that you are going to miss it because of illness. He will arrange for a date and time for a Make-Up quiz/exam. (Generally, the Make-up for a Friday quiz/exam needs to take place on the following Monday or Tuesday. Therefore, it is important to communicate with him right away.( - Go to the Hudson Student Health Center (or some other Medical Professional) to get examined.
- Later, you will need to bring your Mark Barsamian your documentation from the Hudson Student Health Center (or a Medical Professional) showing that you were treated there.

(Observe that **self-diagnosis** of an illness is not a valid documentation of an illness. In other words, you can't just tell Mark Barsamian that you did not come to a Quiz or Exam because you were not feeling well, and expect to get a Make-Up Quiz or Exam. If you are too sick to come to a Quiz or Exam, then you should be sick enough to go to a medical professional to get diagnosed and treated.)

**Missing Quizzes or Exams Because of University Activity: **If you have a University Activity that conflicts with one of our quizzes or exams, you must contact Mark Barsamian well before the quiz or exam to discuss arrangements for a make-up. They will need to see documentation of your activity. If you miss a quiz or an exam because of a University Activity without notifying Mark Barsamian in advance, you will not be given a make-up.

**Missing Quizzes or Exams Because of Religious Observation: **The Ohio University Faculty Handbook states the following:

*Students may be absent for up to three days each academic semester to take time off for reasons of faith or religious or spiritual belief system or participate in organized activities conducted under the auspices of a religious denomination, church, or other religious or spiritual organization. Faculty shall not impose an academic penalty because of a student being absent nor shall faculty question the sincerity of a student's religious or spiritual belief systems. Students are expected to notify faculty in writing of specific dates requested for alternative accommodations no later than fourteen days after the first day of instruction.*

For MATH 2301, this means that if you will be missing any Fall 2023 Quizzes or Exams for religious reasons, and if you want to have a Make-Up Quiz/Exam, **you will need to notify Mark Barsamian no later than Monday, September 11, 2023**. You and he will work out the dates/times of your Make-Up Quiz/Exam. (In general, if you are going to miss a Friday Quiz/Exam, your Ihe will schedule you for a Make-Up on the following Monday or Tuesday.)

**Missing Presentations, Quizzes, or Exams Because of Personal Travel: **This course meets on Mondays, Wednesdays and Fridays, and attendance is required. Your Personal Travel (to home for the weekend, or out of town for vacations, etc) should be scheduled to not conflict with those Monday/Wednesday/Friday meetings. If you miss a Recitation, Quiz, or Exam because of Personal Travel (not an Offical University Activity), you will not be given a make-up. (When you miss a Quiz or Exam and are not given a Make-Up, the missed Quiz or Exam will be considered your one Quiz or Exam score that gets dropped.)

Policy for Electronic communication between MATH 2301 Students and Instructors

- Electronic communication between MATH 2301 Students and Instructors should be done using one of these two methods:
- The
**Official Ohio University e-mail system**. That is, communications should use email addresses ending in*@ohio.edu*. In other words, send your emails from your OU e-mail account, and address them to a recipient's OU e-mail address. (Students: If you use the**Blackboard**system to send an email to your Instructor, this is automatically taken care of.) - The Teams program. (Teams can be used for
**chat**,**voice calling**,**video calling**, and**video meetings**. It is remarkably powerful.

- The
- Do not use a personal email address (such as a gmail address) when sending an email, and do not send emails to a personal email address (such as gmail).
**Students and Instructors should not communicate via***text*messages.**Students and Instructors: It is your reponsibility to check your OU e-mail every day.**(Students: If you are communicating with your Instructor about a time-sensitive issue, such as trying to schedule a Make-Up Quiz or Exam after an illness, your e-mail replies need to be swift. It is not acceptable to let days pass before replying to an important e-mail message, with your excuse being that you had not checked your OU email. If you do this, you will lose the opportunity to have a Make-Up Quiz or Exam.)- It is a good practice to use a descriptive Subject line such as
on your email messages. That way, the recipient will know to give the email message high priority.*Regarding MATH 2301 Section XXX* - It is also a good practice to use a greeting such as
on your email messages, and to identify yourself in your message. And use a closing such as*Hi Elon,**Thanks,*

Jeff Bezos

If cheat on a quiz or exam, you will receive a zero on that quiz or exam and your Instructor will submit a report to the Office of Community Standards and Student Responsibility (CSSR).

If you cheat on another quiz or exam, you will receive a grade of F in the course and your Instructor will again submit a report to the CSSR.

**Calendar: **

Items in **red** are graded.

**Mon Aug 28: **Course Intro and Section 1.3: The Limit of a Function (Lecture Notes)

**Tue Aug 29: **Recitation **R01**: Diagnostic Test and Section 1.3: The Limit of a Function
(Sample Problems for Diagnostic Test)
(Class Drill on Limits)

**Wed Aug 30: **Section 1.4: Calculating Limits (Lecture Notes)(Handout of Limit Laws)

**Fri Sep 1: **Section 1.4: Calculating Limits (Lecture Notes)

**Mon Sep 4: **Holiday

**Tue Sep 5: **Recitation **R02**: Calculating Limits (Section 1.4)(Handout Using the Squeeze Theorem) (Squeeze Theorem Worksheet)

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R02** score will be 5/5. (For this Sep 5 Recitation, students will get 5/5 regardless of whether their solutions are correct. In the future, the scoring will be more stringent.) If they do not present a solution, their **R02** score will be 0/5.

Students find their **Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Bansode,Ankita**Section 101 Student #3:**Bedell,Paris**Section 101 Student #4:**Beegan,Caden**Section 101 Student #5:**Brandt,Roman**Section 101 Student #6:**Earl,Claire-Michael**Section 101 Student #7:**Eisnaugle,Ethan**Section 101 Student #8:**Fogwe,Brandt**Section 101 Student #9:**Frometa,Amelia**Section 101 Student #10:**Jackson,Henry**Section 101 Student #11:**Miller,Taylor**Section 101 Student #12:**Robinson,Alana**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,**Section 101 Student #19:**Unassigned,**Section 101 Student #20:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Fritz,Ronan**Section 102 Student #2:**Herrmann,Mary**Section 102 Student #3:**Hoffman,Sidney**Section 102 Student #4:**Hubbard,Grace**Section 102 Student #5:**Lavender,Kinley**Section 102 Student #6:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**Walsh,Carly**Section 102 Student #14:**White,Anna**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,**Section 102 Student #19:**Unassigned,**Section 102 Student #20:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Gilbert,Wyatt**Section 103 Student #4:**Hains,Amanda**Section 103 Student #5:**Hawley,Frank**Section 103 Student #6:**Kennedy,Quinn**Section 103 Student #7:**Martis,Steve**Section 103 Student #8:**Mikin,Reilly**Section 103 Student #9:**Winterton,Jacob**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,**Section 103 Student #19:**Unassigned,**Section 103 Student #20:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,Joseph**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,**Section 104 Student #19:**Unassigned,**Section 104 Student #20:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Mulholland-Flint,Austin**Section 111 Student #11:**Ngum,Venessa**Section 111 Student #12:**Pickens,Charlee**Section 111 Student #13:**Pinson,Caroline**Section 111 Student #14:**Rasmussen,Cubbie**Section 111 Student #15:**Rodean,Alex**Section 111 Student #16:**Sahr,Griffin**Section 111 Student #17:**Sautter,Jack**Section 111 Student #18:**Scudder,Braedon**Section 111 Student #19:**Wright,Beck**Section 111 Student #20:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Lewis-Baranyai,Enzo**Section 112 Student #14:**Massie,Olivia**Section 112 Student #15:**Miller,Austy**Section 112 Student #16:**Newton,Lilly**Section 112 Student #17:**Shields,Julia**Section 112 Student #18:**Smith,Kaitlyn**Section 112 Student #19:**Whittington,Kelsey**Section 112 Student #20:**Williams,Ava

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Sikora,Daniella**Section 113 Student #16:**Slingluff,Cheyenne**Section 113 Student #17:**Wenning,Luke**Section 113 Student #18:**Unassigned,**Section 113 Student #19:**Unassigned,**Section 113 Student #20:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Blore,Noah**Section 114 Student #4:**Cox,Madelyn**Section 114 Student #5:**Dubois,Aleke**Section 114 Student #6:**Elliott,Maggie**Section 114 Student #7:**Hartzell,Molly**Section 114 Student #8:**Kezele,Ashley**Section 114 Student #9:**Lampa,Andrew**Section 114 Student #10:**Mcclellan,Alex**Section 114 Student #11:**Mcdermitt,Brian**Section 114 Student #12:**Meyer,Morgan**Section 114 Student #13:**Morris,Chase**Section 114 Student #14:**Mueller,Maddy**Section 114 Student #15:**Nguyen,Jim**Section 114 Student #16:**Raynewater,Ty**Section 114 Student #17:**Smith,Riley**Section 114 Student #18:**Sobey,Lily**Section 114 Student #19:**Wall,Logan**Section 114 Student #20:**Young,Kiefer

**Students 1,2: **

(This problem is Exercise 1.4#15, similar to Book Section 1.4 Example 2 and similar to an example done in class on Fri Sep 1)

Find the limit

**Students 3,4: **

(This problem is Exercise 1.4#17, similar to Book Section 1.4 Example 4)

**Students 5,6: **

(This problem is Exercise 1.4#25, an example done in class on Fri Sep 1)

**Students 7,8: **

(This problem is Exercise 1.4#21, similar to Book Section 1.4 Example 5 and similar to an example done in class on Fri Sep 1)

Find the limit

**Students 9,10: **

(This problem is Exercise 1.4#23, similar to Book Section 1.4 Example 5 and similar to an example done in class on Fri Sep 1)

Find the limit

**Students 11,12: **

(This problem is Exercise 1.4#38, similar to Book Section 1.4 Example 7 and similar to an example done in class on Fri Sep 1)

Find the limit

**Students 13,14: ** (1.4#33) Given that for all \(x\),
$$4x-9 \leq f(x) \leq x^2-4x+7$$
find the limit
$$\lim_{x\rightarrow 4}f(x)$$

**Students 15,16: **

(This problem is Exercise 1.4#35, similar to Book Section 1.4 Example 9)

Show that

**Students 17,18: **

(This problem is Exercise 1.4#41, similar to Book Section 1.4 Example 10)

Find the limit

**Students 19,20: **

(This problem is Exercise 1.4#51, similar to Book Section 1.4 Example 10)

Find the limit

**Wed Sep 6: **Section 1.5: Continuity (Lecture Notes)(Handout Using The Intermediate Value Theorem) (Intermediate Value Theorem Worksheets)

**Fri Sep 8: **Section 1.6: Limits Involving Infinity (Lecture Notes)(Last Day to Drop Without a W)(Quiz **Q1**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Three Problems, 10 points each, printed on front & back of one sheet of paper
- One problem based on Suggested Exercises from
**Section 1.3**. - Two problems based on Suggested Exercises from
**Section 1.4**.

- One problem based on Suggested Exercises from

**Mon Sep 11: **Section 1.6 Limits: Involving Infinity (Lecture Notes)

**Tue Sep 12: **Recitation **R03**: Calculating Limits Involving Infinity (Section 1.6)

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R02** score will be in the range 1/5 to 5/5 (depending on how whether they have prepared ahead of time). If they do not present a solution, their **R02** score will be 0/5.

Students find their **Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1 and #11:**Allen,Daylen**Section 101 Student #2 and #12:**Bansode,Ankita**Section 101 Student #3 and #13:**Bedell,Paris**Section 101 Student #4: and #14:**Beegan,Caden**Section 101 Student #5 and #15:**Brandt,Roman**Section 101 Student #6 and #16:**Earl,Claire-Michael**Section 101 Student #7 and #17:**Eisnaugle,Ethan**Section 101 Student #8 and #18:**Frometa,Amelia**Section 101 Student #9 and #19:**Jackson,Henry**Section 101 Student #10 and #20:**Miller,Taylor and Robinson,Alana

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1 and #15:**Fritz,Ronan**Section 102 Student #2 and #16:**Herrmann,Mary**Section 102 Student #3 and #17:**Hoffman,Sidney**Section 102 Student #4 and #18:**Hubbard,Grace**Section 102 Student #5 and #19:**Lavender,Kinley**Section 102 Student #6 and #20:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**Walsh,Carly**Section 102 Student #14:**White,Anna

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1 and #9:**Alder,Ethan**Section 103 Student #2 and #10:**Blower,Carsen**Section 103 Student #3 and #11:**Hains,Amanda**Section 103 Student #4 and #12:**Hawley,Frank**Section 103 Student #5 and #13:**Kennedy,Quinn**Section 103 Student #6 and #14:**Martis,Steve**Section 103 Student #7 and #15:**Mikin,Reilly**Section 103 Student #8 and #16:**Winterton,Jacob**Section 103 Student #17:**Unassigned Challenge Problem: Who can do it?!?**Section 103 Student #18:**Unassigned Challenge Problem: Who can do it?!?**Section 103 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 103 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,Joseph**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned**Section 104 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 104 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Mulholland-Flint,Austin**Section 111 Student #11:**Ngum,Venessa**Section 111 Student #12:**Pickens,Charlee**Section 111 Student #13:**Rasmussen,Cubbie**Section 111 Student #14:**Rodean,Alex**Section 111 Student #15:**Sahr,Griffin**Section 111 Student #16:**Sautter,Jack**Section 111 Student #17:**Scudder,Braedon**Section 111 Student #18:**Wright,Beck**Section 111 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 111 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Lewis-Baranyai,Enzo**Section 112 Student #14:**Massie,Olivia**Section 112 Student #15:**Miller,Austy**Section 112 Student #16:**Newton,Lilly**Section 112 Student #17:**Shields,Julia**Section 112 Student #18:**Smith,Kaitlyn**Section 112 Student #19:**Whittington,Kelsey**Section 112 Student #20:**Williams,Ava

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Sikora,Daniella**Section 113 Student #16:**Slingluff,Cheyenne**Section 113 Student #17:**Wenning,Luke**Section 113 Student #18:**Unassigned**Section 113 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 113 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Blore,Noah**Section 114 Student #4:**Cox,Madelyn**Section 114 Student #5:**Dubois,Aleke**Section 114 Student #6:**Elliott,Maggie**Section 114 Student #7:**Hartzell,Molly**Section 114 Student #8:**Kezele,Ashley**Section 114 Student #9:**Lampa,Andrew**Section 114 Student #10:**Mcclellan,Alex**Section 114 Student #11:**Mcdermitt,Brian**Section 114 Student #12:**Meyer,Morgan**Section 114 Student #13:**Morris,Chase**Section 114 Student #14:**Mueller,Maddy**Section 114 Student #15:**Nguyen,Jim**Section 114 Student #16:**Raynewater,Ty**Section 114 Student #17:**Smith,Riley**Section 114 Student #18:**Sobey,Lily**Section 114 Student #19:**Wall,Logan**Section 114 Student #20:**Young,Kiefer

**Similar Problem from Exercise List:**1.6 #13**Similar Book Example:**Section 1.6 Example 1 is similar to part (b)**Similar Class Example:**Fri Sep 8 Example is similar to part (a) and (b)

We are interested in the following three limits: $$\lim_{x\rightarrow -3^-}\frac{x+2}{x+3} \\ \lim_{x\rightarrow -3^+}\frac{x+2}{x+3} \\ \lim_{x\rightarrow -3}\frac{x+2}{x+3}$$

- Find the limits using the
presented in*expanded definition of limit***Section 1.6**. That is, limits can now include the terminology and notation of. The expanded definition of limit is used in*infinity***Section 1.6 Example 2**. Show all details clearly and use correct notation. - What does the result of (a) tell you about the graph of the rational function?

**Similar Problem from Exercise List:**1.6 #13**Similar Book Example:**Section 1.6 Example 1 is similar to part (b)**Similar Class Example:**Fri Sep 8 Example is similar to part (a) and (b)

We are interested in the following three limits: $$\lim_{x\rightarrow 5^-}\frac{x^2-5x+6}{x-5} \\ \lim_{x\rightarrow 5^+}\frac{x^2-5x+6}{x-5} \\ \lim_{x\rightarrow 5}\frac{x^2-5x+6}{x-5}$$

- Find the limits using the
presented in*expanded definition of limit***Section 1.6**. That is, limits can now include the terminology and notation of. The expanded definition of limit is used in*infinity***Section 1.6 Example 2**. Show all details clearly and use correct notation. - What does the result of (a) tell you about the graph of the rational function?

**Similar Problem from Exercise List:**Exercise 1.4#42 is similar to one of the limits in part (a) and (b)**Similar Book Example:**Section 1.6 Example 1 is similar to one of the limits in part (b)**Similar Class Example:**

We are interested in the following three limits: $$\lim_{x\rightarrow 0^-}\left(\frac{1}{x} - \frac{1}{|x|}\right)\\ \lim_{x\rightarrow 0^+}\left(\frac{1}{x} - \frac{1}{|x|}\right)\\ \lim_{x\rightarrow 0}\left(\frac{1}{x} - \frac{1}{|x|}\right)$$

- Find the limits using the
presented in*expanded definition of limit***Section 1.6**. That is, limits can now include the terminology and notation of. The expanded definition of limit is used in*infinity***Section 1.6 Example 2**. Show all details clearly and use correct notation. - What does the result of (b) tell you about the graph of the function?

**Similar Problem from Exercise List:**1.6 # 19**Similar Book Example:**Section 1.6 Examples 5, 11**Similar Class Example:**

- Find the limit of the rational function using the methods of
**Section 1.6 Examples 5,9**. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty}\frac{7x^5-3x^2+13}{4x^5+199x^3-17}$$ - What does the result of (a) tell you about the graph of the rational function?

**Similar Problem from Exercise List:**1.6 # 19**Similar Book Example:**Section 1.6 Examples 5, 11**Similar Class Example:**

- Find the limit of the rational function using the methods of
**Section 1.6 Examples 5,9**. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty}\frac{7x^5-3x^2+13}{4x^6+199x^3-17}$$ - What does the result of (a) tell you about the graph of the rational function?

**Similar Problem from Exercise List:**1.6 # 19**Similar Book Example:**Section 1.6 Examples 5, 11**Similar Class Example:**

- Find the limit of the rational function using the methods of
**Section 1.6 Examples 5,9**. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty}\frac{7x^8-3x^2+13}{4x^5+199x^3-17}$$ - What does the result of (a) tell you about the graph of the rational function?

**Similar Problem from Exercise List:**1.6 # 25**Similar Book Example:**Section 1.6 Example 6**Similar Class Example:**

- Find the limit of the function using the methods of
**Section 1.6 Example 6**. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty} \left( \sqrt{9x^2+x}-3x\right)$$ - What does the result of (a) tell you about the graph of the function?

**Similar Problem from Exercise List:**1.6 # 29**Similar Book Example:**Section 1.6 Examples 7,8**Similar Class Example:**

- Find the limit $$\lim_{x\rightarrow -\infty} \cos{(x)}$$
- What does the result of (a) tell you about the graph of the function?

**Similar Problem from Exercise List:**1.6 # 35**Similar Book Example:****Similar Class Example:**

- Find the horizontal and vertical asymptotes of the rational function. (Give their
and say if they are horizontal or vertical.) Explain how you determined the asymptotes. $$y=\frac{2x^2+x-1}{x^2+x-2}$$*line equations* - Illustrate your results with a sketch of the graph of the function.

**Similar Problem from Exercise List:**1.6 # 40**Similar Book Example:****Similar Class Example:**

- Find a formula for a function that has vertical asymptotes at \(x=2\) and \(x=5\) and horizontal asymptote \(y=3\). Explain how you determined your function.
- Illustrate your results with a sketch of the graph of the function that you found in (a).

**Wed Sep 13: **Section 2.1: Derivatives and Rates of Change
(Lecture Notes)
(Handout on Rates of Change)

**Fri Sep 15: **Section 2.1: Derivatives and Rates of Change
(Lecture Notes)(Quiz **Q2**)
(Handout on Rates of Change)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Three Problems, 10 points each, printed on front & back of one sheet of paper
- One problem based on Suggested Exercises from
**Section 1.5**. - One problem based on Suggested Exercises from
**Section 1.6**. - One problem based on Suggested Exercises from
**Section 1.6**.

- One problem based on Suggested Exercises from

**Mon Sep 18: **Section 2.2: The Derivative as a Function (Lecture Notes)

**Tue Sep 19: **Recitation **R04**: Derivatives and Rates of Change (2.1) and Calculating Derivatives (2.2)

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R02** score will be in the range 1/5 to 5/5 (depending on how whether they have prepared ahead of time). If they do not present a solution, their **R02** score will be 0/5.

Students find their **Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1 and #11:**Allen,Daylen**Section 101 Student #2 and #12:**Bansode,Ankita**Section 101 Student #3 and #13:**Bedell,Paris**Section 101 Student #4: and #14:**Beegan,Caden**Section 101 Student #5 and #15:**Brandt,Roman**Section 101 Student #6 and #16:**Earl,Claire-Michael**Section 101 Student #7 and #17:**Eisnaugle,Ethan**Section 101 Student #8 and #18:**Frometa,Amelia**Section 101 Student #9 and #19:**Jackson,Henry**Section 101 Student #10 and #20:**Miller,Taylor and Robinson,Alana

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1 and #15:**Fritz,Ronan**Section 102 Student #2 and #16:**Herrmann,Mary**Section 102 Student #3 and #17:**Hoffman,Sidney**Section 102 Student #4 and #18:**Hubbard,Grace**Section 102 Student #5 and #19:**Lavender,Kinley**Section 102 Student #6 and #20:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**Walsh,Carly**Section 102 Student #14:**White,Anna

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1 and #9:**Alder,Ethan**Section 103 Student #2 and #10:**Blower,Carsen**Section 103 Student #3 and #11:**Hains,Amanda**Section 103 Student #4 and #12:**Hawley,Frank**Section 103 Student #5 and #13:**Kennedy,Quinn**Section 103 Student #6 and #14:**Martis,Steve**Section 103 Student #7 and #15:**Mikin,Reilly**Section 103 Student #8 and #16:**Winterton,Jacob**Section 103 Student #17:**Unassigned Challenge Problem: Who can do it?!?**Section 103 Student #18:**Unassigned Challenge Problem: Who can do it?!?**Section 103 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 103 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,Joseph**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned**Section 104 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 104 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Mulholland-Flint,Austin**Section 111 Student #11:**Ngum,Venessa**Section 111 Student #12:**Pickens,Charlee**Section 111 Student #13:**Rasmussen,Cubbie**Section 111 Student #14:**Rodean,Alex**Section 111 Student #15:**Sahr,Griffin**Section 111 Student #16:**Sautter,Jack**Section 111 Student #17:**Scudder,Braedon**Section 111 Student #18:**Wright,Beck**Section 111 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 111 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Lewis-Baranyai,Enzo**Section 112 Student #14:**Massie,Olivia**Section 112 Student #15:**Miller,Austy**Section 112 Student #16:**Shields,Julia**Section 112 Student #17:**Smith,Kaitlyn**Section 112 Student #18:**Whittington,Kelsey**Section 112 Student #19:**Williams,Ava**Section 112 Student #20:**Unassigned

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Sikora,Daniella**Section 113 Student #16:**Slingluff,Cheyenne**Section 113 Student #17:**Wenning,Luke**Section 113 Student #18:**Unassigned**Section 113 Student #19:**Unassigned Challenge Problem: Who can do it?!?**Section 113 Student #20:**Unassigned Challenge Problem: Who can do it?!?

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Blore,Noah**Section 114 Student #4:**Cox,Madelyn**Section 114 Student #5:**Dubois,Aleke**Section 114 Student #6:**Elliott,Maggie**Section 114 Student #7:**Hartzell,Molly**Section 114 Student #8:**Kezele,Ashley**Section 114 Student #9:**Lampa,Andrew**Section 114 Student #10:**Mcclellan,Alex**Section 114 Student #11:**Mcdermitt,Brian**Section 114 Student #12:**Meyer,Morgan**Section 114 Student #13:**Morris,Chase**Section 114 Student #14:**Mueller,Maddy**Section 114 Student #15:**Nguyen,Jim**Section 114 Student #16:**Raynewater,Ty**Section 114 Student #17:**Smith,Riley**Section 114 Student #18:**Sobey,Lily**Section 114 Student #19:**Wall,Logan**Section 114 Student #20:**Young,Kiefer

**Students 1,2: ** (2.1#16) Suppose that a function \(g(x)\) is known to have these properties:

- \(g(5)=-3\)
- \(g'(5)=4\)

**Students 3,4: ** (2.1#18) Suppose that the line that is tangent to the graph of a function \(f(x)\) at the point \((4,3)\) also passes through the point \((0,2)\).

- Find \(f(4)\)
- Find \(f'(4)\)

**Students 5,6: ** The graph of a function \(f(x)\) can be shown by clicking on the button below. Also shown is a tangent line and a secant line, with some given points on those lines. (Notice that the graph is not drawn to scale.) Use the graph to answer the questions below. Project the graph on the screen. (If the projection system is not working, draw the graph on the whiteboard.)

- What is the
*Average Rate of Change of \(f(x)\) from \(x=2\) to \(x=7\)*? Explain. - What is \(f'(2)\)? Explain.

**Students 7,8: ** (2.1#1) For the function \(f(x)=4x-x^2\)

- Find the
*slope*of the line tangent to the graph at \(x=1\). Show all details clearly and explain key steps.

**Remark:**When finding derivatives, use the$$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$ That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use*Definition of the Derivative*that you may have learned in previous courses.*Derivative Rules* - Find the
*equation*of the line tangent to the graph at \(x=1\). Show all details clearly and explain key steps. - Sketch a graph of \(f(x)\) and draw the tangent line. Label key points with their \((x,y)\) coordinates.

**Students 9,10: ** (2.1#5) For the function \(f(x)=\sqrt{x}\)

- Find the
*slope*of the line tangent to the graph at \(x=4\). Show all details clearly and explain key steps.

**Remark:**When finding derivatives, use the$$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$ That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use*Definition of the Derivative*that you may have learned in previous courses.*Derivative Rules* - Find the
*equation*of the line tangent to the graph at \(x=4\). Show all details clearly and explain key steps. - Sketch a graph of \(f(x)\) and draw the tangent line. Label key points with their \((x,y)\) coordinates.

**Students 11,12: ** (2.1#1) A ball is thrown into the air. Its height (in feet) after \(t\) seconds is given by the equation
$$y=40t-16t^2$$
Find the *velocity* when \(t=2\). Show all details clearly and explain key steps.

**Remark: **When finding derivatives, use the ** Definition of the Derivative**
$$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$
That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use

**Students 13,14: **(This is the messiest problem. Sorry!) (2.1#27) For the function
$$f(t)=\frac{2t+1}{t+3}$$

- Find \(f'(2)\).

**Remark:**When finding derivatives, use the$$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$ That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use*Definition of the Derivative*that you may have learned in previous courses.*Derivative Rules* - What is the slope of the line tangent to the graph of \(f(t)\) at \(t=2\)? Explain.

**Students 15,16: **(2.2#19) For the function
$$f(x)=3x-5$$

- Find \(f'(x)\) using the
$$f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ That is, build the limit and find its value. (Do not use*Definition of the Derivative*that you may have learned in previous courses.) Show all steps clearly and explain key steps.*Derivative Rules* - What is the slope of the line tangent to the graph of \(f(x)\) at \(x=7\)? Explain, using a graph of \(f(x)\).

**Students 17,18: ** (2.2#22) For the function
$$g(t)=\frac{1}{\sqrt{t}}$$

- Find \(g'(t)\) using the
$$g'(t)=\lim_{h\rightarrow 0} \frac{g(t+h)-g(t)}{x}$$ That is, build the limit and find its value. (Do not use*Definition of the Derivative*that you may have learned in previous courses.) Show all steps clearly and explain key steps.*Derivative Rules* - What is the slope of the line tangent to the graph of \(f(x)\) at \(x=9\)? Explain.

**Students 19,20 ** (2.2#23) For the function
$$g(x)=\frac{1}{x}$$

- Find \(g'(x)\) using the
$$g'(x)=\lim_{h\rightarrow 0} \frac{g(x+h)-g(x)}{x}$$ That is, build the limit and find its value. (Do not use*Definition of the Derivative*that you may have learned in previous courses.) Show all steps clearly and explain key steps.*Derivative Rules* - What is the slope of the line tangent to the graph of \(g(x)\) at \(x=5\)? Explain.

**Wed Sep 20: **Section 2.2: The Derivative as a Function
(Lecture Notes from Section 100 (Isaac Agyei))
(Lecture Notes from Section 110 (Kenny So))

**Fri Sep 22: Exam ****X1** **Covering Through Section 2.2**

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**The Exam will last the full duration of the class period.****No books, notes, calculators, or phones**- Eight problems, 25 points each, printed on front & back of three sheets of paper. All problems are based on suggested exercises.
- A problem about limits, based on suggested exercises from Section 1.3
- A problem about calculating limits, based on suggested exercises from Section 1.4
- A problem using the concept of continuity, based on suggested exercises from Section 1.5
- A problem about calculating a limit involving infinity, based on suggested exercises from Section 1.6
- A problem about calculating a limit involving infinity, based on suggested exercises from Section 1.6
- A problem about derivatives and rates of change, based on suggested exercises from Section 2.1
- A problem about calculating a derivative, based on suggested exercises from Section 2.2
- A problem involving a tangent line, based on suggested exercises from Sections 2.1 and 2.2

**Mon Sep 25: **Section 2.3: Basic Differentiation Formulas (Lecture Notes)

**Tue Sep 26: **Recitation **R05**: Using Basic Differentiation Formulas (Section 2.5)

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R05** score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their **R05** score will be 0/5.

**Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Bansode,Ankita**Section 101 Student #3:**Bedell,Paris**Section 101 Student #4:**Beegan,Caden**Section 101 Student #5:**Brandt,Roman**Section 101 Student #6:**Earl,Claire-Michael**Section 101 Student #7:**Eisnaugle,Ethan**Section 101 Student #8:**Frometa,Amelia**Section 101 Student #9:**Jackson,Henry**Section 101 Student #10:**Miller,Taylor**Section 101 Student #11:**Robinson,Alana**Section 101 Student #12:**Unassigned,**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,**Section 101 Student #19:**Unassigned,**Section 101 Student #20:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Fritz,Ronan**Section 102 Student #2:**Herrmann,Mary**Section 102 Student #3:**Hoffman,Sidney**Section 102 Student #4:**Hubbard,Grace**Section 102 Student #5:**Lavender,Kinley**Section 102 Student #6:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**Walsh,Carly**Section 102 Student #14:**White,Anna**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,**Section 102 Student #19:**Unassigned,**Section 102 Student #20:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Hains,Amanda**Section 103 Student #4:**Hawley,Frank**Section 103 Student #5:**Kennedy,Quinn**Section 103 Student #6:**Martis,Steve**Section 103 Student #7:**Mikin,Reilly**Section 103 Student #8:**Winterton,Jacob**Section 103 Student #9:**Unassigned,**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,**Section 103 Student #19:**Unassigned,**Section 103 Student #20:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,J**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,**Section 104 Student #19:**Unassigned,**Section 104 Student #20:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Mulholland-Flint,Austin**Section 111 Student #11:**Ngum,Venessa**Section 111 Student #12:**Pickens,Charlee**Section 111 Student #13:**Rasmussen,Cubbie**Section 111 Student #14:**Rodean,Alex**Section 111 Student #15:**Sahr,Griffin**Section 111 Student #16:**Sautter,Jack**Section 111 Student #17:**Scudder,Braedon**Section 111 Student #18:**Wright,Beck**Section 111 Student #19:**Unassigned,**Section 111 Student #20:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Lewis-Baranyai,Enzo**Section 112 Student #14:**Massie,Olivia**Section 112 Student #15:**Miller,Austy**Section 112 Student #16:**Unassigned,**Section 112 Student #17:**Shields,Julia**Section 112 Student #18:**Smith,Kaitlyn**Section 112 Student #19:**Whittington,Kelsey**Section 112 Student #20:**Williams,Ava

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Sikora,Daniella**Section 113 Student #16:**Slingluff,Cheyenne**Section 113 Student #17:**Wenning,Luke**Section 113 Student #18:**Unassigned,**Section 113 Student #19:**Unassigned,**Section 113 Student #20:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Blore,Noah**Section 114 Student #4:**Cox,Madelyn**Section 114 Student #5:**Dubois,Aleke**Section 114 Student #6:**Elliott,Maggie**Section 114 Student #7:**Hartzell,Molly**Section 114 Student #8:**Kezele,Ashley**Section 114 Student #9:**Lampa,Andrew**Section 114 Student #10:**Mcclellan,Alex**Section 114 Student #11:**Mcdermitt,Brian**Section 114 Student #12:**Meyer,Morgan**Section 114 Student #13:**Morris,Chase**Section 114 Student #14:**Mueller,Maddy**Section 114 Student #15:**Nguyen,Jim**Section 114 Student #16:**Raynewater,Ty**Section 114 Student #17:**Smith,Riley**Section 114 Student #18:**Sobey,Lily**Section 114 Student #19:**Wall,Logan**Section 114 Student #20:**Young,Kiefer

**Derivative of a Constant Function** If \(c\) is a constant, then
$$\frac{d}{dx}(c)=0$$

**The Power Rule** If \(n\) is any real number, then
$$\frac{d}{dx}\left(x^n \right)=nx^{n-1}$$

**The Sum Constant Multiple Rule** If \(a\) and \(b\) are constants and \(f\) and \(g\) are differentiable functions, then
$$\frac{d}{dx}\left[af(x) +bg(x)\right]=a\frac{d}{dx}f(x)+b\frac{d}{dx}g(x)$$

**The Sine and Cosine Rules (Not discussed in class Monday, but simple enough.)**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$

**Students 1,2 (first problem)(You'll have another problem later.)** (2.3#2) Find the derivative of the function
$$f(x) = \pi^2$$
Show all details clearly and use correct notation.

**Students 3,4 (first problem)(You'll have another problem later.)** (2.3#3) Find the derivative of the function
$$f(t)=2-\frac{2}{3}t$$
Show all details clearly and use correct notation.

**Students 5,6 (first problem)(You'll have another problem later.)** (2.3#4) For the function \(F(x)=\frac{3}{4}x^8\)

- Find \(F(2)\)
- Find \(F'(x)\)
- Find \(F'(2)\)
- Find the
*height*of the graph of \(F(x)\) at \(x=2\). - Find the
*slope*of the graph of \(F(x)\) at \(x=2\).

**Students 7,8 (first problem)(You'll have another problem later.)** (2.3#5) For the function \(f(x)=x^3-4x+6\)

- Find \(F(3)\)
- Find \(F'(x)\)
- Find \(F'(3)\)
- Find the
*height*of the graph of \(f(x)\) at \(x=3\). - Find the
*slope*of the graph of \(f(x)\) at \(x=3\).

**Students 9,10 (first problem)(You'll have another problem later.)** (2.3#7) For the function \(f(x)=3x^2-2\cos{(x)}\)

- Find \(F(\pi)\)
- Find \(F'(x)\)
- Find \(F'(\pi)\)
- Find the
*height*of the graph of \(f(x)\) at \(x=\pi\). (Give an exact answer in symbols, not a decimal approximation.) - Find the
*slope*of the graph of \(f(x)\) at \(x=\pi\). (Give an exact answer in symbols, not a decimal approximation.)

**Students 11,12 (first problem)(You'll have another problem later.)** (2.3#9) Find the derivative of the function
$$g(x)=x^2(1-2x)$$
Show all details clearly and use correct notation.

**Students 13,14 (first problem)(You'll have another problem later.)** For the function \(f(x)=2x^{1/3}\)

- Find \(f(8)\)
**(no calculators!)** - Find \(f'(x)\)
- Find \(f'(8)\)
**(no calculators!)** - Find the
*height*of the graph of \(f(x)\) at \(x=8\). - Find the
*slope*of the graph of \(f(x)\) at \(x=8\).

**Students 15,16 (first problem)(You'll have another problem later.)** (2.3#11) Find the derivative of the function
$$f(t)=\frac{2}{t^{3/4}}$$
Show all details clearly and use correct notation

**Students 17,18 (first problem)(You'll have another problem later.)** (2.3#19) For the function
$$f(x)=\frac{x^2+4x+3}{\sqrt{x}}$$

- Rewrite \(f(x)\) in
. That is, write it in the form $$f(x)=ax^p+bx^q+cx^r$$ where \(a,b,c,p,q,r\) are real numbers.*power function form* - Find \(f'(x)\)

**Students 19,20 (first problem)(You'll have another problem later.)** (2.3#21) For the function
$$v=t^2-\frac{1}{\sqrt[4]{t^3}}$$

- Rewrite the function in
. That is, write it in the form $$v(t)=at^p+bt^q$$ where \(a,b,p,q\) are real numbers.*power function form* - Find \(v'(t)\)

Remember that the **line tangent to the graph of \(f(x)\) at \(x=a\)** is the line that has these two properties

- The line touches the graph of \(f(x)\) at \(x=a\). So the line contains the point \((x,y)=(a,f(a))\), called the
*point of tangency* - The line has slope \(m=f'(a)\)

A new thing, the **line normal to the graph of \(f(x)\) at \(x=a\)**, is the line that has these two properties

- The line touches the graph of \(f(x)\) at \(x=a\). So the line contains the point \((x,y)=(a,f(a))\)
- The line is perpendicular to the line that is tangent to the graph at that point. That is,
- If the tangent line has slope \(m_T\neq 0\), then the normal line has slope $$m_N=-\frac{1}{m_T}$$
- If the tangent line has slope \(m_T = 0\), which indicates that the tangent is
*horizontal*, then the normal line is*vertical*.

**Students 1- 12 (second problem)** (2.3#27) For the function
$$f(x)=2\sin{(x)}$$

**Students 1,2:**Find the equation for the line*tangent to the graph of \(f(x)\) at \(x=\frac{\pi}{3}\)*. Draw the graph and draw your tangent line. Label important stuff.**Students 3,4:**Find the equation for the line*normal to the graph of \(f(x)\) at \(x=\frac{\pi}{3}\)*. Draw the graph and draw your normal line. Label important stuff.**Students 5,6:**Find the equation for the line*tangent to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\)*. Draw the graph and draw your tangent line. Label important stuff.**Students 7,8:**Find the equation for the line*normal to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\)*. Draw the graph and draw your normal line. Label important stuff.**Students 9,10:**Find the equation for the line*tangent to the graph of \(f(x)\) at \(x=\frac{3\pi}{4}\)*. Draw the graph and draw your tangent line. Label important stuff.

**Students 11 - 20 (second problem)** (2.3#29) For the function
$$f(x)=-x^2+8x=-x(x-8)$$

**Students 11,12:**Find the equation for the line*tangent to the graph of \(f(x)\) at \(x=2\)*. Draw the graph and draw your tangent line. Label important stuff.**Students 13,14:**Find the equation for the line*normal to the graph of \(f(x)\) at \(x=2\)*. Draw the graph and draw your normal line. Label important stuff.**Students 15,16:**Find the equation for the line*tangent to the graph of \(f(x)\) at \(x=4\)*. Draw the graph and draw your tangent line. Label important stuff.**Students 17,18:**Find the equation for the line*normal to the graph of \(f(x)\) at \(x=4\)*. Draw the graph and draw your normal line. Label important stuff.**Students 19,20:**Find the equation for the line*tangent to the graph of \(f(x)\) at \(x=5\)*. Draw the graph and draw your tangent line. Label important stuff.

**Wed Sep 27: **Section 2.3: Basic Differentiation Formulas (Lecture Notes)
(Class Drill on Rewriting Function Before Differentiating)

**Fri Sep 29: **Section 2.4: The Product and Quotient Rules (Lecture Notes)(Quiz **Q3**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Three Problems, 10 points each, about using the Basic Differentiation Formulas, based on Suggested Exercises from Section 2.3, printed on front & back of one sheet of paper.

**Mon Oct 2: **Section 2.5: The Chain Rule (Lecture Notes)

**Tue Oct 3: **Recitation **R06**: Using Differentiation Formulas (Sections 2.3, 2.4, 2.5)

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Bansode,Ankita**Section 101 Student #3:**Bedell,Paris**Section 101 Student #4:**Beegan,Caden**Section 101 Student #5:**Brandt,Roman**Section 101 Student #6:**Earl,Claire-Michael**Section 101 Student #7:**Eisnaugle,Ethan**Section 101 Student #8:**Frometa,Amelia**Section 101 Student #9:**Jackson,Henry**Section 101 Student #10:**Miller,Taylor**Section 101 Student #11:**Robinson,Alana**Section 101 Student #12:**Unassigned,**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,**Section 101 Student #19:**Unassigned,**Section 101 Student #20:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Fritz,Ronan**Section 102 Student #2:**Herrmann,Mary**Section 102 Student #3:**Hoffman,Sidney**Section 102 Student #4:**Hubbard,Grace**Section 102 Student #5:**Lavender,Kinley**Section 102 Student #6:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**Walsh,Carly**Section 102 Student #14:**White,Anna**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,**Section 102 Student #19:**Unassigned,**Section 102 Student #20:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Hains,Amanda**Section 103 Student #4:**Hawley,Frank**Section 103 Student #5:**Kennedy,Quinn**Section 103 Student #6:**Martis,Steve**Section 103 Student #7:**Mikin,Reilly**Section 103 Student #8:**Winterton,Jacob**Section 103 Student #9:**Unassigned,**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,**Section 103 Student #19:**Unassigned,**Section 103 Student #20:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,J**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,**Section 104 Student #19:**Unassigned,**Section 104 Student #20:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Mulholland-Flint,Austin**Section 111 Student #11:**Ngum,Venessa**Section 111 Student #12:**Pickens,Charlee**Section 111 Student #13:**Rasmussen,Cubbie**Section 111 Student #14:**Rodean,Alex**Section 111 Student #15:**Sahr,Griffin**Section 111 Student #16:**Sautter,Jack**Section 111 Student #17:**Scudder,Braedon**Section 111 Student #18:**Wright,Beck**Section 111 Student #19:**Unassigned,**Section 111 Student #20:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Lewis-Baranyai,Enzo**Section 112 Student #14:**Massie,Olivia**Section 112 Student #15:**Miller,Austy**Section 112 Student #16:**Unassigned,**Section 112 Student #17:**Shields,Julia**Section 112 Student #18:**Smith,Kaitlyn**Section 112 Student #19:**Whittington,Kelsey**Section 112 Student #20:**Williams,Ava

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Sikora,Daniella**Section 113 Student #16:**Slingluff,Cheyenne**Section 113 Student #17:**Wenning,Luke**Section 113 Student #18:**Unassigned,**Section 113 Student #19:**Unassigned,**Section 113 Student #20:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Blore,Noah**Section 114 Student #4:**Cox,Madelyn**Section 114 Student #5:**Dubois,Aleke**Section 114 Student #6:**Elliott,Maggie**Section 114 Student #7:**Hartzell,Molly**Section 114 Student #8:**Kezele,Ashley**Section 114 Student #9:**Lampa,Andrew**Section 114 Student #10:**Mcclellan,Alex**Section 114 Student #11:**Mcdermitt,Brian**Section 114 Student #12:**Meyer,Morgan**Section 114 Student #13:**Morris,Chase**Section 114 Student #14:**Mueller,Maddy**Section 114 Student #15:**Nguyen,Jim**Section 114 Student #16:**Raynewater,Ty**Section 114 Student #17:**Smith,Riley**Section 114 Student #18:**Sobey,Lily**Section 114 Student #19:**Wall,Logan**Section 114 Student #20:**Young,Kiefer

**Derivative of a Constant Function** If \(c\) is a constant, then
$$\frac{d}{dx}(c)=0$$

**The Power Rule** If \(n\) is any real number, then
$$\frac{d}{dx}\left(x^n \right)=nx^{n-1}$$

**The Sum Constant Multiple Rule** If \(a\) and \(b\) are constants and \(f\) and \(g\) are differentiable functions, then
$$\frac{d}{dx}\left[af(x) +bg(x)\right]=a\frac{d}{dx}f(x)+b\frac{d}{dx}g(x)$$

**The Product Rule**
$$\frac{d}{dx}\left(\text{left}(x) \cdot \text{right}(x)\right)=\left(\frac{d}{dx}\text{left}(x)\right) \cdot \text{right}(x)+\text{left}(x) \cdot \left(\frac{d}{dx}\text{right}(x)\right)$$

**The Quotient Rule**
$$\frac{d}{dx}\left(\frac{\text{top}(x)}{\text{bottom}(x)}\right)=\frac{\left(\frac{d}{dx}\text{top}(x)\right) \cdot \text{bottom}(x)-\text{top}(x) \cdot \left(\frac{d}{dx}\text{bottom}(x)\right)}{(\text{bottom}(x))^2}$$

**The Chain Rule**
$$\frac{d}{dx}\text{outer}(\text{inner}(x))=\text{outer}'(\text{inner}(x))\cdot\text{inner}'(x)$$

**Derivatives of Trig Functions**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$
$$\frac{d}{dx}\tan(x)=(\sec(x))^2$$
$$\frac{d}{dx}\csc{(x)}=-\csc{(x)}\cot{(x)}$$
$$\frac{d}{dx}\sec{(x)}=\sec{(x)}\tan{(x)}$$
$$\frac{d}{dx}\cot{(x)}=-(\csc(x))^2$$

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R06** score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their **R07** score will be 0/5.

**Students 1,2 ** (2.4#3) Find the derivative of the function
$$g(t)=t^4\cos{(t)}$$
Show all details clearly and use correct notation.

**Students 3,4 ** (2.4#13) Find the derivative of the function
$$f(x)=\frac{x^3}{5-x^2}$$
Show all details clearly and use correct notation.

**Students 5,6 ** (2.4#16) Find the derivative of the function
$$g(t)=\frac{t-\sqrt{t}}{t^{2/3}}$$
Show all details clearly and use correct notation. **Hint: **The function is presented as a *quotient*, but the derivative is *very hard* if you use the *Quotient Rule*. Simplify the function by first rewriting it in *power function form*, and then finding the derivative using *simpler rules*.

**Students 7,8 ** (2.4#17) Find the derivative of the function
$$f(t)=\frac{5t}{5+\sqrt{t}}$$
Show all details clearly and use correct notation.

**Students 9,10 ** (2.4#19) Find the derivative of the function
$$f(x)=\frac{x}{3-\tan{(x)}}$$
Show all details clearly and use correct notation.

**Students 11,12 ** (2.4#27) Find the equation of the line tangent to the graph of
$$f(x)=\frac{x^2-1}{x^2+x+1}$$
at \(x=1\). Present your line equation in **slope intercept form**. Show all details clearly and use correct notation.

**Students 13,14 ** (2.4#31) Find the equation of the line tangent to the graph of
$$f(x)=\frac{1}{1+x^2}$$
at \(x=-1\). Present your line equation in **slope intercept form**. Show all details clearly and use correct notation.

**Students 15,16 **(2.5#1) Find the derivative of
$$f(x)=\sqrt[3]{1+4x}$$
Show all details clearly and use correct notation.

**Students 17,18 **(2.5#13) Find the derivative of
$$f(x)=\cos{(a^3+x^3)}$$
Show all details clearly and use correct notation.

**Students 19,20 **(2.5#51) Find the \((x,y)\) coordinates of all points on the graph of
$$f(x)=2\sin{(x)}+\sin^2{(x)}$$
that have horizontal tangent lines. Show all details clearly and use correct notation.

**Instructor Ask Question #1 for the Class:** Frick and Frack have been asked the following:

They are arguing about the result.

- Frick says that the slope is \( 3x^2 \) because the
*derivative*is the*tangent line*. - Frack that the the slope is $$ m=\frac{f(6)-f(5)}{6-5}=\frac{216-125}{1}=91$$

Frick and Frack are ** both** wrong!

Frick says that the derivative is the tangent line. But this is not correct. The objective is to find the *slope of the tangent line*. This will be a *number*. The *derivative* is a *function*, not a *number*. (The *derivative* is a *function* that can be used to *find* the *number* that is the *slope of the tangent line*.)

Frack is also wrong. Frack computed the *slope of a secant line*.

The correct procedure to find the *slope* of the line *tangent* to the graph of \(f(x)=x^3\) at \(x=5\) is as follows.

**Step 1: **Find \(f'(x)\). The result is

**Step 2: **Substitute \(x=5\) into \(f'(x)\) to get \(m=f'(5)\). The result is

**Instructor Ask Question #2 for the Class:** Wacky Jack has been asked the following:

Their answer was $$y=2x^3-5x^2+4x-11$$ Which of these three statements is true?

- Wacky Jack's answer is correct.
- Wacky Jack's answer is incorrect.
- There is not enough information to be able to say whether Wacky Jack's answer is correct or incorrect. One needs to know the function \(g(x)\) in order to judge.

At first, you might think that of course one would need more information before being able to say whether Wacky Jack's answer is correct or incorrect. But in fact, it is easy to see immediately that **Wacky Jack's answer is incorrect**.

The key is to remember that Wacky Jack was asked to find *the equation of a line*. That means that their result must be in the form
$$y=mx+b$$
where \(m\) and \(b\) are *numbers*. Since Wacky Jack's answer is not in that form, their answer is incorrect.

This example illustrates one kind of *quick check* on problems involving finding the equation of a tangent line. You will encounter problems of that sort where the calculations get quite messy. But the end result should always be an equation of the form \(y=mx+b\).

**Instructor Ask Question #3 for the Class:** For the function

- the \(y\) intercept of \(f(x)\)
- the \(y\) intercept of \(f'(x)\)
- the \(y\) intercept of the the line tangent to \(f(x)\) at \(x=2\)

**Take-away from this exercise: **Observe that these three \(y\) intercepts are three different things. In tangent line problems, a few of you mistakenly use the \(y\) intercept of \(f(x)\), or the \(y\) intercept of \(f'(x)\), as \(y\) intercept of the the line tangent to \(f(x)\) at \(x=a\).

**Wed Oct 4: **Section 2.6: Implicit Differentiation (Lecture Notes)

**Fri Oct 6: **Section 2.7: Related Rates
(Lecture Notes)
(Handout on Implicit Differentiation and related Rates)
(Quiz **Q4**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- four Problems, printed on front & back of one sheet of paper
- One Product Rule problem based on Suggested Exercises from
**Section 2.4**. - One Quotient Rule problem based on Suggested Exercises from
**Section 2.4**. - One Chain Rule problem based on Suggested Exercises from
**Section 2.5**. - One Implicit Differentiation problem based on Suggested Exercises from
**Section 2.6**.

- One Product Rule problem based on Suggested Exercises from

**Mon Oct 9: **Section 2.8: Linear Approx & Differentials
(Lecture Notes)
(Handout on Linearizations and the Method of Using a Linear Approximation)

**Tue Oct 10: **Recitation **R07**: Related Rates and Linearizations (Sections 2.7 and 2.8)

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Bansode,Ankita**Section 101 Student #3:**Bedell,Paris**Section 101 Student #4:**Beegan,Caden**Section 101 Student #5:**Brandt,Roman**Section 101 Student #6:**Earl,Claire-Michael**Section 101 Student #7:**Eisnaugle,Ethan**Section 101 Student #8:**Frometa,Amelia**Section 101 Student #9:**Jackson,Henry**Section 101 Student #10:**Miller,Taylor**Section 101 Student #11:**Robinson,Alana**Section 101 Student #12:**Unassigned,**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,**Section 101 Student #19:**Unassigned,**Section 101 Student #20:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Fritz,Ronan**Section 102 Student #2:**Herrmann,Mary**Section 102 Student #3:**Hoffman,Sidney**Section 102 Student #4:**Hubbard,Grace**Section 102 Student #5:**Lavender,Kinley**Section 102 Student #6:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**Walsh,Carly**Section 102 Student #14:**White,Anna**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,**Section 102 Student #19:**Unassigned,**Section 102 Student #20:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Hains,Amanda**Section 103 Student #4:**Hawley,Frank**Section 103 Student #5:**Kennedy,Quinn**Section 103 Student #6:**Martis,Steve**Section 103 Student #7:**Mikin,Reilly**Section 103 Student #8:**Winterton,Jacob**Section 103 Student #9:**Unassigned,**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,**Section 103 Student #19:**Unassigned,**Section 103 Student #20:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,J**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,**Section 104 Student #19:**Unassigned,**Section 104 Student #20:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Mulholland-Flint,Austin**Section 111 Student #11:**Ngum,Venessa**Section 111 Student #12:**Pickens,Charlee**Section 111 Student #13:**Rasmussen,Cubbie**Section 111 Student #14:**Rodean,Alex**Section 111 Student #15:**Sahr,Griffin**Section 111 Student #16:**Sautter,Jack**Section 111 Student #17:**Scudder,Braedon**Section 111 Student #18:**Wright,Beck**Section 111 Student #19:**Unassigned,**Section 111 Student #20:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Lewis-Baranyai,Enzo**Section 112 Student #14:**Massie,Olivia**Section 112 Student #15:**Miller,Austy**Section 112 Student #16:**Unassigned,**Section 112 Student #17:**Shields,Julia**Section 112 Student #18:**Smith,Kaitlyn**Section 112 Student #19:**Whittington,Kelsey**Section 112 Student #20:**Williams,Ava

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Sikora,Daniella**Section 113 Student #16:**Slingluff,Cheyenne**Section 113 Student #17:**Wenning,Luke**Section 113 Student #18:**Unassigned,**Section 113 Student #19:**Unassigned,**Section 113 Student #20:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Blore,Noah**Section 114 Student #4:**Cox,Madelyn**Section 114 Student #5:**Dubois,Aleke**Section 114 Student #6:**Elliott,Maggie**Section 114 Student #7:**Hartzell,Molly**Section 114 Student #8:**Kezele,Ashley**Section 114 Student #9:**Lampa,Andrew**Section 114 Student #10:**Mcclellan,Alex**Section 114 Student #11:**Mcdermitt,Brian**Section 114 Student #12:**Meyer,Morgan**Section 114 Student #13:**Morris,Chase**Section 114 Student #14:**Mueller,Maddy**Section 114 Student #15:**Nguyen,Jim**Section 114 Student #16:**Raynewater,Ty**Section 114 Student #17:**Smith,Riley**Section 114 Student #18:**Sobey,Lily**Section 114 Student #19:**Wall,Logan**Section 114 Student #20:**Young,Kiefer

**Derivative of a Constant Function** If \(c\) is a constant, then
$$\frac{d}{dx}(c)=0$$

**The Power Rule** If \(n\) is any real number, then
$$\frac{d}{dx}\left(x^n \right)=nx^{n-1}$$

**The Sum Constant Multiple Rule** If \(a\) and \(b\) are constants and \(f\) and \(g\) are differentiable functions, then
$$\frac{d}{dx}\left[af(x) +bg(x)\right]=a\frac{d}{dx}f(x)+b\frac{d}{dx}g(x)$$

**The Product Rule**
$$\frac{d}{dx}\left(\text{left}(x) \cdot \text{right}(x)\right)=\left(\frac{d}{dx}\text{left}(x)\right) \cdot \text{right}(x)+\text{left}(x) \cdot \left(\frac{d}{dx}\text{right}(x)\right)$$

**The Quotient Rule**
$$\frac{d}{dx}\left(\frac{\text{top}(x)}{\text{bottom}(x)}\right)=\frac{\left(\frac{d}{dx}\text{top}(x)\right) \cdot \text{bottom}(x)-\text{top}(x) \cdot \left(\frac{d}{dx}\text{bottom}(x)\right)}{(\text{bottom}(x))^2}$$

**The Chain Rule**
$$\frac{d}{dx}\text{outer}(\text{inner}(x))=\text{outer}'(\text{inner}(x))\cdot\text{inner}'(x)$$

**Derivatives of Trig Functions**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$
$$\frac{d}{dx}\tan(x)=(\sec(x))^2$$
$$\frac{d}{dx}\csc{(x)}=-\csc{(x)}\cot{(x)}$$
$$\frac{d}{dx}\sec{(x)}=\sec{(x)}\tan{(x)}$$
$$\frac{d}{dx}\cot(x)=-(\csc(x))^2$$

**Students 1,2 ** (2.7#4) The length of a rectangle is increasing at a rate of \(8\) cm/s and its width is increasing at a rate of \(3\) cm/s. When the length is \(20\) cm and the width is \(10\) cm, how fast is the area of the rectangle increasing?
Make a good drawing and use correct units in your answer.

**Students 3,4 ** (2.7#5) A cylindrical tank with radius \(5\)m is being filled with water at a rate of \(3\) m^{3}/min. How fast is the height of the water increasing?
Make a good drawing and use correct units in your answer.

**Hint: **Make sure that you start with the correct equation describing the relationship between the radius, height, and volume of a cylinder! Look it up to make sure that you have it right.

**Students 5,6 ** (2.7#11) A snowball melts so that its surface area decreases at a rate of \(1\) cm^{3}/min. Find the rate at which the diameter decreases when the diameter is \(10\) cm.
Make a good drawing and use correct units in your answer.

**Hint: **You'll have to start by coming up with an equation describing the relationship between the ** surface area of a sphere** and

**Students 7,8 ** (2.7#13) A plane flying horizontally at an altitude of \(1\) mi and a speed of \(500\) mi/h passes directly over a radar station.
Find the rate at which the distance from the plane to the station is increasing when it is \(2\) mi away from the station.
Make a good drawing and use correct units in your answer.
(Observe that this problem is not clearly written. The phrase ** distance from the plane to the station** refers to the

**Hint: **Make a right triangle with base \(b\), height \(h\), and hypotenuse \(L\). Identify the given information in terms of \(b\), \(h\), and \(L\). Observe that you are being asked to find \(L'\). Use the **Pythagorean Theorem** to get an equation that expresses a relationship between \(b\), \(h\), and \(L\). Then use **Implicit Differentiation** to get a new equation that expresses a relationship between \(b,h,L,b’,h’,L'\). Solve this equation for \(L'\). Then plug in known values to get a value for \(L'\).

**Students 9,10 ** (2.7#15) Two cars start moving from the same point. One travels south at \(60\) mi/h and the other travels west at \(25\) mi/h. At what rate is the distance between the cars increasing two hours later?
Make a good drawing and use correct units in your answer.

**Hint: **Make a right triangle with base \(b\), height \(h\), and hypotenuse \(L\). Identify the given information in terms of \(b\), \(h\), and \(L\). Observe that you are being asked to find \(h'\). Use the **Pythagorean Theorem** to get an equation that expresses a relationship between \(b\), \(h\), and \(L\). Then use **Implicit Differentiation** to get a new equation that expresses a relationship between \(b,h,L,b’,h’,L'\). Solve this equation for \(L'\). Then plug in known values to get a value for \(L'\).

**Students 11,12 ** (2.7#25) A trough is \(10\) ft long and its ends have the shape of isosceles triangles that are \(3\) ft across the top and have a height of \(1\) ft. The trough is being filled with water at a rate of \(12\) ft^{3}/min. How fast is the water level rising when the water is 6 inches deep?
Make a good drawing and use correct units in your answer.

**Hint: ** Notice that the problem statement uses a mixture of units for length: **feet** and **inches**. This is stupid, but it is done on purpose: You will usually have to deal with inconvenient units when you encounter math problems any real situation. My advice is: convert everything to one unit of length, either **feet** or **inches**, and work the problem that unit.

**Students 13,14 ** (2.7#27) Gravel is being dumped from a conveyor belt at a rate of \(30\) ft^{3}/min, forming a pile in the shape of a right circular cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is \(10\) ft high?
Make a good drawing and use correct units in your answer.

**Hint: **A similar problem was an example in a recent *Lecture*.

**Students 15,16 ** (2.7#28) A kite \(100\) ft above the ground moves horizontally at a speed of \(8\) ft/s. At what rate is the angle between the string and the horizontal decreasing when \(200\) ft of string have been let out? (angles in radians)
Make a good drawing and use correct units in your answer.

**Hint: **Make a right triangle with base \(b\), height \(h\), and hypotenuse \(L\), and important angle \(theta\). Identify the given information in terms of \(b,h,L\). Observe that you are being asked to find \(\theta'\). Find a **Trig Formula** to get an equation that expresses a relationship between \(b\), \(h\), and \(\theta\). Then use **Implicit Differentiation** to get a new equation that expresses a relationship between \(b,h,\theta,b',h',\theta'\). Solve this equation for \(\theta'\). Then plug in known values to get a value for \(\theta'\).

**Students 17,18 ** A ladder \(10\) ft long is leaning against a vertical wall. The foot of the ladder is sliding away from the wall a rate of \(2\) ft/s. How fast is the top of the ladder sliding down the wall at the instant when the foot of the ladder is \(6\) ft from the wall?
Make a good drawing and use correct units in your answer.

**Hint: **Make a right triangle with base \(b\), height \(h\), and hypotenuse \(L\). Identify the given information in terms of \(b\), \(h\), and \(L\). Observe that you are being asked to find \(h'\). Use the **Pythagorean Theorem** to get an equation that expresses a relationship between \(b\), \(h\), and \(L\). Then use **Implicit Differentiation** to get a new equation that expresses a relationship between \(b,h,L,b’,h’,L'\). Solve this equation for \(h'\). Then plug in known values to get a value for \(h'\).

**Students 19,20** (2.7#31) A ladder is leaning against a vertical wall. The top of a ladder slides down the wall at a rate of \(0.15\) m/s. At the moment when the ladder is \(3\) m from the wall, it slides away from the wall at a rate of \(0.2\) m/s. How long is the ladder?
Make a good drawing and use correct units in your answer.

**Hint: **Make a right triangle with base \(b\), height \(h\), and hypotenuse \(L\). Identify the given information in terms of \(b,h,L\). Observe that you are being asked to find \(L\). But it will be simpler to first find a value for the height \(h\). Use the **Pythagorean Theorem** to get an equation that expresses a relationship between \(b,h,L\). Then use **Implicit Differentiation** to get a new equation that expresses a relationship between \(b,h,L,b’,h’,L'\). Solve this equation for \(h\). Then plug in known values to get a value for \(h\). Finally, use the stuff that you know to find a value for \(L\).

**Students 1,2 ** (Similar to Exercise 2.8#5) The goal is to use a *Linear Approximation* to estimate the number \(\sqrt{15.9}\). Answer questions (a) - (f) below.

**Students 3,4 ** (Similar to Exercise 2.8#5) The goal is to use a *Linear Approximation* to estimate the number \(\sqrt{16.1}\). Answer questions (a) - (f) below.

**Students 5,6 ** (Similar to Exercise 2.8#11) The goal is to use a *Linear Approximation* to estimate the number \(2.9^4\). Answer questions (a) - (f) below.

**Students 7,8 ** (Similar to Exercise 2.8#11) The goal is to use a *Linear Approximation* to estimate the number \(3.1^4\). Answer questions (a) - (f) below.

**Students 9,10 ** (Similar to Exercise 2.8#13) The goal is to use a *Linear Approximation* to estimate the number \(7.9^{2/3}\). Answer questions (a) - (f) below.

**Students 11,12 ** (Similar to Exercise 2.8#13) The goal is to use a *Linear Approximation* to estimate the number \(8.1^{2/3}\). Answer questions (a) - (f) below.

**Students 13,14 ** (Similar to Exercise 2.8#17) The goal is to use a *Linear Approximation* to estimate the number \(\sin(-0.1)\). (angles in radians) Answer questions (a) - (f) below.

**Students 15,16 ** (Similar to Exercise 2.8#17) The goal is to use a *Linear Approximation* to estimate the number \(\sin(0.1)\). (angles in radians) Answer questions (a) - (f) below.

**Students 17,18 ** (Similar to Exercise 2.8#17) The goal is to use a *Linear Approximation* to estimate the number \(\cos(-0.1)\). (angles in radians) Answer questions (a) - (f) below.

**Students 19,20 ** (Similar to Exercise 2.8#17) The goal is to use a *Linear Approximation* to estimate the number \(\cos(0.1)\). (angles in radians) Answer questions (a) - (f) below.

- What is the related
*function*, \(f(x)\)? - What is the
*inconvenient \(x\) value*, \(\hat{x}\)? - What is a
*convenient nearby \(x\) value*, \(a\)? - Build the
*Linearization of \(f\) at \(a\)*. That is, build the function $$L(x)=f(a)+f’(a)\cdot(x-a)$$ .
- Use your
*linearization*to find \(L(\hat{x})\). That is, find the value $$L(\hat{x})=f(a)+f’(a)\cdot(\hat{x}-a)$$ This is the estimate that was the goal of the problem. - While it might not be possible to write an exact value for \(f(\hat{x})\), you can use a calculator to get a very precise (but not exact) decimal value for \(f(\hat{x})\). Do that, and see how it compares to your
*estimate*from part (e).

**Wed Oct 11: **Exponential Functions, Inverse Functions, Logarithms (Sections 3.1, 3.2) (Lecture Notes)(Quiz **Q5**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Two Problems, 15 points each, printed on front & back of one sheet of paper
- One Related Rates problem based on Suggested Exercises from
**Section 2.7**. - One Linearization problem problem based on Suggested Exercises from
**Section 2.8**.

- One Related Rates problem based on Suggested Exercises from

**Fri Oct 13: **Holiday

**Mon Oct 16: **Section 3.3: Derivatives of Logarithmic and Exponential Functions (Lecture Notes)

**Tue Oct 17: **Recitation **R08**: Derivatives of Logarithmic and Exponential Functions (Section 3.3)

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R07** score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their **R07** score will be 0/5.

**Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Bedell,Paris**Section 101 Student #3:**Beegan,Caden**Section 101 Student #4:**Brandt,Roman**Section 101 Student #5:**Earl,Claire-Michael**Section 101 Student #6:**Eisnaugle,Ethan**Section 101 Student #7:**Frometa,Amelia**Section 101 Student #8:**Jackson,Henry**Section 101 Student #9:**Miller,Taylor**Section 101 Student #10:**Robinson,Alana**Section 101 Student #11:**Unassigned,**Section 101 Student #12:**Unassigned,**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,**Section 101 Student #19:**Unassigned,**Section 101 Student #20:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Fritz,Ronan**Section 102 Student #2:**Herrmann,Mary**Section 102 Student #3:**Hoffman,Sidney**Section 102 Student #4:**Hubbard,Grace**Section 102 Student #5:**Lavender,Kinley**Section 102 Student #6:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**White,Anna**Section 102 Student #14:**Unassigned,**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,**Section 102 Student #19:**Unassigned,**Section 102 Student #20:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Hains,Amanda**Section 103 Student #4:**Hawley,Frank**Section 103 Student #5:**Kennedy,Quinn**Section 103 Student #6:**Martis,Steve**Section 103 Student #7:**Mikin,Reilly**Section 103 Student #8:**Winterton,Jacob**Section 103 Student #9:**Unassigned,**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,**Section 103 Student #19:**Unassigned,**Section 103 Student #20:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:55pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,J**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,**Section 104 Student #19:**Unassigned,**Section 104 Student #20:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Ngum,Venessa**Section 111 Student #11:**Pickens,Charlee**Section 111 Student #12:**Rasmussen,Cubbie**Section 111 Student #13:**Rodean,Alex**Section 111 Student #14:**Sahr,Griffin**Section 111 Student #15:**Sautter,Jack**Section 111 Student #16:**Scudder,Braedon**Section 111 Student #17:**Wright,Beck**Section 111 Student #18:**Unassigned,**Section 111 Student #19:**Unassigned,**Section 111 Student #20:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Massie,Olivia**Section 112 Student #14:**Miller,Austy**Section 112 Student #15:**Shields,Julia**Section 112 Student #16:**Smith,Kaitlyn**Section 112 Student #17:**Williams,Ava**Section 112 Student #18:**Unassigned,**Section 112 Student #19:**Unassigned,**Section 112 Student #20:**Unassigned,

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Slingluff,Cheyenne**Section 113 Student #16:**Wenning,Luke**Section 113 Student #17:**Unassigned,**Section 113 Student #18:**Unassigned,**Section 113 Student #19:**Unassigned,**Section 113 Student #20:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Dubois,Aleke**Section 114 Student #4:**Elliott,Maggie**Section 114 Student #5:**Hartzell,Molly**Section 114 Student #6:**Kezele,Ashley**Section 114 Student #7:**Lampa,Andrew**Section 114 Student #8:**Mcclellan,Alex**Section 114 Student #9:**Mcdermitt,Brian**Section 114 Student #10:**Meyer,Morgan**Section 114 Student #11:**Morris,Chase**Section 114 Student #12:**Mueller,Maddy**Section 114 Student #13:**Nguyen,Jim**Section 114 Student #14:**Raynewater,Ty**Section 114 Student #15:**Smith,Riley**Section 114 Student #16:**Sobey,Lily**Section 114 Student #17:**Wall,Logan**Section 114 Student #18:**Young,Kiefer**Section 114 Student #19:**Unassigned,**Section 114 Student #20:**Unassigned,

**Derivative of a Constant Function** If \(c\) is a constant, then
$$\frac{d}{dx}(c)=0$$

**The Power Rule** If \(n\) is any real number, then
$$\frac{d}{dx}\left(x^n \right)=nx^{n-1}$$

**The Sum Constant Multiple Rule** If \(a\) and \(b\) are constants and \(f\) and \(g\) are differentiable functions, then
$$\frac{d}{dx}\left[af(x) +bg(x)\right]=a\frac{d}{dx}f(x)+b\frac{d}{dx}g(x)$$

**The Product Rule**
$$\frac{d}{dx}\left(\text{left}(x) \cdot \text{right}(x)\right)=\left(\frac{d}{dx}\text{left}(x)\right) \cdot \text{right}(x)+\text{left}(x) \cdot \left(\frac{d}{dx}\text{right}(x)\right)$$

**The Quotient Rule**
$$\frac{d}{dx}\left(\frac{\text{top}(x)}{\text{bottom}(x)}\right)=\frac{\left(\frac{d}{dx}\text{top}(x)\right) \cdot \text{bottom}(x)-\text{top}(x) \cdot \left(\frac{d}{dx}\text{bottom}(x)\right)}{(\text{bottom}(x))^2}$$

**The Chain Rule**
$$\frac{d}{dx}\text{outer}(\text{inner}(x))=\text{outer}'(\text{inner}(x))\cdot\text{inner}'(x)$$

**Derivatives of Trig Functions**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$
$$\frac{d}{dx}\tan(x)=(\sec(x))^2$$
$$\frac{d}{dx}\csc{(x)}=-\csc{(x)}\cot{(x)}$$
$$\frac{d}{dx}\sec{(x)}=\sec{(x)}\tan{(x)}$$
$$\frac{d}{dx}\cot{(x)}=-(\csc(x))^2$$

**Derivatives of Logarithmic Functions**
$$\frac{d}{dx}\ln{(x)}=\frac{1}{x}\text{ restricted to the domain }x\gt 0$$
$$\frac{d}{dx}\log_b{(x)}=\frac{1}{x\ln{(b)}}\text{ restricted to the domain }x\gt 0$$
$$\frac{d}{dx}\ln{(|x|)}=\frac{1}{x}$$

**Derivatives of Exponential Functions**
$$\frac{d}{dx}e^{(x)}=e^{(x)}$$
$$\frac{d}{dx}b^{(x)}=b^{(x)}\ln{(b)}$$

**Students 1,2 ** (3.3#1) Differentiate the function.
$$f(x)=\log_{10}\left(x^3+5x^2+7x+11\right)$$

**Students 3,4 ** (3.3#3) Differentiate the function.
$$f(x)=\sin\left(\ln{(x)}\right)$$

**Students 5,6 ** (3.3#4) Differentiate the function.
$$f(x)=\ln\left(\sin^2{(x)}\right)$$

**Students 7,8 ** (3.3#6) Differentiate the function.
$$y=\frac{1}{\ln{(x)}}$$

**Students 9,10 ** (3.3#9) Differentiate the function.
$$g(x)=\ln\left(\frac{a-x}{a+x}\right)$$
**Hint: **This looks like a problem that would involve three rules: The **Chain Rule** (to deal with the nested function), the **Logarithm Rule** (to deal with the derivative of the outer function), and the **Quotient Rule** (to deal with the derivative of the innner function). While the problem can be done that way, there is a way to make the derivative much simpler. Before taking the derivative, use a rule of logarithms to rewrite the function \(g(x)\) so that it is not the logarithm of a quotient. Then find the derivative of the rewritten function.

**Students 11,12 ** (3.3#13) Differentiate the function.
$$G(x)=\ln\left(\frac{(2x+1)^5}{\sqrt{x^2+1}}\right)$$
**Hint: **This looks like a problem that would involve many rules: The **Chain Rule** (to deal with the nested function), the **Logarithm Rule** (to deal with the derivative of the outer function), the **Quotient Rule** and **Chain Rule** (again!) (to deal with the derivative of the innner function). While the problem can be done that way, there is a way to make the derivative much simpler. Before taking the derivative, use a rule of logarithms to rewrite the function \(G(x)\) so that it is not the logarithm of a quotient and then use another rule of logarithms to rewrite \(G(x)\) so that the inside functions are simple polynomials, not nested functions. Then find the derivative of the rewritten \(G(x)\).

**Students 13,14 ** (3.3#20) Differentiate the function.
$$g(x)=\sqrt{x}e^{(x)}$$

**Students 15,16 ** (3.3#26) Differentiate the function.
$$y=10^\left(1-x^2\right)$$

**Students 17,18 ** (3.3#31) Differentiate the function.
$$f(t)=\tan{\left(e^{(t)}\right)}+e^{\tan{(t)}}$$

**Students 19,20 ** (3.3#35) Differentiate the function.
$$y=2x\log_{10}{\left(\sqrt{x}\right)}$$
**Hint: **This looks like a problem that would involve many rules: The **Product Rule** (to deal with the product), the **Logarithm base \(b\) Rule** (to deal with the \(\log_b\)), the **Chain Rule** (to deal with the nested function), and the **Power Rule** (to deal with the square root). While the problem can be done that way, there is a way to make the derivative much simpler. Before taking the derivative, use a rule of logarithms to rewrite the function so that it is not the logarithm of a square root. Then find the derivative of the rewritten function.

**Students 1,2,3,4 ** (3.3#41) Find \(y'\) and \(y''\)
$$y=e^{(\alpha x)}\sin{(\beta x)}$$

**Students 5,6,7,8 ** (3.3#45) Find the equation of the line tangent to the graph of \(y=\ln{\left(x^2-4x+5\right)}\) at \(x=3\).

**Students 9,10,11,12 ** For the function \(f(x)=e^{\left(-x^2+2x-1\right)}\)

- Find \(f'(x)\).
- Find the
*slope*of the line tangent to the graph of \(f(x)\) at \(x=0\). - Find the \(x\) coordinates of all points on the graph of \(f(x)\) that have
*horizontal tangent lines*. - Illustrate your results from (b),(c) using a graph of \(f(x)\). Feel free to get a graph from Desmos. What famous shape is this graph?

**Students 13,14,15,16 ** (3.3#55) Use **logarithmic differentiation** to find the derivative.
$$y=x^x$$

**Students 17,18,19,20** (3.3#57) Use **logarithmic differentiation** to find the derivative.
$$y=\left(\cos{(x)}\right)^x$$

**Wed Oct 18: **Section 3.4: Exponential Growth & Decay (Lecture Notes)

**Fri Oct 20: Exam ****X2** **Covering Section 2.3 through Chapter 3**

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**The Exam will last the full duration of the class period.****No books, notes, calculators, or phones**- Eight problems, 25 points each, printed on front & back of three sheets of paper. All problems are based on suggested exercises.
- Four problems about finding derivatives using various methods that we have studied (in sections 2.3, 2.4, 2.5, 2.6, 3.3)
- Four problems about using derivatives to find things.
- Related rates (Section 2.7)
- Exponential Growth in Biology or Exponential Decay of Radioactive Substance (Section 3.4)
- Velocity & Acceleration (Problems about this appear in Sections 2.3, 2.4, 2.5.)
- Slope or Equation of the Tangent Line and/or Normal Line. (Problems about this appear in Sections 2.3, 2.4, 2.5, 3.3.)

**Mon Oct 23: **Section 4.1: Maximum and Minimum Values (Lecture Notes)

**Tue Oct 24: **Recitation **R09**: Extrema and Critical Numbers (Section 4.1)

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R07** score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their **R07** score will be 0/5.

**Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Bedell,Paris**Section 101 Student #3:**Beegan,Caden**Section 101 Student #4:**Brandt,Roman**Section 101 Student #5:**Earl,Claire-Michael**Section 101 Student #6:**Eisnaugle,Ethan**Section 101 Student #7:**Jackson,Henry**Section 101 Student #8:**Miller,Taylor**Section 101 Student #9:**Robinson,Alana**Section 101 Student #10:**Unassigned,**Section 101 Student #11:**Unassigned,**Section 101 Student #12:**Unassigned,**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,**Section 101 Student #19:**Unassigned,**Section 101 Student #20:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Fritz,Ronan**Section 102 Student #2:**Herrmann,Mary**Section 102 Student #3:**Hoffman,Sidney**Section 102 Student #4:**Hubbard,Grace**Section 102 Student #5:**Lavender,Kinley**Section 102 Student #6:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**White,Anna**Section 102 Student #14:**Unassigned,**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,**Section 102 Student #19:**Unassigned,**Section 102 Student #20:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Hains,Amanda**Section 103 Student #4:**Hawley,Frank**Section 103 Student #5:**Kennedy,Quinn**Section 103 Student #6:**Martis,Steve**Section 103 Student #7:**Mikin,Reilly**Section 103 Student #8:**Winterton,Jacob**Section 103 Student #9:**Unassigned,**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,**Section 103 Student #19:**Unassigned,**Section 103 Student #20:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:55pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,J**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,**Section 104 Student #19:**Unassigned,**Section 104 Student #20:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Mckinney,Kaia**Section 111 Student #10:**Ngum,Venessa**Section 111 Student #11:**Pickens,Charlee**Section 111 Student #12:**Rasmussen,Cubbie**Section 111 Student #13:**Rodean,Alex**Section 111 Student #14:**Sahr,Griffin**Section 111 Student #15:**Sautter,Jack**Section 111 Student #16:**Scudder,Braedon**Section 111 Student #17:**Wright,Beck**Section 111 Student #18:**Unassigned,**Section 111 Student #19:**Unassigned,**Section 111 Student #20:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Massie,Olivia**Section 112 Student #14:**Miller,Austy**Section 112 Student #15:**Shields,Julia**Section 112 Student #16:**Smith,Kaitlyn**Section 112 Student #17:**Williams,Ava**Section 112 Student #18:**Unassigned,**Section 112 Student #19:**Unassigned,**Section 112 Student #20:**Unassigned,

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Slingluff,Cheyenne**Section 113 Student #16:**Wenning,Luke**Section 113 Student #17:**Unassigned,**Section 113 Student #18:**Unassigned,**Section 113 Student #19:**Unassigned,**Section 113 Student #20:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Dubois,Aleke**Section 114 Student #4:**Elliott,Maggie**Section 114 Student #5:**Hartzell,Molly**Section 114 Student #6:**Kezele,Ashley**Section 114 Student #7:**Lampa,Andrew**Section 114 Student #8:**Mcclellan,Alex**Section 114 Student #9:**Mcdermitt,Brian**Section 114 Student #10:**Meyer,Morgan**Section 114 Student #11:**Morris,Chase**Section 114 Student #12:**Mueller,Maddy**Section 114 Student #13:**Nguyen,Jim**Section 114 Student #14:**Raynewater,Ty**Section 114 Student #15:**Smith,Riley**Section 114 Student #16:**Sobey,Lily**Section 114 Student #17:**Wall,Logan**Section 114 Student #18:**Young,Kiefer**Section 114 Student #19:**Unassigned,**Section 114 Student #20:**Unassigned,

Remember the definition of ** Critical Number** from the Monday March 20 Lecture. (The wording of Barsamian's definition differs from the wording of the book's definition, but the underlying meaning is the same.)

**Definition: **A ** Critical Number** of a function \(f(x)\) is an \(x=c\) that satisfies both of these requirements:

- \(f(c)\) exists. (That is, \(x=c\) is in the
of \(f(x)\).*domain* - \(f'(c)=0\) or \(f'(c)\)
*does not exist*.

For each function \(f(x)\), answer the following questions:

- Find the
*domain*of \(f(x)\). - Find \(f'(x)\).
- Find the
*domain*of \(f'(x)\). - Find all \(x\) values that are in the
*domain*of \(f(x)\) but that are*not*in the domain of \(f'(x)\). That is, find all \(x\) values such that \(f(x)\) exists but \(f'(x)\) does not exist. Explain clearly. - Find all \(x\) values where \(f'(x)=0\). Explain clearly.
- Find all
*critical numbers*of \(f(x)\). Explain clearly.

**Students 1,2** (4.1#25)
$$f(x)=2x^3-3x^2-36x$$

**Students 3,4** (4.1#25)
$$f(x)=2x^3-3x^2-36x$$

**Students 5,6** (similar to 4.1#25)
$$f(x)=x^4-6x^2+5$$

**Students 7,8** (4.1#29)
$$f(x)=\frac{x-1}{x^2-x+1}$$

**Students 9,10** (similar to 4.1#35, but easier)
$$f(x)=xe^{(-3x)}$$

**Students 11,12** (4.1#35)
$$f(x)=x^2e^{(-3x)}$$

**Students 13,14** (4.1#43)
$$f(x)=x\sqrt{4-x^2}$$

**Students 15,16** (4.1#47)
$$f(x)=xe^{(-x^2/8)}$$

**Students 17,18** (4.1#49)
$$f(x)=\ln(x^2+x+1)$$

**Students 19,20** (Similar to Section 4.1 Example 5 on p. 207)
$$f(x)=x^{2/5}(x-7)$$

Students do this Class Drill about Identifying Extrema

**Wed Oct 25: **Section 4.1: Maximum and Minimum Values (Lecture Notes)

**Fri Oct 27: **Section 4.2: The Mean Value Theorem (Lecture Notes)(Quiz **Q6**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Two Problems, 15 points each, about Maximum and Minimum Values, based on Suggested Exercises from
**Section 4.1**, printed on front & back of one sheet of paper.

**Mon Oct 30: **Section 4.3: Derivatives and the Shapes of Graphs (Lecture Notes)

**Tue Oct 31: **Recitation **R10**: Sections 4.2 and 4.3

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R07** score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their **R07** score will be 0/5.

**Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Bedell,Paris**Section 101 Student #3:**Beegan,Caden**Section 101 Student #4:**Brandt,Roman**Section 101 Student #5:**Earl,Claire-Michael**Section 101 Student #6:**Eisnaugle,Ethan**Section 101 Student #7:**Jackson,Henry**Section 101 Student #8:**Miller,Taylor**Section 101 Student #9:**Robinson,Alana**Section 101 Student #10:**Unassigned,**Section 101 Student #11:**Unassigned,**Section 101 Student #12:**Unassigned,**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Fritz,Ronan**Section 102 Student #2:**Herrmann,Mary**Section 102 Student #3:**Hoffman,Sidney**Section 102 Student #4:**Hubbard,Grace**Section 102 Student #5:**Lavender,Kinley**Section 102 Student #6:**Lindsay,Tamryn**Section 102 Student #7:**Mccoy,Caleb**Section 102 Student #8:**Osterlink,Bianca**Section 102 Student #9:**Richardson,Ryan**Section 102 Student #10:**Rickey,Jacqueline**Section 102 Student #11:**Roberts,Madachi**Section 102 Student #12:**Voegele,Brooklynne**Section 102 Student #13:**White,Anna**Section 102 Student #14:**Unassigned,**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Hains,Amanda**Section 103 Student #4:**Hawley,Frank**Section 103 Student #5:**Kennedy,Quinn**Section 103 Student #6:**Martis,Steve**Section 103 Student #7:**Mikin,Reilly**Section 103 Student #8:**Winterton,Jacob**Section 103 Student #9:**Unassigned,**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:55pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,J**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Ngum,Venessa**Section 111 Student #10:**Pickens,Charlee**Section 111 Student #11:**Rasmussen,Cubbie**Section 111 Student #12:**Rodean,Alex**Section 111 Student #13:**Sahr,Griffin**Section 111 Student #14:**Sautter,Jack**Section 111 Student #15:**Wright,Beck**Section 111 Student #16:**Unassigned,**Section 111 Student #17:**Unassigned,**Section 111 Student #28:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Massie,Olivia**Section 112 Student #14:**Miller,Austy**Section 112 Student #15:**Shields,Julia**Section 112 Student #16:**Smith,Kaitlyn**Section 112 Student #17:**Williams,Ava**Section 112 Student #18:**Unassigned,

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Slingluff,Cheyenne**Section 113 Student #16:**Wenning,Luke**Section 113 Student #17:**Unassigned,**Section 113 Student #18:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Dubois,Aleke**Section 114 Student #4:**Elliott,Maggie**Section 114 Student #5:**Hartzell,Molly**Section 114 Student #6:**Kezele,Ashley**Section 114 Student #7:**Lampa,Andrew**Section 114 Student #8:**Mcclellan,Alex**Section 114 Student #9:**Mcdermitt,Brian**Section 114 Student #10:**Meyer,Morgan**Section 114 Student #11:**Morris,Chase**Section 114 Student #12:**Mueller,Maddy**Section 114 Student #13:**Nguyen,Jim**Section 114 Student #14:**Raynewater,Ty**Section 114 Student #15:**Smith,Riley**Section 114 Student #16:**Sobey,Lily**Section 114 Student #17:**Young,Kiefer**Section 114 Student #18:**Unassigned,

**Rolle's Theorem: **If a function \(f\) satisfies the following three requirements (the ** hypotheses**)

- \(f\) is
*continuous*on the*closed interval*\([a,b]\). - \(f\) is
*differentiable*on the*open interval*\((a,b)\). - \(f(a)=f(b)\).

There is at least one number \(x=c\) with \(a \lt c \lt b\) such that $$f'(c)=0$$ In other words, $$\text{the slope of the tangent line at }x=c\text{ is }m=f’(c)=0$$

**Remark: **The theorem does not give you the *value* of \(c\). If \(c\) *exists*, you'll have to figure out its value.

**Students 1,2: **Consider the function \(f(x)=x^3-3x+5\) on the interval \(\left[-\sqrt{3},\sqrt{3}\right]\).

- Verify that the function and the interval satisfy the three
*hypotheses*of Rolle's Theorem. Explain clearly. - Find all numbers \(c\) that satisfy the
*conclusion*of Rolle's Theorem. Show all details clearly. - Draw a graph of \(f(x)\) on the interval and draw a tangent line to illustrate your result.

**Students 3,4: **Consider the function \(\cos{(x)}\) on the interval \([\frac{\pi}{6},\frac{13\pi}{6}]\).

- Verify that the function and the interval satisfy the three
*hypotheses*of Rolle's Theorem. Explain clearly. - Find all numbers \(c\) that satisfy the
*conclusion*of Rolle's Theorem. Show all details clearly. - Draw a graph of \(f(x)\) on the interval and draw a tangent line to illustrate your result.

**Students 5,6: **Consider the function \(f(x)=x+\frac{1}{x}\) on the interval \([\frac{1}{3},3]\).

- Verify that the function and the interval satisfy the three
*hypotheses*of Rolle's Theorem. Explain clearly. - Find all numbers \(c\) that satisfy the
*conclusion*of Rolle's Theorem. Show all details clearly. - Draw a graph of \(f(x)\) on the interval and draw a tangent line to illustrate your result.

**The Mean Value Theorem: **If a function \(f\) satisfies the following three requirements (the ** hypotheses**)

- \(f\) is
*continuous*on the*closed interval*\([a,b]\). - \(f\) is
*differentiable*on the*open interval*\((a,b)\).

There is at least one number \(x=c\) with \(a \lt c \lt b\) such that $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ In other words, $$\text{slope of the tangent line at }c \ \text{ is equal to the slope of the secant line from }a\text{ to }b$$

**Remark: **The theorem does not give you the *value* of \(c\). If \(c\) *exists*, you'll have to figure out its value.

**Students 7,8: **Consider the function \(f(x)=x^3-3x+2\) on the interval \([-2,2]\).

- Verify that the function and the interval satisfy the two
*hypotheses*of the Mean Value Theorem. Explain clearly. - Find all numbers \(c\) that satisfy the
*conclusion*of the Mean Value Theorem. Show all details clearly. - Draw a graph of \(f(x)\) on the interval and draw a tangent line and a secant line to illustrate your result.

**Students 9,10: **Consider the function \(f(x)=\ln{(x)}\) on the interval \([1,4]\).

- Verify that the function and the interval satisfy the two
*hypotheses*of the Mean Value Theorem. Explain clearly. - Find all numbers \(c\) that satisfy the
*conclusion*of the Mean Value Theorem. Show all details clearly. - Draw a graph of \(f(x)\) on the interval and draw a tangent line and a secant line to illustrate your result.

**Students 11,12: **Consider the function \(f(x)=\frac{1}{x}\) on the interval \([1,3]\).

- Verify that the function and the interval satisfy the two
*hypotheses*of the Mean Value Theorem. Explain clearly. - Find all numbers \(c\) that satisfy the
*conclusion*of the Mean Value Theorem. Show all details clearly. - Draw a graph of \(f(x)\) on the interval and draw a tangent line and a secant line to illustrate your result.

- If \(f'(x)\) is
*positive*on an interval \( (a,b) \) then \(f(x)\) is*increasing*on the interval \( (a,b) \). - If \(f'(x)\) is
*negative*on an interval \( (a,b) \) then \(f(x)\) is*decreasing*on the interval \( (a,b) \). - If \(f'(x)\) is
*zero*on a whole interval \( (a,b) \) then \(f(x)\) is*constant*on the interval \( (a,b) \).

**Test 1:**\(f'(c)=0\) or \(f'(c) DNE\). (If the number \(c\) passes**Test 1**, then \(c\) is called afor \(f'(x)\).)*partition number***Test 2:**\(f(c)\) exists. (If the number \(c\) passes both**Test 1**and**Test 2**, then \(c\) is called afor \(f(x)\).)*critical number***Test 3:**\(f(x)\) isat \(c\).*continuous***Test 4:**\(f'(x)\)at \(c\).(If the number \(c\) passes*changes sign***Tests 1,2,3,4**, then \(x=c\) is the location of aor*local max*of \(f(x)\). The corresponding \(y\) value, \(f(c)\), is called the*local min*or*local max value*.)*local max value*

- If \(f''(x)\) is
*positive*on an interval \( (a,b) \) then \(f'(x)\) is*increasing*on the interval \( (a,b) \), which in turn means that \(f(x)\) is*concave up*on the interval \((a,b)\). - If \(f''(x)\) is
*negative*on an interval \( (a,b) \) then \(f'(x)\) is*decreasing*on the interval \( (a,b) \), which in turn means that \(f(x)\) is*concave cown*on the interval \((a,b)\).

**Related terminology: **An ** inflection point** is a point on the graph of a function where the
function is continuous and the concavity changes (from up to down or from down to up).

**Students 13,14: **Consider the function \(f(x)=\sin{(x)}-\cos{(x)}\) and the interval \([-2,2]\).

- Find the intervals on which \(f\) is
*increasing*or*decreasing*. - Find the
*local maximum values*and*local minimum values*of \(f\). - Find the intervals on which \(f\) is
*concave up*or*concave down*. - Find the \((x,y)\) coordinates of all
*inflection points*of \(f\).

**Students 15,16: **Consider the function \(f(x)=xe^{(-x)}\).

- Find the intervals on which \(f\) is
*increasing*or*decreasing*. - Find the
*local maximum values*and*local minimum values*of \(f\). - Find the intervals on which \(f\) is
*concave up*or*concave down*. - Find the \((x,y)\) coordinates of all
*inflection points*of \(f\).

**Students 17,18: **Consider the function \(f(x)=x^4-2x^2+3\).

- Find the intervals on which \(f\) is
*increasing*or*decreasing*. - Find the
*local maximum values*and*local minimum values*of \(f\). - Find the intervals on which \(f\) is
*concave up*or*concave down*. - Find the \((x,y)\) coordinates of all
*inflection points*of \(f\).

**Wed Nov 1: **Section 4.4: Curve Sketching (Handout on Graphing Strategy)(Lecture Notes)

**Fri Nov 3: **Section 4.5: Optimization (Lecture Notes)(Last Day to Drop)(Quiz **Q7**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Three Problems, 10 points each, printed on front & back of one sheet of paper.
- One Mean Value Theorem problem based on Suggested Exercises from
**Section 4.2**. - One problem about Derivatives and the Shapes of Graphs, based on Suggested Exercises from
**Section 4.3**. - One problem about Curve Sketching, based on Suggested Exercises from
**Section 4.4**.

- One Mean Value Theorem problem based on Suggested Exercises from

**Mon Nov 6: **Section 4.6: Newton's Method (Class Drill on Newton's Method)(Lecture Notes)

**Tue Nov 7: **Recitation **R11**: Optimization; Newton's Method (Sections 4.5, 4.6)

**Scoring: **If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their **R07** score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their **R07** score will be 0/5.

**Student Number** in the lists below. The problems to be solved are listed farther down the page.

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei)

**Section 101 Student #1:**Allen,Daylen**Section 101 Student #2:**Beegan,Caden**Section 101 Student #3:**Brandt,Roman**Section 101 Student #4:**Earl,Claire-Michael**Section 101 Student #5:**Eisnaugle,Ethan**Section 101 Student #6:**Jackson,Henry**Section 101 Student #7:**Miller,Taylor**Section 101 Student #8:**Robinson,Alana**Section 101 Student #9:**Unassigned,**Section 101 Student #10:**Unassigned,**Section 101 Student #11:**Unassigned,**Section 101 Student #12:**Unassigned,**Section 101 Student #13:**Unassigned,**Section 101 Student #14:**Unassigned,**Section 101 Student #15:**Unassigned,**Section 101 Student #16:**Unassigned,**Section 101 Student #17:**Unassigned,**Section 101 Student #18:**Unassigned,

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei)

**Section 102 Student #1:**Herrmann,Mary**Section 102 Student #2:**Hoffman,Sidney**Section 102 Student #3:**Hubbard,Grace**Section 102 Student #4:**Lavender,Kinley**Section 102 Student #5:**Lindsay,Tamryn**Section 102 Student #6:**Mccoy,Caleb**Section 102 Student #7:**Osterlink,Bianca**Section 102 Student #8:**Richardson,Ryan**Section 102 Student #9:**Rickey,Jacqueline**Section 102 Student #10:**Roberts,Madachi**Section 102 Student #11:**Voegele,Brooklynne**Section 102 Student #12:**White,Anna**Section 102 Student #13:**Unassigned,**Section 102 Student #14:**Unassigned,**Section 102 Student #15:**Unassigned,**Section 102 Student #16:**Unassigned,**Section 102 Student #17:**Unassigned,**Section 102 Student #18:**Unassigned,

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei)

**Section 103 Student #1:**Alder,Ethan**Section 103 Student #2:**Blower,Carsen**Section 103 Student #3:**Hains,Amanda**Section 103 Student #4:**Hawley,Frank**Section 103 Student #5:**Kennedy,Quinn**Section 103 Student #6:**Martis,Steve**Section 103 Student #7:**Mikin,Reilly**Section 103 Student #8:**Winterton,Jacob**Section 103 Student #9:**Unassigned,**Section 103 Student #10:**Unassigned,**Section 103 Student #11:**Unassigned,**Section 103 Student #12:**Unassigned,**Section 103 Student #13:**Unassigned,**Section 103 Student #14:**Unassigned,**Section 103 Student #15:**Unassigned,**Section 103 Student #16:**Unassigned,**Section 103 Student #17:**Unassigned,**Section 103 Student #18:**Unassigned,

Recitation Section 104 (meets Tue 2:00pm – 2:55pm in Morton 318 with Instructor Isaac Agyei)

**Section 104 Student #1:**Akpofure,Alexander**Section 104 Student #2:**Armstrong,Graci**Section 104 Student #3:**Benton,Kaleb**Section 104 Student #4:**Bersagel,Via**Section 104 Student #5:**Burns,J**Section 104 Student #6:**Graham,Taylor**Section 104 Student #7:**Griffiths,Kristen**Section 104 Student #8:**Huntley,Lauren**Section 104 Student #9:**King,Mason**Section 104 Student #10:**Lopinsky,Iliana**Section 104 Student #11:**Lucas,Madison**Section 104 Student #12:**Maag,Stacie**Section 104 Student #13:**Mcculloch,Thomas**Section 104 Student #14:**Mcgannon,Jane**Section 104 Student #15:**Neal,Daniel**Section 104 Student #16:**Nestor,Nicholas**Section 104 Student #17:**Vivo,Nicholas**Section 104 Student #18:**Unassigned,

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So)

**Section 111 Student #1:**Blankenship,Conner**Section 111 Student #2:**Chaney,Alyssa**Section 111 Student #3:**Christy,Carly**Section 111 Student #4:**Henely,Lydia**Section 111 Student #5:**Keener,Mckensie**Section 111 Student #6:**Kessler,Crosley**Section 111 Student #7:**Leary,Austin**Section 111 Student #8:**Locke,Tyler**Section 111 Student #9:**Ngum,Venessa**Section 111 Student #10:**Pickens,Charlee**Section 111 Student #11:**Rasmussen,Cubbie**Section 111 Student #12:**Rodean,Alex**Section 111 Student #13:**Sahr,Griffin**Section 111 Student #14:**Sautter,Jack**Section 111 Student #15:**Wright,Beck**Section 111 Student #16:**Unassigned,**Section 111 Student #17:**Unassigned,**Section 111 Student #18:**Unassigned,

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So)

**Section 112 Student #1:**Alsko,Adam**Section 112 Student #2:**Altiere,David**Section 112 Student #3:**Beya,Mimi**Section 112 Student #4:**Byrd,Iana**Section 112 Student #5:**Collins,Kian**Section 112 Student #6:**Gonzales,Solana**Section 112 Student #7:**Hellmich,Adam**Section 112 Student #8:**Horgan,Ruby**Section 112 Student #9:**Ijoma,Lillian**Section 112 Student #10:**Jones,Cate**Section 112 Student #11:**Jotia,Zinzi**Section 112 Student #12:**Lenz,Wyatt**Section 112 Student #13:**Massie,Olivia**Section 112 Student #14:**Miller,Austy**Section 112 Student #15:**Shields,Julia**Section 112 Student #16:**Smith,Kaitlyn**Section 112 Student #17:**Williams,Ava**Section 112 Student #18:**Unassigned,

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So)

**Section 113 Student #1:**Berry,Jaden**Section 113 Student #2:**Brand,Kylee**Section 113 Student #3:**Cattani,Ella**Section 113 Student #4:**Davis,Ethan**Section 113 Student #5:**Duncan,Ellora**Section 113 Student #6:**Espinueva,Shirleen**Section 113 Student #7:**Fisher,Hunter**Section 113 Student #8:**Frizzell,Leah**Section 113 Student #9:**Hagstrom,Steven**Section 113 Student #10:**Ingraham,Emma**Section 113 Student #11:**Johnson,Josh**Section 113 Student #12:**Mccall,Lauren**Section 113 Student #13:**Miles,Abby**Section 113 Student #14:**Mullins,Kaitlyn**Section 113 Student #15:**Slingluff,Cheyenne**Section 113 Student #16:**Wenning,Luke**Section 113 Student #17:**Unassigned,**Section 113 Student #18:**Unassigned,

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So)

**Section 114 Student #1:**Angerstien,Blake**Section 114 Student #2:**Blair,Natalie**Section 114 Student #3:**Dubois,Aleke**Section 114 Student #4:**Elliott,Maggie**Section 114 Student #5:**Hartzell,Molly**Section 114 Student #6:**Kezele,Ashley**Section 114 Student #7:**Lampa,Andrew**Section 114 Student #8:**Mcclellan,Alex**Section 114 Student #9:**Mcdermitt,Brian**Section 114 Student #10:**Meyer,Morgan**Section 114 Student #11:**Morris,Chase**Section 114 Student #12:**Mueller,Maddy**Section 114 Student #13:**Nguyen,Jim**Section 114 Student #14:**Raynewater,Ty**Section 114 Student #15:**Smith,Riley**Section 114 Student #16:**Sobey,Lily**Section 114 Student #17:**Young,Kiefer**Section 114 Student #18:**Unassigned,

**Students 1,2: **(Suggested Exercise 4.5#2) Find two numbers whose difference is 100 and whose product is a minimum. (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 3,4: **(Suggested Exercise 4.5#7) Find the dimensions of a rectangle with perimeter 100m whose area is as large as possible. (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 5,6: **(Suggested Exercise 4.5#11) If 1200 cm^{2} of material is available to make a box with a square base and an open top, find the largest possible volume of the box. (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 7,8: **(Suggested Exercise 4.5#15) Find the point on the line \(y=2x+3\) that is closest to the origin. (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 9,10: **(Suggested Exercise 4.5#17) Find the points on the ellipse \(4x^2+y^2=4\) that are farthest away from the point \((1,0)\) (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 11,12: **(Suggested Exercise 4.5#22)Find the area of the largest rectange that can be inscribed in a right triangle with legs of lengths 3cm and 2cm if two sides of the rectangle lie along the legs. (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 13,14: **(Suggested Exercise 4.5#25) A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 ft, find the dimensions of the window that has the greatest area. (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 15,16: **(Suggested Exercise 4.5#30) A cone-shaped paper drinking cup is to be made to hold 27 cm^{3} of water. Find the height and radius of the cup that will use the smallest amount of paper. (You must use calculus and show all details clearly. No credit for just guessing values.)

**Students 17,18: **(Suggested Exercise 4.5#39) Find an equation of the line through the point \((3,5)\) that cuts off the least area from the first quadrant. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students work in pairs on this (Class Drill on Using Newton's Method)

**Wed Nov 8: **Section 4.7: Antiderivatives (Lecture Notes)

**Fri Nov 10: **Holiday

**Mon Nov 13: Exam ** **X3** **Covering Section Chapter 4 **

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**The Exam will last the full duration of the class period.****No books, notes, calculators, or phones**- Five problems, printed on front & back of three sheets of paper. All problems are based on suggested exercises.
- A problem about Maximum and Minimum Values (Section 4.1)
- A problem about Derivatives and the Shapes of Graphs and/or Curve Sketching (Sections 4.3 and 4.4)
- A problem about Optimization (Section 4.5)
- A problem about Newton's Method (Section 4.6)
**Note:**I will not give you the formula to use for Newton's Method. The formula is presented in the Book, and it was presented in Lecture and in Recitation. You should learn the formula by doing exercises that require you to*use*the formula. There are exercises of this type in the*Homework List*, and you did*Class Drills*in Class on Mon Nov 6 and in Recitation on Tue Nov 7. You can see those*Class Drills*in the calendar entries for those days.

- A problem about Antiderivatives (Section 4.7)
**Note:**I will not give you the give you the**Basic Antidifferentiation Formulas**. The formulas are presented in the Book, and they were presented in Lecture. You should learn those formulas by doing exercises that require you to*use*those formulas. There are exercises of this type in the*Homework List*.

- The most important issue is,
Therefore, the centerpiece of your studying should be,**Can you successfully write down the solution to a problem?****Practicing writing down the solutions to problems.** - Mathematical concepts get presented to you in the
**Book**and in**Lecture**. Examples are also presented in both places. I generally try to**not**present a particular type of example in class if a similar example is already presented well in the book. Rather, I try to present examples in class that are different from the examples that are presented in the book. Therefore, for your studying, you should**be sure to study not just the examples that I do in class, but also the examples that are presented in the book!** - The book and my lectures are not supposed to present examples similar to all of the kinds of problems that you need to know how to solve. The idea is that the book and my lectures teach you the concepts and show you some examples, and from there, you need to be able to
and solve different problems.*generalize* - In writing my
**Quizzes**and**Exams**, I aim to include a mixture of- Problems that are based on problems from the
and that are*Homework List*.*similar to a class example* - Problems that are based on problems from the
and that are*Homework List*, but that are*not similar to a class example*.*similar to a book example* - Problems that are based on problems from the
but that are*Homework List*.*not similar to any class or book example*

- Problems that are based on problems from the

**Tue Nov 14: **Recitation **R12**: Antiderivatives, Position, Velocity, Acceleration (Section 4.7 Leftovers)

All students in the room work on problem **[1]** for about 5 minutes, then the Instructor discusses that problem. Then all students work on problem **[2]** for 5 minutes, and the Instructor discusses that problem, and so on. Some problems will need more time.

**[1] (Suggested Exercise 4.7 #15)** Let
$$f(x)=7x-3x^5$$

- Find the
, \(F(x)\).*General Antiderivative* - Find the
that satisfies \(F(1)=5\).*Particular Antiderivative*

**[2] (not like a book exercise)** Let
$$f(t)=3e^t-4$$

- Find the
, \(F(t)\).*General Antiderivative* - Find the
that satisfies \(F(0)=8\).*Particular Antiderivative*

**[3] (review of prerequisites)** Draw the first quadrant of the ** unit circle**, with important famous angles \(\theta=0,\pi/6,\pi/4,\pi/3,\pi/2 \) shown, along with the \((x,y)\) coordinates of the points where the rays of those angles intersect the circle.

**[4] (4.7#27)** Find \(f(t)\) such that
$$f'(t)=10\cos t - \sec^2 t \ \text{ for } \ -\pi/2 \lt t \lt \pi/2 \ \text{ and that } \ f(\pi/3)=13$$

**[5] (4.7#20)** Suppose that
$$f''(x)=30x-\sin x$$
Find \(f(x)\).

Remember that for an object moving in one dimension, the ** velocity**, \(v(t)\), is the

Therefore, ** position**, \(s(t)\), is an

Also remember that the ** acceleration**, \(a(t)\), is the

Therefore, ** velocity**, \(v(t)\), is an

Furthermore, recall that when an object falls freely under the influence of ** gravity**, it is known that the object will have

**[6] (based on 4.7#40, similar to 4.7#47) **
Suppose that an object is moving in one dimension with velocity
$$v(t)=9\sqrt{t} \ \text{ ft/s}$$

- Find the
, \(s(t)\). That is, find the*general form of the position function*of \(v(t)\), but instead of calling it \(V(t)\), call it \(s(t)\).*General Antiderivative* - Suppose that it is also known that the initial position is \(s(0)=13 \ \text{ft}\). Find the
. That is, find the*position function*that satisfies \(s(0)=13\).*Particular Antiderivative*

**[7] (based on 4.7#43) ** Suppose that an stone is dropped off a tower that is 400 feet tall and falls freely. Let ** position** be defined to be

- What is the value of the
*initial position*of the stone, \(s(0)\)? - What is the value of the
*initial velocity*of the stone, \(v(0)\)? - The
*acceleration*of the stone is constant. What is the value of the the*acceleration*, \(a\)? - Given that the
, \(v(t)\), is an*velocity*of the*antiderivative*, find the*acceleration*, \(v(t)\). That is, find the*general form of the velocity function*of \(a(t)\), but instead of calling it \(A(t)\), call it \(v(t)\).*General Antiderivative* - Knowing what you know about the
*initial velocity*, \(v(0)\), find the, \(v(t)\). That is, find the*particular form of the velocity function*of \(a(t)\) that satisfies $$v(0) = \text{ initial velocity that you identified earlier}$$*Particular Antiderivative* - Given that the
, \(s(t)\), is an*position*of the*antiderivative*, find the*velocity*, \(s(t)\). That is, find the*general form of the position function*of \(v(t)\), but instead of calling it \(V(t)\), call it \(s(t)\).*General Antiderivative* - Knowing what you know about the
*initial position*, \(s(0)\), find the, \(s(t)\). That is, find the*particular form of the position function*of \(v(t)\) that satisfies $$s(0) = \text{ initial position that you identified earlier}$$ The formula that you have found for the*Particular Antiderivative**position function*, \(s(t)\) gives the position of the stone above ground level at time \(t\). - What is the time when the stone reaches ground level?
- What is the
of the stone when it strike the ground?*speed*

**Wed Nov 15: **Section 5.1: Areas and Distances (Lecture Notes)(Class Drill on Riemann Sums)

**Fri Nov 17: **Section 5.2: The Definite Integral (Lecture Notes)

**Mon Nov 20: **Section 5.3: Evaluating Definite Integrals (Lecture Notes)(Quiz **Q8**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Two Problems, 15 points each, printed on front & back of one sheet of paper.
- A Problem about Areas and Distances, based on Suggested Exercises from
**Section 5.1**. - A problem about Definite Integrals, based on Suggested Exercises from
**Section 5.2**.

- A Problem about Areas and Distances, based on Suggested Exercises from

**Tue Nov 21: **Recitation **R13**: The Definite Integral (Section 5.2) and Evaluating Definite Integrals (Section 5.3)

All students in the room work on problem **[1]** for about 10 minutes, then the Instructor discusses that problem. Then all students work on problem **[2]** for 10 minutes, and the Instructor discusses that problem, and so on. Some problems will need more time.

**[1]: **Students work in pairs on this **Class Drill**:
Definite Integrals for a Graph Made Up of Geometric Shapes

**[2]: **Students work in pairs on this **Class Drill**:
Computing Definite Integrals Using Geometry

(the relationship between ** definite integrals** and

If \(f(x)\) is continuous on the interval \([a,b]\), then
$$\int_a^bf(x)dx\underset{\text{ET}}{=}F(b)-F(a)$$
where \(F(x)\) is any ** antiderivative** of \(f(x)\).

Use the ** Evaluation Theorem** to evaluate the integrals. Show all details clearly and use correct notation.

**[3]: **(5.3#3)$$\int_{-2}^{0}\left(\frac{1}{2}t^4+\frac{1}{4}t^3-t\right)dt$$

**[4]: **(5.3#13)$$\int_{1}^{2}\left(\frac{x}{2}-\frac{2}{x}\right)dx$$

**[5]: **(5.3#7)$$\int_{0}^{\pi}\left(5e^x+3\sin x\right)dx$$

**[6]: **(5.3#18)(A lot of rewriting on this one, but it results in a very simple integral!) $$\int_{0}^{\pi/3}\left(\frac{\sin \theta +\sin \theta \tan^2 \theta}{\sec^2 \theta}\right)d\theta$$

**[7]: **(5.3#9)
$$\int_{1}^{4}\left(\frac{4+6u}{\sqrt{u}}\right)du$$

**[8]: **(5.3#23)$$\int_{1}^{e}\left(\frac{x^2+x+1}{x}\right)dx$$

**Wed Nov 22: **Holiday

**Fri Nov 24: **Holiday

**Mon Nov 27: **Section 5.3: Evaluating Definite Integrals (Lecture Notes)

**Tue Nov 28: **Recitation **R14**: Evaluating Definite Integrals (Section 5.3)

All students in the room work on problem **[1]** for about 10 minutes, then the Instructor discusses that problem. Then all students work on problem **[2]** for 10 minutes, and the Instructor discusses that problem, and so on. Some problems will need more time.

(the relationship between ** definite integrals** and

If \(f(x)\) is continuous on the interval \([a,b]\), then
$$\int_a^bf(x)dx\underset{\text{ET}}{=}F(b)-F(a)$$
where \(F(x)\) is any ** antiderivative** of \(f(x)\).

(the relationship between ** definite integrals** and

If \(f(x)\) is continuous on the interval \([a,b]\), then

$$\int_a^bf(x)dx\underset{\text{ET}}{=}\left. \left(\int f(x)dx\right)\right\vert_a^b$$**[1]: **(5.3#11)
$$\int_{0}^{1}x\left(\sqrt[3]{x}+\sqrt[4]{x}\right)dx$$
Give an *exact answer* and a *decimal approximation*, rounded to 3 decimal places.
**Hint: **You’ll have to rewrite the integrand as a sum of power functions before integrating.

**[2]: **(5.3#15)
$$\int_{0}^{1}\left(x^{10}+10^x\right)dx$$
Give an *exact answer* and a *decimal approximation*, rounded to 3 decimal places.
**Hint: **You’ll have to do some sleuth work to figure out one of the antiderivatives. Try checking your book in Section 5.3.

**[3]: **(5.3#29)
$$\int_{-1}^{2}\left(x-2|x|\right)dx$$
Give an *exact answer*.
**Hint: **Remember that the function \(|x|\) is a *piecewise-defined* function. That is, the formula for \(|x|\) depends on which piece of the domain that you are in. That will mean that you will need to break up this definite integral on the interval \([-1,2]\) into *two* definite integrals, each on a smaller interval.

(the ** integral of a rate of change of a quantity** is the

If \(F(x)\) is differentiable on the interval \([a,b]\), then $$\int_a^bF'(x)dx\underset{\text{NCT}}{=}F(b)-F(a)$$

**[4]: **(5.3#51) If \(w'(t)\) is the rate of growth of a child in pounds per year, what does the integral below represent?
$$\int_{5}^{10}w'(t)dt$$

**[5]: **(5.3#52) If oil leaks from a tank at a rate of \(r(t)\) gallons per minute at time \(t\), what does the integral below represent?
$$\int_{0}^{120}r(t)dt$$

**[6]: **(5.3#59) An object moves along a line with velocity
$$v(t)=3t-5 \ \text{ for } \ 0\leq t \leq 3$$

- Find the
*displacement*of the object during the time interval. Give an*exact answer*. - Illustrate your result for (a) using a graph of the velocity \(v(t)\).
- Find the
*distance traveled*by the object during the time interval. Give an*exact answer*and a*decimal approximation*, rounded to 3 decimal places.

**[7]: **(5.3#60) An object moves along a line with velocity
$$v(t)=t^2-2t-3\ \text{ for } \ 0\leq t \leq 6$$

- Find the
*displacement*of the object during the time interval \([2,5]\). Give an*exact answer*. - Illustrate your result for (a) using a graph of the velocity \(v(t)\).
- Find the
*distance traveled*by the object during the time interval \([2,5]\). Give an*exact answer*and a*decimal approximation*, rounded to 3 decimal places.

**Wed Nov 29: **Section 5.4: The Fundamental Theorem of Calculus (Lecture Notes)(Class Drill: Area Function)

**Fri Dec 1: **Section 5.4: The Fundamental Theorem of Calculus (Lecture Notes)(Quiz **Q9**)

- Section 100: Sit in Alternate Seats and rows in Morton 235
- Section 110: Sit in Alternate Seats and rows in Morton 237
**20 Minutes at the end of class****No books, notes, calculators, or phones**- Three Problems, 10 points each, printed on front & back of one sheet of paper.
- A Problem about Evaluating Definite Integrals, based on Suggested Exercises from
**Section 5.3**. - A Problem about Evaluating Definite Integrals, based on Suggested Exercises from
**Section 5.3**. - A problem about the Fundamental Theorem of Calculus, based on Suggested Exercises from
**Section 5.4**.

- A Problem about Evaluating Definite Integrals, based on Suggested Exercises from

**Mon Dec 4 **Section 5.5: The Substitution Rule (Lecture Notes)(Handout on Substitution Method)

**Tue Dec 5: **Recitation **R15**: The Fundamental Theorem of Calculus and the Substitution Rule (Sections 5.4 and 5.5)

If \(f\) is continuous on the interval \([a,b]\), then $$\frac{d}{dx}\left(\int_a^xf(t)dt\right)\underset{\text{FTC1}}{=}f(x) \text{ for } \ a \lt x \lt b$$

**[1]: **(5.4#6) The function \(g(x)\) is defined by the integral:
$$g(x)=\int_{3}^{x}e^{t^2-t} \ dt$$
Find \(g'(x)\).

**[2]: **(5.4#10) The function \(g(x)\) is defined by the integral:
$$g(x)=\int_{0}^{x}\sqrt{1+\sqrt{t}} \ dt$$
Find \(g'(x)\).

**[3]: **(5.4#10) The function \(h(x)\) is defined by the integral:
$$h(x)=\int_{0}^{\tan x}\sqrt{1+\sqrt{t}} \ dt$$
(Hint: You will need the ** Chain Rule**.)

If \(f(x)\) is continuous on the interval \([a,b]\), then the **Average Value of \(f(x)\) on the interval \([a,b]\) ** is defined to be the number

**[4]: ** Find the average value of the function
\(f(x)= \frac{1}{x}\)
on the interval
\([1,4]\).
Simplify your answer.

**[5]: ** Find the average value of the function
\(f(x)= \sin (x) \)
on the interval
\([0,\pi]\).
Simplify your answer.

**[6]: ** Find the average value of the function
\(f(x)= \sec^2(\theta)\)
on the interval
\([0,\pi/4]\).
Simplify your answer.

**[7]: (5.5#3)** Find the Indefinite Integral by using the *Substitution Method*.
$$\int x^4 \sqrt{1+x^5} \ dx$$

**[8]: (5.5#13)** Find the Indefinite Integral by using the *Substitution Method*.
$$\int \frac{1}{7-5x} \ dx$$

**[9]: (5.5#19)** Find the Indefinite Integral by using the *Substitution Method*.
$$\int e^x \sqrt{1+e^x} \ dx$$

**[10]: (5.5#20)** Find the Indefinite Integral by using the *Substitution Method*.
$$\int \cos^4 \theta \sin \theta \ d\theta$$

**[11]: (5.5#27)**

- (
**Review**) Use theor the*Chain Rule*to find the derivative of \(\sec x\). (Either rule can be used. One is simpler.)*Quotient Rule* - Find the Indefinite Integral by using the
*Substitution Method*. $$\int \sec^3 x \tan x \ dx$$ (**Hint:**Rewrite theas \(\sec^2 x \sec x \tan x \).)*integrand*

**Wed Dec 6: **Section 5.5: The Substitution Rule (Lecture Notes)

**Fri Dec 8: **Review (Lecture Notes)

**Thu Dec 14: Combined Final Exam ****FX**** from 2:30pm – 4:30pm**

- Section 100 (Mon, Wed, Fri 8:35am) will have its final in its usual room, Morton 235. Seating will be in Alternate Seats and Alternate Rows.
- Section 110 (Mon, Wed, Fri 10:45am) will have its final in its usual room, Morton 237. Seating will be in Alternate Seats and Alternate Rows.

page maintained by Mark Barsamian, last updated Tue Dec 12, 2023