**Campus: **Ohio University, Athens Campus

**Department: **Mathematics

**Instructor: **Mark Barsamian (Contact Information and Office Hours are on his Web Page.)

Section 110 is a Lecture Section, taught by Mark Barsamian.

- Lecture Section 110 meets Mon Wed, Fri 11:50am – 12:45pm in Morton Hall 237.

Associated to this Lecture Section are four Recitation Sections, led by Delfino Nolasco

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

Delfino Nolasco's Office Hours:

- Tue 5:00pm – 7:00pm in Morton 313
- Wed 6:00pm – 8:00pm in Morton 313

Email the instructors by clicking the links below.

- Send Mark Barsamian an email by clicking this link: barsamia@ohio.edu
- Send Delfino Nolasco an email by clicking this link: dn751620@ohio.edu

**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)

**Special Needs: **If you have physical, psychiatric, or learning disabilities that require accommodations, please let me know as soon as possible so that your needs may be appropriately met.

**Final Exam Date: **All Athens Campus Sections of MATH 2301 have a Common Final Exam on Tue May 2, 2023, from 4:40pm – 6:40pm 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 Information: **

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 $20.48 plus Ohio sales tax, for a total of about $22. That cost will be automatically billed to your Ohio University Student Account. If you drop the course before the drop deadline (Friday, Jan 27, 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 $104 plus Ohio sales tax for a total of about $110. Note that the Retail Price is significantly higher than the Inclusive Access Price. (Some students prefer reading printed books rather than eBooks. When you access the Online Course Materials, you will find information about an optional add-on of a printed copy of the textbook for a discounted price.)(The book that we will be using is James Stewart, Essential Calculus: Early Transcendentals, 2nd Edition.)

**Exercises: **

(from Stewart Essential Calculus Early Transcendentals 2nd Edition)

Your goal should be to write solutions to all 319 exercises in this list.

Printable PDF of the Exercise List

- 1.3 The Limit of a Function: 1, 5, 7, 11, 15, 23, 33, 39
- 1.4 Calculating Limits: 5, 7, 11, 15, 17, 21, 23, 25, 27, 31, 33, 35, 38, 49, 51
- 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, 27, 31, 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, 41, 45, 55, 57
- 3.4 Exponential Growth & Decay: 1, 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, 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, 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, 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 the Recitation Instructor (Delfino Nolasco), or me, to look over your written work with you.

**Grading: **

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

**Recitation:**14 Tuesday Recitation Activities @ 5 points each = 70 points possible**Quizzes:**9 quizzes @ 30 points each = 270 points possible**Exams:**3 Exams @ 150 points each = 450 points possible**Final Exam:**210 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

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**Exercises:**There is a list of Suggested 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. You are also encouraged to type the answers to the exercises into the online*WebAssign*system, which will give you confirmation of whether or not your answers are correct. The WebAssign system will give you scores for the homework that you do in WebAssign, but those scores are not part of your course grade.

During the semester, you can keep track of your scores on all the ** Graded Items: Recitations, Quizzes, Exams, Final Exam**. (You can also find scores for all those items in the

Here is a sample of a partially filled out Grade Calculation Worksheet.

**Calendar: **

Items in **red** are graded.

**Mon Jan 16: **Holiday

**Tue Jan 17: **Recitation **R01**: Course Introduction and Diagnostic Test
Sample Problems for Diagnostic Test

**Wed Jan 18: **Section 1.3: The Limit of a Function (Meeting Notes)

**Fri Jan 20: **Section 1.4: Calculating Limits (Meeting Notes)

**Mon Jan 23: **Section 1.4: Calculating Limits (Meeting Notes)

**Tue Jan 24: **Recitation **R02**: Calculating Limits (Section 1.4)

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

Three or four pairs of students should be able to work simultaneously, at white boards around Ellis 107. There may be two or three rounds. Instructor Delfino Nolasco 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 Delfino will lead a discussion of the solutions on the white board. He and the students in the class will talk about what is good and not so good in each solution. Then the white boards will be erased and the next round of students will come to the white 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 Jan 24 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.

**Section 111 Student #1:**Ankita Bansode**Section 111 Student #2:**Madison Clark**Section 111 Student #3:**Aubrey Dearwester**Section 111 Student #4:**Seth Hammond**Section 111 Student #5:**Will Hanzel**Section 111 Student #6:**Alaina Hatton**Section 111 Student #7:**Isabella Huff**Section 111 Student #8:**Kaci Jeffries**Section 111 Student #9:**Allison Johnston**Section 111 Student #10:**Claire Kingsley**Section 111 Student #11:**Madison Klobucar**Section 111 Student #12:**Lauren Lietzke**Section 111 Student #13:**Mallory Lindsay**Section 111 Student #14:**Sean Little**Section 111 Student #15:**Eric Mcculloch**Section 111 Student #16:**Olivia Ohms**Section 111 Student #17:**Lillian Peters**Section 111 Student #18:**Allison Rayburg**Section 111 Student #19:**Jacob Sizemore**Section 111 Student #20:**Jillian Zeigler

**Section 112 Student #1:**Miriam Churchin**Section 112 Student #2:**Kate Cunningham**Section 112 Student #3:**Rilee Davis**Section 112 Student #4:**Tyler Edgar**Section 112 Student #5:**Olivia Fanelli**Section 112 Student #6:**Jack Golla**Section 112 Student #7:**Sofia Hoffman**Section 112 Student #8:**Olivia Hopper**Section 112 Student #9:**Isabelle Isco**Section 112 Student #10:**Morgan Isla**Section 112 Student #11:**Rachel Jackson**Section 112 Student #12:**Austin Kiggins**Section 112 Student #13:**Katie McGrath**Section 112 Student #14:**Ray Olin**Section 112 Student #15:**Addison Richards**Section 112 Student #16:**Taylor Wagner**Section 112 Student #17:**Emily Williams**Section 112 Student #18:**Gavin Wolfe

**Section 113 Student #1:**Cindy Childers**Section 113 Student #2:**Mia Collier**Section 113 Student #3:**Emily Croy**Section 113 Student #4:**Chloe Fulmer**Section 113 Student #5:**Shelby Gorman**Section 113 Student #6:**Ellie Hofmeister**Section 113 Student #7:**William Kilbane**Section 113 Student #8:**Paulina Kramarczyk**Section 113 Student #9:**Finn Lennerth**Section 113 Student #10:**Sarah Lockwood**Section 113 Student #11:**Constantine Nonno**Section 113 Student #12:**Katherine O'loughlin**Section 113 Student #13:**Charlee Pickens**Section 113 Student #14:**Allie Pierce**Section 113 Student #15:**Carson Rivera-Gebeau**Section 113 Student #16:**Mercedes Santamaria**Section 113 Student #17:**Rachel Stern**Section 113 Student #18:**Ezra Taylor**Section 113 Student #19:**Delaney Watson**Section 113 Student #20:**Kaitlyn Welch**Section 114 Student #1:**Wilona Acquah

**Section 114 Student #2:**Aria Carter**Section 114 Student #3:**Ranlyn Chowdhury**Section 114 Student #4:**Christopher Crabtree**Section 114 Student #5:**Tracie Crookston**Section 114 Student #6:**Ethan Davis**Section 114 Student #7:**Ella Guthrie**Section 114 Student #8:**Brady Kuntz**Section 114 Student #9:**Andrew Lampa**Section 114 Student #10:**Olivia Moores**Section 114 Student #11:**Spencer Nadeau**Section 114 Student #12:**Savanna Nichols**Section 114 Student #13:**Lauren Nuske**Section 114 Student #14:**Riley Pocisk**Section 114 Student #15:**Hanna Popp**Section 114 Student #16:**Ty Raynewater**Section 114 Student #17:**Daniel Richter**Section 114 Student #18:**Dominik Spencer**Section 114 Student #19:**Brice Wood

**Students 1,2: ** (1.4#15) Find the limit

**Students 3,4: ** (1.4#17) Find the limit

**Students 5,6: ** (1.4#25) Find the limit

**Students 7,8: ** (1.4#21) Find the limit

**Students 9,10: ** (1.4#23) Find the limit

**Students 11,12: ** (1.4#38) 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: ** (1.4#35) Prove that
$$\lim_{x\rightarrow 0}x^2\cos{(20\pi x)}=0$$

**Students 17,18: ** (1.4#49) Find the limit

**Students 19,20: ** (1.4#51) Find the limit

**Wed Jan 25: **Section 1.5: Continuity (Meeting Notes)

**Fri Jan 27: **Section 1.6: Limits Involving Infinity (Meeting Notes)(Quiz **Q1**)(Last Day to Drop Without a W)

**Fri Jan 27 Quiz Q1 Covering Sections 1.3 and 1.4**

- Sit in Alternate Seats
**20 Minutes at the end of class****No Calculators, No Cell 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 Jan 30: **Section 1.6 Limits: Involving Infinity (Meeting Notes)

**Tue Jan 31: **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 whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Three or four pairs of students should be able to work simultaneously, at white boards around Ellis 107. There may be two or three rounds. Instructor Delfino Nolasco 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 Delfino will lead a discussion of the solutions on the white board. He and the students in the class will talk about what is good and not so good in each solution. Then the white boards will be erased and the next round of students will come to the white 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 Jan 24 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.

**Section 111 Student #1:**Ankita Bansode**Section 111 Student #2:**Madison Clark**Section 111 Student #3:**Aubrey Dearwester**Section 111 Student #4:**Seth Hammond**Section 111 Student #5:**Will Hanzel**Section 111 Student #6:**Alaina Hatton**Section 111 Student #7:**Kaci Jeffries**Section 111 Student #8:**Allison Johnston**Section 111 Student #9:**Claire Kingsley**Section 111 Student #10:**Madison Klobucar**Section 111 Student #11:**Lauren Lietzke**Section 111 Student #12:**Mallory Lindsay**Section 111 Student #13:**Sean Little**Section 111 Student #14:**Eric Mcculloch**Section 111 Student #15:**Olivia Ohms**Section 111 Student #16:**Lillian Peters**Section 111 Student #17:**Allison Rayburg**Section 111 Student #18:**Jacob Sizemore**Section 111 Student #19:**Jillian Zeigler

**Section 112 Student #1:**Miriam Churchin**Section 112 Student #2:**Kate Cunningham**Section 112 Student #3:**Rilee Davis**Section 112 Student #4:**Tyler Edgar**Section 112 Student #5:**Olivia Fanelli**Section 112 Student #6:**Jack Golla**Section 112 Student #7:**Sofia Hoffman**Section 112 Student #8:**Olivia Hopper**Section 112 Student #9:**Isabelle Isco**Section 112 Student #10:**Morgan Isla**Section 112 Student #11:**Rachel Jackson**Section 112 Student #12:**Austin Kiggins**Section 112 Student #13:**Katie McGrath**Section 112 Student #14:**Ray Olin**Section 112 Student #15:**Addison Richards**Section 112 Student #16:**Taylor Wagner**Section 112 Student #17:**Emily Williams**Section 112 Student #18:**Gavin Wolfe

**Section 113 Student #1:**Cindy Childers**Section 113 Student #2:**Mia Collier**Section 113 Student #3:**Emily Croy**Section 113 Student #4:**Shelby Gorman**Section 113 Student #5:**Ellie Hofmeister**Section 113 Student #6:**William Kilbane**Section 113 Student #7:**Paulina Kramarczyk**Section 113 Student #8:**Finn Lennerth**Section 113 Student #9:**Sarah Lockwood**Section 113 Student #10:**Constantine Nonno**Section 113 Student #11:**Katherine O'loughlin**Section 113 Student #12:**Charlee Pickens**Section 113 Student #13:**Allie Pierce**Section 113 Student #14:**Carson Rivera-Gebeau**Section 113 Student #15:**Mercedes Santamaria**Section 113 Student #16:**Rachel Stern**Section 113 Student #17:**Ezra Taylor**Section 113 Student #18:**Delaney Watson**Section 113 Student #19:**Kaitlyn Welch

**Section 114 Student #1:**Wilona Acquah**Section 114 Student #2:**Aria Carter**Section 114 Student #3:**Ranlyn Chowdhury**Section 114 Student #4:**Christopher Crabtree**Section 114 Student #5:**Tracie Crookston**Section 114 Student #6:**Ella Guthrie**Section 114 Student #7:**Brady Kuntz**Section 114 Student #8:**Andrew Lampa**Section 114 Student #9:**Olivia Moores**Section 114 Student #10:**Spencer Nadeau**Section 114 Student #11:**Lauren Nuske**Section 114 Student #12:**Riley Pocisk**Section 114 Student #13:**Hanna Popp**Section 114 Student #14:**Ty Raynewater**Section 114 Student #15:**Daniel Richter**Section 114 Student #16:**Brice Wood

**Students 1,2: ** (1.6#13) Find the limits using the methods of **Section 1.6 Example 2**. Show all details clearly and use correct notation.

**Students 3,4: ** (1.6#19)

- 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?

**Students 5,6: ** (1.6#19)

- 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?

**Students 7,8: ** (1.6#19)

- 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?

**Students 9,10: ** (1.6#25)

- 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 rational function?

**Students 11,12: ** (1.6#29) Find the limit

**Students 13,14: ** (1.6#35) Find the horizontal and vertical asymptotes of the rational function. (Give their ** line equations** and say if they are horizontal or vertical. Illustrate your results with a sketch of the graph of the function.
$$y=\frac{2x^2+x-1}{x^2+x-2}$$

**Students 15,16: ** (1.6#40) Find a formula for a function that has vertical asymptotes at \(x=2\) and \(x=5\) and horizontal asymptote \(y=3\). Illustrate with a sketch of the graph of the function.

**Students 17,18: ** (1.6#41) Find a formula for a function \(f\) that satisfies all of the following conditions.

- \(lim_{x\rightarrow \pm \infty} f(x) = 0\)
- \(lim_{x\rightarrow 0} f(x) = -\infty \)
- \(f(2)=0\)
- \(lim_{x\rightarrow 3^-} f(x) = \infty \)
- \(lim_{x\rightarrow 3^+} f(x) = -\infty \)

**Students 19,20 ** Do Exercise **1.6#49**.

**Wed Feb 1: **Section 2.1: Derivatives and Rates of Change (Meeting Notes)

**Fri Feb 3: **Section 2.1: Derivatives and Rates of Change (Meeting Notes)(Quiz **Q2**)

**Fri Feb 3 Quiz Q2 Covering Sections 1.6 and 2.1**

- Sit in Alternate Seats
**20 Minutes at the end of class****No Calculators, No Cell Phones**- Two Problems, 15 points each, printed on front & back of one sheet of paper
- One problem based on Suggested Exercises from
**Section 1.6**. - One problem based on Suggested Exercises from
**Section 2.1**.

- One problem based on Suggested Exercises from

**Mon Feb 6: **Section 2.2: The Derivative as a Function (Meeting Notes)

**Tue Feb 7: **Recitation **R04**: Calculating Derivatives (Section 2.2)

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

Three or four pairs of students should be able to work simultaneously, at white boards around Ellis 107. There may be two or three rounds. Instructor Delfino Nolasco 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 Delfino will lead a discussion of the solutions on the white board. He and the students in the class will talk about what is good and not so good in each solution. Then the white boards will be erased and the next round of students will come to the white board and write the solution to their problems.

**Scoring: **Students will be graded on a scale from 0 - 5 for the quality of their work

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

**Section 111 Student #1:**Bansode, Ankita**Section 111 Student #2:**Clark, Madison**Section 111 Student #3:**Dearwester, Aubrey**Section 111 Student #4:**Hammond, Seth**Section 111 Student #5:**Hanzel, Will**Section 111 Student #6:**Hatton, Alaina**Section 111 Student #7:**Jeffries, Kaci**Section 111 Student #8:**Johnston, Allison**Section 111 Student #9:**Kingsley, Claire**Section 111 Student #10:**Klobucar, Madison**Section 111 Student #11:**Lietzke, Lauren**Section 111 Student #12:**Lindsay, Mallory**Section 111 Student #13:**Little, Sean**Section 111 Student #14:**Mcculloch, Eric**Section 111 Student #15:**Ohms, Olivia**Section 111 Student #16:**Peters, Lillian**Section 111 Student #17:**Rayburg, Allison**Section 111 Student #18:**Sizemore, Jacob**Section 111 Student #19:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Richards, Addison**Section 112 Student #16:**Wagner, Taylor**Section 112 Student #17:**Williams, Emily**Section 112 Student #18:**Wolfe, Gavin

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Collier, Mia**Section 113 Student #3:**Croy, Emily**Section 113 Student #4:**Gorman, Shelby**Section 113 Student #5:**Hofmeister, Ellie**Section 113 Student #6:**Kilbane, William**Section 113 Student #7:**Kramarczyk, Paulina**Section 113 Student #8:**Lennerth, Finn**Section 113 Student #9:**Lockwood, Sarah**Section 113 Student #10:**Nonno, Constantine**Section 113 Student #11:**O'loughlin, Katherine**Section 113 Student #12:**Pickens, Charlee**Section 113 Student #13:**Pierce, Allie**Section 113 Student #14:**Rivera-Gebeau, Carson**Section 113 Student #15:**Santamaria, Mercedes**Section 113 Student #16:**Stern, Rachel**Section 113 Student #17:**Taylor, Ezra**Section 113 Student #18:**Watson, Delaney**Section 113 Student #19:**Welch, Kaitlyn

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #4:**Crabtree, Christopher**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Lampa, Andrew**Section 114 Student #9:**Moores, Olivia**Section 114 Student #10:**Nadeau, Spencer**Section 114 Student #11:**Nuske, Lauren**Section 114 Student #12:**Popp, Hanna**Section 114 Student #13:**Raynewater, Ty**Section 114 Student #14:**Richter, Daniel**Section 114 Student #15:**Wood, Brice

**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. - 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. - 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.

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

- Find \(f'(2)\) using 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.*Definition of the Derivative* - 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. Show all steps clearly and explain key steps.*Definition of the Derivative* - 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. Show all steps clearly and explain key steps.*Definition of the Derivative* - 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. Show all steps clearly and explain key steps.*Definition of the Derivative* - What is the slope of the line tangent to the graph of \(g(x)\) at \(x=5\)? Explain.

**Wed Feb 8: **Section 2.2: The Derivative as a Function (Meeting Notes)

**Fri Feb 10: **Exam **X1** Covering Through Section 2.2

**Mon Feb 13: **Section 2.3: Basic Differentiation Formulas (Meeting Notes)

**Tue Feb 14: **Recitation **R05**: Using Basic Differentiation Formulas (Section 2.5)

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

**Scoring: **Students will be graded on a scale from 0 - 5 for the quality of their work

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

**Section 111 Student #1:**Bansode, Ankita**Section 111 Student #2:**Clark, Madison**Section 111 Student #3:**Dearwester, Aubrey**Section 111 Student #4:**Hammond, Seth**Section 111 Student #5:**Hanzel, Will**Section 111 Student #6:**Hatton, Alaina**Section 111 Student #7:**Jeffries, Kaci**Section 111 Student #8:**Johnston, Allison**Section 111 Student #9:**Kingsley, Claire**Section 111 Student #10:**Klobucar, Madison**Section 111 Student #11:**Lietzke, Lauren**Section 111 Student #12:**Lindsay, Mallory**Section 111 Student #13:**Little, Sean**Section 111 Student #14:**Mcculloch, Eric**Section 111 Student #15:**Ohms, Olivia**Section 111 Student #16:**Peters, Lillian**Section 111 Student #17:**Rayburg, Allison**Section 111 Student #18:**Sizemore, Jacob**Section 111 Student #19:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Richards, Addison**Section 112 Student #16:**Wagner, Taylor**Section 112 Student #17:**Williams, Emily**Section 112 Student #18:**Wolfe, Gavin

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Collier, Mia**Section 113 Student #3:**Croy, Emily**Section 113 Student #4:**Gorman, Shelby**Section 113 Student #5:**Hofmeister, Ellie**Section 113 Student #6:**Kilbane, William**Section 113 Student #7:**Kramarczyk, Paulina**Section 113 Student #8:**Lennerth, Finn**Section 113 Student #9:**Lockwood, Sarah**Section 113 Student #10:**Nonno, Constantine**Section 113 Student #11:**O'loughlin, Katherine**Section 113 Student #12:**Pickens, Charlee**Section 113 Student #13:**Pierce, Allie**Section 113 Student #14:**Rivera-Gebeau, Carson**Section 113 Student #15:**Santamaria, Mercedes**Section 113 Student #16:**Stern, Rachel**Section 113 Student #17:**Taylor, Ezra**Section 113 Student #18:**Watson, Delaney**Section 113 Student #19:**Welch, Kaitlyn

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #4:**Crabtree, Christopher**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Lampa, Andrew**Section 114 Student #9:**Moores, Olivia**Section 114 Student #10:**Nadeau, Spencer**Section 114 Student #11:**Nuske, Lauren**Section 114 Student #12:**Popp, Hanna**Section 114 Student #13:**Raynewater, Ty**Section 114 Student #14:**Richter, Daniel**Section 114 Student #15:**Wood, Brice

**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 Feb 15: **Section 2.3: Basic Differentiation Formulas (Meeting Notes)

**Fri Feb 16: **Section 2.4: The Product and Quotient Rules (Meeting Notes)(Quiz **Q3**)

**Mon Feb 20: **Section 2.5: The Chain Rule (Meeting Notes)

**Tue Feb 21: **Recitation **R06**: Using the Product, Quotient, and Chain Rules (Sections 2.4, 2.5)

**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**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$
$$\frac{d}{dx}\tan(x)=(\sec(x))^2$$

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Collier, Mia**Section 113 Student #3:**Croy, Emily**Section 113 Student #4:**Gorman, Shelby**Section 113 Student #5:**Hofmeister, Ellie**Section 113 Student #6:**Kilbane, William**Section 113 Student #7:**Kramarczyk, Paulina**Section 113 Student #8:**Lennerth, Finn**Section 113 Student #9:**Lockwood, Sarah**Section 113 Student #10:**Nonno, Constantine**Section 113 Student #11:**O'loughlin, Katherine**Section 113 Student #12:**Pickens, Charlee**Section 113 Student #13:**Pierce, Allie**Section 113 Student #14:**Rivera-Gebeau, Carson**Section 113 Student #15:**Santamaria, Mercedes**Section 113 Student #16:**Stern, Rachel**Section 113 Student #17:**Taylor, Ezra**Section 113 Student #18:**Watson, Delaney**Section 113 Student #19:**Welch, Kaitlyn**Section 113 Student #20:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #4:**Crabtree, Christopher**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Lampa, Andrew**Section 114 Student #9:**Moores, Olivia**Section 114 Student #10:**Nadeau, Spencer**Section 114 Student #11:**Nuske, Lauren**Section 114 Student #12:**Popp, Hanna**Section 114 Student #13:**Raynewater, Ty**Section 114 Student #14:**Richter, Daniel**Section 114 Student #15:**Wood, Brice

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

Four or Five pairs of students should be able to work simultaneously, at white boards around Ellis 107. There may be two or three rounds. Instructor Delfino Nolasco 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 Delfino will lead a discussion of the solutions on the white board. He and the students in the class will talk about what is good and not so good in each solution. Then the white boards will be erased and the next round of students will come to the white board and write the solution to their problems.

**Scoring: **Students will be graded on a scale from 0 - 5 for the quality of their work

**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.

**Delfino 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

**Delfino 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 \(y=mx+b\).

**Delfino 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 Feb 22: **Section 2.6: Implicit Differentiation (Meeting Notes)

**Fri Feb 24: **Section 2.7: Related Rates (Meeting Notes)(Quiz **Q4**)

- Sit in Alternate Seats
**20 Minutes at the end of class****No Calculators, No Cell Phones**- Three Problems, 10 points each, 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 Product Rule problem based on Suggested Exercises from

**Mon Feb 27: **Section 2.8: Linear Approx & Differentials (Meeting Notes)

**Tue Feb 28: **Recitation **R07**: Implicit Differentiation and Related Rates (Sections 2.6 and 2.7)

**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**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$
$$\frac{d}{dx}\tan(x)=(\sec(x))^2$$

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Collier, Mia**Section 113 Student #3:**Croy, Emily**Section 113 Student #4:**Gorman, Shelby**Section 113 Student #5:**Hofmeister, Ellie**Section 113 Student #6:**Kilbane, William**Section 113 Student #7:**Kramarczyk, Paulina**Section 113 Student #8:**Lennerth, Finn**Section 113 Student #9:**Lockwood, Sarah**Section 113 Student #10:**Nonno, Constantine**Section 113 Student #11:**O'loughlin, Katherine**Section 113 Student #12:**Pickens, Charlee**Section 113 Student #13:**Pierce, Allie**Section 113 Student #14:**Rivera-Gebeau, Carson**Section 113 Student #15:**Santamaria, Mercedes**Section 113 Student #16:**Stern, Rachel**Section 113 Student #17:**Taylor, Ezra**Section 113 Student #18:**Watson, Delaney**Section 113 Student #19:**Welch, Kaitlyn**Section 113 Student #20:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #4:**Crabtree, Christopher**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Lampa, Andrew**Section 114 Student #9:**Moores, Olivia**Section 114 Student #10:**Nadeau, Spencer**Section 114 Student #11:**Nuske, Lauren**Section 114 Student #12:**Popp, Hanna**Section 114 Student #13:**Raynewater, Ty**Section 114 Student #14:**Richter, Daniel**Section 114 Student #15:**Wood, Brice

**Students 1,2 ** (2.6#5) Find \(dy/dx\) by implicit differentiation.
$$x^2+xy-y^2=4$$

**Students 3,4 ** (2.6#7) Find \(dy/dx\) by implicit differentiation.
$$y\cos{(x)}=x^2+y^2$$

**Students 5,6 ** (2.6#9) Find \(dy/dx\) by implicit differentiation.
$$4\cos{(x)}sin{(y)}=1$$

**Students 7,8 ** (2.6#11) Find \(dy/dx\) by implicit differentiation.
$$\tan{(x/y)}=x+y$$

**Students 9,10 ** (2.6#13) Find \(dy/dx\) by implicit differentiation.
$$\sqrt{xy}=1+x^2y$$

**Students 11,12 ** (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 13,14 ** (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.

**Students 15,16 ** (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.

**Students 17,18 ** (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 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.
- This problem is not clearly written. It is not clear whether the \(2\) mi is referring to the
*base*of the triangle, or the*hypotenuse*. Which did you assume? Why? The authors' answer to this problem is \(250\sqrt{3}\) mi/h. Do you think that they used the \(2\) mi as the*base*of the triangle, or the*hypotenuse*? Explain.

**Students 19,20 ** (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.

**Students 1,2,3,4 ** (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.

**Students 5,6,7,8 ** (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.

**Students 9,10,11,12 ** (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 strng and the horizontal decreasing when \(200\) ft of string have been let out?
Make a good drawing and use correct units in your answer.

**Students 13,14,15,16 ** 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.

**Students 17,18,19,20** (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.

**Wed Mar 1: **Section 3.1: Exponential Functions (Meeting Notes)

**Fri Mar 3: **Section 3.2: Inverse Functions, Logarithms (Meeting Notes)(Quiz **Q5**)

- Sit in Alternate Seats
**20 Minutes at the end of class****No Calculators, No Cell 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

**Mon Mar 6: **Section 3.3: Derivatives of Logarithmic and Exponential Functions (Meeting Notes)

**Tue Mar 7: **Recitation **R08**: Derivatives of Logarithmic and Exponential Functions (Section 3.3)

**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**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$
$$\frac{d}{dx}\tan(x)=(\sec(x))^2$$

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

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

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Collier, Mia**Section 113 Student #3:**Croy, Emily**Section 113 Student #4:**Gorman, Shelby**Section 113 Student #5:**Hofmeister, Ellie**Section 113 Student #6:**Kilbane, William**Section 113 Student #7:**Kramarczyk, Paulina**Section 113 Student #8:**Lennerth, Finn**Section 113 Student #9:**Lockwood, Sarah**Section 113 Student #10:**Nonno, Constantine**Section 113 Student #11:**O'loughlin, Katherine**Section 113 Student #12:**Pickens, Charlee**Section 113 Student #13:**Pierce, Allie**Section 113 Student #14:**Rivera-Gebeau, Carson**Section 113 Student #15:**Santamaria, Mercedes**Section 113 Student #16:**Stern, Rachel**Section 113 Student #17:**Taylor, Ezra**Section 113 Student #18:**Watson, Delaney**Section 113 Student #19:**Welch, Kaitlyn**Section 113 Student #20:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #4:**Crabtree, Christopher**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Lampa, Andrew**Section 114 Student #9:**Moores, Olivia**Section 114 Student #10:**Nadeau, Spencer**Section 114 Student #11:**Nuske, Lauren**Section 114 Student #12:**Popp, Hanna**Section 114 Student #13:**Raynewater, Ty**Section 114 Student #14:**Richter, Daniel**Section 114 Student #15:**Wood, Brice

**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.
$$G(x)=\ln\left(\frac{(2y+1)^5}{\sqrt{y^2+1}}\right)$$
**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 \(G(y)\) so that it is not the logarithm of a quotient. Then find the derivative of the rewritten function.

**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 \(y=\ln{\left(x^2-3x+1\right)}\) at \(x=3\).

**Students 9,10,11,12 ** (3.3#45) Find the equation of the line tangent to the graph of \(y=\frac{e^{(x)}}{x}\) at \(x=1\).

**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 Mar 8: **Section 3.4: Exponential Growth & Decay (Meeting Notes)

**Fri Mar 10: **Exam **X2** Covering Section 2.3 through Chapter 3

**The Exam will last the full duration of the class period.****No Calculators, No Cell Phones**- Sit in Alternate Seats
- Eight problems, 20 points each, printed on front & back of two sheets of paper
- 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 problems in Biology (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.)

- Notice that there are actually 160 points possible on Exam X2. Exams are worth 150 points in the grading scheme for this course. Any points scored over 150 on Exam X2 will be considered Extra Credit.

**Mon Mar 13 - Fri Mar 17: **Holiday

**Mon Mar 20: **Section 4.1: Maximum and Minimum Values (Meeting Notes)

**Tue Mar 21: **Recitation **R09**: Extrema and Critical Numbers (Section 4.1)

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Collier, Mia**Section 113 Student #3:**Croy, Emily**Section 113 Student #4:**Gorman, Shelby**Section 113 Student #5:**Hofmeister, Ellie**Section 113 Student #6:**Kilbane, William**Section 113 Student #7:**Kramarczyk, Paulina**Section 113 Student #8:**Lennerth, Finn**Section 113 Student #9:**Lockwood, Sarah**Section 113 Student #10:**Nonno, Constantine**Section 113 Student #11:**O'loughlin, Katherine**Section 113 Student #12:**Pickens, Charlee**Section 113 Student #13:**Pierce, Allie**Section 113 Student #14:**Rivera-Gebeau, Carson**Section 113 Student #15:**Santamaria, Mercedes**Section 113 Student #16:**Stern, Rachel**Section 113 Student #17:**Taylor, Ezra**Section 113 Student #18:**Watson, Delaney**Section 113 Student #19:**Welch, Kaitlyn**Section 113 Student #20:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #4:**Crabtree, Christopher**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Lampa, Andrew**Section 114 Student #9:**Moores, Olivia**Section 114 Student #10:**Nadeau, Spencer**Section 114 Student #11:**Nuske, Lauren**Section 114 Student #12:**Popp, Hanna**Section 114 Student #13:**Raynewater, Ty**Section 114 Student #14:**Richter, Daniel**Section 114 Student #15:**Wood, Brice

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)
$$f(x)=x^{2/5}(x-7)$$

**Wed Mar 22: **Section 4.1: Maximum and Minimum Values (Meeting Notes)

**Fri Mar 24: **Section 4.2: The Mean Value Theorem (Meeting Notes)(Quiz **Q6**)

**Mon Mar 27: **Section 4.3: Derivatives and the Shapes of Graphs (Meeting Notes)

**Tue Mar 28: **Recitation **R10**: Sections 4.2 and 4.3

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Collier, Mia**Section 113 Student #3:**Croy, Emily**Section 113 Student #4:**Gorman, Shelby**Section 113 Student #5:**Hofmeister, Ellie**Section 113 Student #6:**Kilbane, William**Section 113 Student #7:**Lennerth, Finn**Section 113 Student #8:**Lockwood, Sarah**Section 113 Student #9:**Nonno, Constantine**Section 113 Student #10:**O'loughlin, Katherine**Section 113 Student #11:**Pickens, Charlee**Section 113 Student #12:**Pierce, Allie**Section 113 Student #13:**Rivera-Gebeau, Carson**Section 113 Student #14:**Santamaria, Mercedes**Section 113 Student #15:**Stern, Rachel**Section 113 Student #16:**Taylor, Ezra**Section 113 Student #17:**Watson, Delaney**Section 113 Student #18:**Welch, Kaitlyn**Section 113 Student #29:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #4:**Crabtree, Christopher**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Moores, Olivia**Section 114 Student #9:**Nadeau, Spencer**Section 114 Student #10:**Nuske, Lauren**Section 114 Student #11:**Popp, Hanna**Section 114 Student #12:**Raynewater, Ty**Section 114 Student #13:**Richter, Daniel**Section 114 Student #14:**Wood, Brice

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

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

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

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

**Students 1,2: **Consider the function \(f(x)=5-12x+3x^2\) and the interval \([1,3]\)

- 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.

**Students 3,4: **Consider the function \(f(x)=x^3-x^2-6x+2\) and the interval \([0,3]\)

- 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.

**Students 5,6: **Consider the function \(f(x)=\sqrt{x}-\frac{1}{5}x\) and the interval \([0,25]\)

- 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 7,8: **Consider the function \(f(x)=\cos(2x)\) and the interval \([\frac{\pi}{8},\frac{7\pi}{8}]\)

- 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 9,10: **Consider the function \(f(x)=x+\frac{1}{x}\) and the interval \([\frac{1}{2},2]\)

- 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 11,12: **Consider the function \(f(x)=2x^2-3x+1\) and the interval \([0,2]\)

- 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 13,14: **Consider the function \(f(x)=x^3-3x+2\) and 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 15,16: **Consider the function \(f(x)=\ln(x)\) and 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 17,18: **Consider the function \(f(x)=\frac{1}{x}\) and the interval \([1,3]\)

- Find all numbers \(c\) that satisfy the
*conclusion*of the Mean Value Theorem. Show all details clearly.

**Students 19,20: **Consider the function \(f(x)=\sqrt{x}\) and the interval \([4,25]\)

- Find all numbers \(c\) that satisfy the
*conclusion*of the Mean Value Theorem. Show all details clearly.

**Students 1,2: **Let \(f(x)=2x^3+3x^2-36x\)

- 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.

**Students 3,4: **Let \(f(x)=4x^3+3x^2-6x+1\).

- 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.

**Students 5,6: **Let \(f(x)=x^4-2x^2+3\).

- 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.

**Students 7,8: **Let \(f(x)=\frac{x}{x^2+1}\).

- 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.

**Students 9,10: **Let \(f(x)=\sin(x)+\cos(x)\) on the interval \(0 \leq x \leq 2\pi\).

- 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.

**Students 11,12: **Let \(f(x)=\cos^2(x)-2\sin(x)\) on the interval \(0 \leq x \leq 2\pi\).

- 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.

**Students 13,14: **Let \(f(x)=e^{2x}+e^{-x}\).

- 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.

**Students 15,16: **Let \(f(x)=x^2\ln x\).

- 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.

**Students 17,18: **Let \(f(x)=x^2-x-\ln x\).

- 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.

**Students 19,20: **Let \(f(x)=x^4e^{-x}\).

- 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.

**Wed Mar 29: **Section 4.4: Curve Sketching (Meeting Notes)

**Fri Mar 31: **Section 4.4: Curve Sketching (Meeting Notes)(Quiz **Q7**)(Last Day to Drop)

**Mon Apr 3: **Section 4.5: Optimization (Meeting Notes)

**Tue Apr 4: **Recitation **R11**: Optimization Problems (Sections 4.5)

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Croy, Emily**Section 113 Student #3:**Gorman, Shelby**Section 113 Student #4:**Hofmeister, Ellie**Section 113 Student #5:**Kilbane, William**Section 113 Student #6:**Lennerth, Finn**Section 113 Student #7:**Lockwood, Sarah**Section 113 Student #8:**Nonno, Constantine**Section 113 Student #9:**O'loughlin, Katherine**Section 113 Student #10:**Pickens, Charlee**Section 113 Student #11:**Pierce, Allie**Section 113 Student #12:**Rivera-Gebeau, Carson**Section 113 Student #13:**Santamaria, Mercedes**Section 113 Student #14:**Stern, Rachel**Section 113 Student #15:**Taylor, Ezra**Section 113 Student #16:**Watson, Delaney**Section 113 Student #17:**Welch, Kaitlyn**Section 113 Student #18:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Moores, Olivia**Section 114 Student #9:**Nadeau, Spencer**Section 114 Student #10:**Nuske, Lauren**Section 114 Student #11:**Popp, Hanna**Section 114 Student #12:**Raynewater, Ty**Section 114 Student #13:**Richter, Daniel**Section 114 Student #14:**Wood, Brice

**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 shap 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 drnking 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.)

**Wed Apr 5: **Section 4.7: Antiderivatives (Meeting Notes)

**Fri Apr 7: **Exam **X3** Covering Chapter 4 Sections 1,3,4,5,7 (Not Sections 2 or 6)

**The Exam will last the full duration of the class period.****No Calculators, No Cell Phones**- Sit in Alternate Seats
- Five problems, 30 points each, printed on front & back of two sheets of paper
- A problem on Max and Min Values (Section 4.1)
- A problem about Derivatives and the Shapes of Graphs (Section 4.3)
- A problem about Curve Sketchng (Section 4.4)
- A problem about Optimization (Section 4.5)
- A problem about Antiderivatives (Section 4.7)

**Mon Apr 10: **Section 5.1: Areas and Distances (Meeting Notes)

**Tue Apr 11: **Recitation **R12**: Areas and Distances (Section 5.1)

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Croy, Emily**Section 113 Student #3:**Gorman, Shelby**Section 113 Student #4:**Hofmeister, Ellie**Section 113 Student #5:**Kilbane, William**Section 113 Student #6:**Lennerth, Finn**Section 113 Student #7:**Lockwood, Sarah**Section 113 Student #8:**Nonno, Constantine**Section 113 Student #9:**O'loughlin, Katherine**Section 113 Student #10:**Pickens, Charlee**Section 113 Student #11:**Pierce, Allie**Section 113 Student #12:**Rivera-Gebeau, Carson**Section 113 Student #13:**Santamaria, Mercedes**Section 113 Student #14:**Stern, Rachel**Section 113 Student #15:**Taylor, Ezra**Section 113 Student #16:**Watson, Delaney**Section 113 Student #17:**Welch, Kaitlyn**Section 113 Student #18:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Moores, Olivia**Section 114 Student #9:**Nadeau, Spencer**Section 114 Student #10:**Nuske, Lauren**Section 114 Student #11:**Popp, Hanna**Section 114 Student #12:**Raynewater, Ty**Section 114 Student #13:**Richter, Daniel**Section 114 Student #14:**Wood, Brice

Each group will be given a function \(f(x)\), and interval \([a,b]\), and a positive integer \(n\). Their job will be to do these six tasks. **(No Calculators!)**

- Sketch three graphs of \(f(x)\) on the interval \([a,b]\).
- On the left graph, draw in the \(n\)
on the interval \([a,b]\) and shade them.*left rectangles* - On the middle graph, shade in the region between the graph and the \(x\) axis on the interval \([a,b]\) .
- On the right graph, draw in the \(n\)
on the interval \([a,b]\) and shade them.*right rectangles* - Compute the value of the
**Riemann Sum with \(n\) Left Rectangles, \(L_n\)**. - Compute the value of the
**Riemann Sum with \(n\) Right Rectangles, \(R_n\)**.

**Students 1,2:**Let \(f(x)=1+x^2\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 3,4:**Let \(f(x)=2+x^2\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 5,6:**Let \(f(x)=3+x^2\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 7,8:**Let \(f(x)=1+x^3\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 9,10:**Let \(f(x)=2+x^3\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 11,12:**Let \(f(x)=3+x^3\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 13,14:**Let \(f(x)=37-x^2\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 15,16:**Let \(f(x)=38-x^2\), use interval \([a,b]=[0,6]\), and let \(n=3\).**Students 17,18:**Let \(f(x)=39-x^2\), use interval \([a,b]=[0,6]\), and let \(n=3\).

Each group will be given a positive integer \(n\). Their job will be to do these two tasks. **(No Calculators!)**

- Compute the sum \(1+2+3+\dots+n\) directly, by just adding up a bunch of numbers.
- Compute ghe value of the expression \(\frac{n(n+1)}{2}\)
- Do the two results numbers agree? If so, yell
**“Eureka”!**

**Students 1,2:**Use \(n=9\).**Students 3,4:**Use \(n=10\).**Students 5,6:**Use \(n=11\).

Each group will be given a positive integer \(n\). Their job will be to do these two tasks. **(No Calculators!)**

- Compute the sum \(1^2+2^2+3^2+\dots+n^2\) directly, by just adding up a bunch of numbers.
- Compute the value of the expression \(\frac{n(n+1)(2n+1)}{6}\)
- Do the two results numbers agree? If so, yell
**“Eureka”!**

**Students 7,8:**Use \(n=5\).**Students 9,10:**Use \(n=6\).**Students 11,12:**Use \(n=7\).

Each group will be given a positive integer \(n\). Their job will be to do these two tasks. **(No Calculators!)**

- Compute the sum \(1^3+2^3+3^3+\dots+n^3\) directly, by just adding up a bunch of numbers.
- Compute the value of the expression \(\left[\frac{n(n+1)}{2}\right]^2\)
- Do the two results numbers agree? If so, yell
**“Eureka”!**

**Students 13,14:**Use \(n=3\).**Students 15,16:**Use \(n=4\).**Students 17,18:**Use \(n=5\).

**Wed Apr 12: **Section 5.2: The Definite Integral (Meeting Notes)

**Fri Apr 14: **Section 5.3: Evaluating Definite Integrals (Meeting Notes)(Quiz **Q8**)

**No Calculators, No Cell Phones**- Sit in Alternate Seats
- One problem, printed on front and back of one sheet of paper.
- Problem is on Section 5.1 concepts. To prepare for the Quiz, read Section 5.1 and do the Suggested Exercises from Section 5.1.

**Mon Apr 17: **Section 5.3: The Net Change Theorem (Meeting Notes)

**Tue Apr 18: **Recitation **R13**: The Evaluation Theorem and The Net Change Theorem (Section 5.3)

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Croy, Emily**Section 113 Student #3:**Gorman, Shelby**Section 113 Student #4:**Hofmeister, Ellie**Section 113 Student #5:**Kilbane, William**Section 113 Student #6:**Lennerth, Finn**Section 113 Student #7:**Lockwood, Sarah**Section 113 Student #8:**Nonno, Constantine**Section 113 Student #9:**O'loughlin, Katherine**Section 113 Student #10:**Pickens, Charlee**Section 113 Student #11:**Pierce, Allie**Section 113 Student #12:**Rivera-Gebeau, Carson**Section 113 Student #13:**Santamaria, Mercedes**Section 113 Student #14:**Stern, Rachel**Section 113 Student #15:**Taylor, Ezra**Section 113 Student #16:**Watson, Delaney**Section 113 Student #17:**Welch, Kaitlyn**Section 113 Student #18:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Moores, Olivia**Section 114 Student #9:**Nadeau, Spencer**Section 114 Student #10:**Nuske, Lauren**Section 114 Student #11:**Popp, Hanna**Section 114 Student #12:**Raynewater, Ty**Section 114 Student #13:**Richter, Daniel**Section 114 Student #14:**Wood, Brice

(the relationship between ** definite integrals** and

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

$$\int_a^bf(x)dx\underset{\text{ET}}{=}\left. \left(\int f(x)dx\right)\right\vert_a^b$$Use the ** Evaluation Theorem** to evaluate the integrals. Show all details clearly and use correct notation.

**Students 1,2:**(5.3#3)$$\int_{-2}^{0}\left(\frac{1}{2}t^4+\frac{1}{4}t^3-t\right)dt$$**Students 3,4:**(5.3#7)$$\int_{0}^{\pi}\left(5e^x+3\sin x\right)dx$$**Students 5,6:**(5.3#9)$$\int_{1}^{4}\left(\frac{4+6u}{\sqrt{u}}\right)du$$**Students 7,8:**(5.3#11)$$\int_{0}^{1}x\left(\sqrt[3]{x}+\sqrt[4]{x}\right)dx$$**Students 9,10:**(5.3#13)$$\int_{1}^{2}\left(\frac{x}{2}-\frac{2}{x}\right)dx$$**Students 11,12:**(5.3#15)$$\int_{0}^{1}\left(x^{10}+10^x\right)dx$$**Students 13,14:**(5.3#18)$$\int_{0}^{\pi/4}\left(\frac{\sin \theta +\sin \theta \tan^2 \theta}{\sec^2 \theta}\right)d\theta$$**Students 15,16:**(5.3#23)$$\int_{1}^{e}\left(\frac{x^2+x+1}{x}\right)dx$$**Students 17,18:**(5.3#29)$$\int_{-1}^{2}\left(x-2|x|\right)dx$$

(The ** definite integral of the rate of change** equals the

Each group will use the ** Net Change Theorem** to solve a problem that is a variation on Suggested Exercise 5.3#59. (Each group will have a slightly different

(5.3#59) A particle moves along a line with velocity \(v(t)\) for the time interval \(0\leq t\leq 15\).

- Find the displacement of the particle during that time interval.
- Find the distance travelled by the particle during that time interval.
- Graph your velocity function \(v(t)\) and illustrate your results using that graph.

**Students 1,2:**Use \(v(t)=2t-22\)**Students 3,4:**Use \(v(t)=2t-20\)**Students 5,6:**Use \(v(t)=2t-18\)**Students 7,8:**Use \(v(t)=2t-16\)**Students 9,10:**Use \(v(t)=2t-14\)**Students 11,12:**Use \(v(t)=2t-12\)**Students 13,14:**Use \(v(t)=2t-10\)**Students 15,16:**Use \(v(t)=2t-8\)**Students 17,18:**Use \(v(t)=2t-6\)

**Wed Apr 19: **Section 5.4: The Fundamental Theorem of Calculus (Meeting Notes)

**Fri Apr 21: **Section 5.4: The Fundamental Theorem of Calculus (Meeting Notes)(Quiz **Q9**)

- Quiz Q9 will be over Section 5.3 (The Evaluation Theorem and the Net Change Theorem).

**Mon Apr 24: **Section 5.5: The Substitution Rule (Meeting Notes)

**Tue Apr 25: **Recitation **R13**: Sections 5.3 and 5.4

**Section 111 Student #1:**Clark, Madison**Section 111 Student #2:**Dearwester, Aubrey**Section 111 Student #3:**Hammond, Seth**Section 111 Student #4:**Hanzel, Will**Section 111 Student #5:**Hatton, Alaina**Section 111 Student #6:**Jeffries, Kaci**Section 111 Student #7:**Johnston, Allison**Section 111 Student #8:**Kingsley, Claire**Section 111 Student #9:**Klobucar, Madison**Section 111 Student #10:**Lietzke, Lauren**Section 111 Student #11:**Lindsay, Mallory**Section 111 Student #12:**Little, Sean**Section 111 Student #13:**Mcculloch, Eric**Section 111 Student #14:**Ohms, Olivia**Section 111 Student #15:**Peters, Lillian**Section 111 Student #16:**Rayburg, Allison**Section 111 Student #17:**Sizemore, Jacob**Section 111 Student #18:**Zeigler, Jillian

**Section 112 Student #1:**Churchin, Miriam**Section 112 Student #2:**Cunningham, Kate**Section 112 Student #3:**Davis, Rilee**Section 112 Student #4:**Edgar, Tyler**Section 112 Student #5:**Fanelli, Olivia**Section 112 Student #6:**Golla, Jack**Section 112 Student #7:**Hoffman, Sofia**Section 112 Student #8:**Hopper, Olivia**Section 112 Student #9:**Isco, Isabelle**Section 112 Student #10:**Isla, Morgan**Section 112 Student #11:**Jackson, Rachel**Section 112 Student #12:**Kiggins, Austin**Section 112 Student #13:**McGrath, Katie**Section 112 Student #14:**Olin, Ray**Section 112 Student #15:**Wagner, Taylor**Section 112 Student #16:**Wolfe, Gavin**Section 112 Student #17:****Section 112 Student #18:**

**Section 113 Student #1:**Childers, Cindy**Section 113 Student #2:**Croy, Emily**Section 113 Student #3:**Gorman, Shelby**Section 113 Student #4:**Hofmeister, Ellie**Section 113 Student #5:**Kilbane, William**Section 113 Student #6:**Lennerth, Finn**Section 113 Student #7:**Lockwood, Sarah**Section 113 Student #8:**Nonno, Constantine**Section 113 Student #9:**O'loughlin, Katherine**Section 113 Student #10:**Pickens, Charlee**Section 113 Student #11:**Pierce, Allie**Section 113 Student #12:**Rivera-Gebeau, Carson**Section 113 Student #13:**Santamaria, Mercedes**Section 113 Student #14:**Stern, Rachel**Section 113 Student #15:**Taylor, Ezra**Section 113 Student #16:**Watson, Delaney**Section 113 Student #17:**Welch, Kaitlyn**Section 113 Student #18:**Bansode, Ankita

**Section 114 Student #1:**Acquah, Wilona**Section 114 Student #2:**Carter, Aria**Section 114 Student #3:**Chowdhury, Ranlyn**Section 114 Student #5:**Crookston, Tracie**Section 114 Student #6:**Guthrie, Ella**Section 114 Student #7:**Kuntz, Brady**Section 114 Student #8:**Moores, Olivia**Section 114 Student #9:**Nadeau, Spencer**Section 114 Student #10:**Nuske, Lauren**Section 114 Student #11:**Popp, Hanna**Section 114 Student #12:**Raynewater, Ty**Section 114 Student #13:**Richter, Daniel**Section 114 Student #14:**Wood, Brice

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

Each team of students will find the avearage value of a given function on a given interval.

$$\text{Students 1,2 find the average value of the function }f(x)=\sqrt[3]{x} \text{ on the interval }[1,8]$$ $$\text{Students 3,4 find the average value of the function }f(x)=\frac{5}{\sqrt{x}} \text{ on the interval }[16,49]$$ $$\text{Students 5,6 find the average value of the function }f(x)=\frac{3}{x} \text{ on the interval }[1,5]$$ $$\text{Students 7,8 find the average value of the function }f(x)=\cos{(x)} \text{ on the interval }[0,\pi/2]$$ $$\text{Students 9,10 find the average value of the function }f(x)=\sin{(x)} \text{ on the interval }[0,\pi]$$ $$\text{Students 11,12 find the average value of the function }f(\theta)=\sec^2{(\theta)} \text{ on the interval }[0,\pi/4]$$ $$\text{Students 13,14 find the average value of the function }f(\theta)=\sec{(\theta)}\tan{(\theta)} \text{ on the interval }[0,\pi/4]$$ $$\text{Students 15,16 find the average value of the function }f(x)=x^2 \text{ on the interval }[0,5]$$ $$\text{Students 17,18 find the average value of the function }f(x)=e^x \text{ on the interval }[0,1]$$Remember that the *Chain Rule for Derivatives* is used for taking the *derivative of nested functions*

The goal now is to find the *general antiderivative* of a function \(f(x)\) that involves a nested function. That is, we wish to find the indefinite integral
$$\int f(x)dx$$
where the integrand \(f(x)\) involves a nested function. This is not always possible. But sometimes it is, using the ** Substitution Method**.

for finding the indefinite integral $$F(x)=\int f(x)dx$$ where the integrand \(f(x)\) involves a nested function.

**Step 1: Identify the inner function and call it \(u\). **

Write the equation $$inner(x)=u$$ (using the actual inner function from your integrand) to introduce the single letter \(u\) that will represent the inner function. Circle the equation.

**Step 2: Build the equation \(dx=\frac{1}{u’}du\). **

To do this, first find \(u’\), then use it to build equation $$dx=\frac{1}{u’}du$$ Circle the equation.

**Step 3: Substitute, Cancel, Simplify.**

In steps (1) and (2) you have two circled equations. **Substitute** these into the integrand of your indefinite integral. **Cancel** as much as possible and **simplify** by using the *Constant Multiple Rule*. The result should be a new basic integral involving just the variable \(u\). (See **Remarks about Step 3** below.)

**Step 4: Integrate.**

Find the new indefinite integral by using the indefinite integral rules. The result should be a *function form* involving just the variable \(u\) and \(+C\).

**Step 5: Substitute Back.**

Substitute \(u=inner(x)\) into your function from Step (4) The result will be a new function form involving just the variable \(x\)and \(+C\). This is the \(F(x)\) that you seek. Present the result clearly as $$\int f(x)dx=F(x)$$ and circle it.

**Remarks about Step 3: **The result of Step 3 should be a new indefinite integral with an integrand that is a function involving the variable \(u\). There are three important things to check at the end of Step 3:

- There should be no \(x\). in the new indefinite integral. It should involve only \(u\)..
- The new indefinite integral should
*not*involve a nested function, and it should be a*basic integral*that can be integrated using our indefinite integral rules. - If the above two items are not satisfied, then either you made a mistake, or the original integral might be one for which the Substitution Method cannot be used.

Each group will use the ** Substitution Method** to find an

**Wed Apr 26: **Section 5.5: The Substitution Rule (Meeting Notes)

**Fri Apr 28: **Section 5.5 The Substitution Rule (Meeting Notes)

**Tue May 2: **Combined Final Exam **FX** from 4:40pm – 6:40pm. Our section (Secton 110) will have its exam in our usual room, Morton 237. Seating will be in Alternate Seats and Alternate Rows.)

page maintained by Mark Barsamian, last updated Tue May 2, 2023