**Final Exam, Mon Dec 5, 2:30pm - 4:30pm
**

- Students of Shadik, Eisworth and Barsamian in Morton 235
- Students of Wu and Regan in Morton 237

**Course: **MATH 2301 Calculus I

**Meetings: **

**Lecture:**MATH 2301 Section 172 Lecture taught by Mark Barsamian meets M,W,F 12:55pm - 1:50pm in Porter 100**Recitation:**MATH 2301 Sections 173, 174, 175, 176 Lecture taught by Delfino Nolasco- Section 173 Meets Tue 9:30am - 10:25am in Ellis 013
- Section 174 Meets Tue 11:00am - 11:55am in Ellis 013
- Section 175 Meets Tue 2:00pm - 2:55pm in Ellis 013
- Section 176 Meets Tue 3:30pm - 4:25pm in Ellis 013

**Campus: **Ohio University, Athens Campus

**Department: **Mathematics

**Academic Year: **2022 - 2023

**Term: **Fall Semester

**Instructor: **Mark Barsamian

**Contact Information: **My contact information is posted on my web page.

**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: **MATH 2301 has a Common Final Exam on Monday, December 5, 2022, from 2:30pm to 4:30pm.

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

**Textbook Information: **

- website
- Recommended: html version: Free. Works well on phones, tablets, and computers. Has some interactive content and auto-corrected exercises. 5th edition.
- pdf version: Free. Useful if you want to print portions, make notes in the text on a tablet, or work offline.
**Note:**The pdf is the 4th edition, whereas the html above is the 5th edition. Either is fine to learn from, but I will be using the html version. - Link to 4th Edition PDF
- If you like a hard copy: Print Version

**Grading: **

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

**Presentations:**5 Presentations (during Meetings and Meetings) @ 10 points each = 50 points possible**Quizzes:**10 quizzes @ 20 points each = 200 points possible**Exams:**3 Exams @ 180 points each for a total of 540 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. But these are not graded and are not part of your course grade.

**List of Exercises** displayed on this web page

**Section 1.1: **#2,3,4,7,9,11,13,15,21,23,27
**Section 1.3: **#1,2,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,21,23,25,27,29,31,33,35,36,37,38,39,43
**Section 1.4: **#1,2,3,4,5,6,7,8,9,10,11,12,13,15,17,19,21
**Section 1.5: **#1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,27,29,31,33,35,37
**Section 1.6: **#1,2,3,4,5,6,7,8,9,10,11,12,13,14,19,20,21,22,23,24,25,26,27,28

**List of Exercises** on a printable PDF document.

**List of Reference Pages **

- R01: Definition of Limit was distributed on Tue Aug 23.
- R02: Worksheet about the Squeeze Theorem was distributed on Mon Aug 29.
- R03: Worksheet about the Intermediate Value Theorem
- R04: Limits Involving Infinity was discussed Fri Sep 9.
- Reference R05: Graphing Strategy was discussed Wed Oct 19.
- Reference R06: Steps for Computing Riemann Sums was discussed Fri Nov 18.
- Reference R07: The Method of Integration by Substitution was discussed Wed Nov 30.

**List of Group Work**

- GW01: Limits for a Function Given by a Graph, Tue Aug 23.
- GW02: Guessing Limits for a Function Given by a Formula by Plugging in Numbers, Tue Aug 23
- GW03: Function Values and Limits for a Rational Function, Fri Aug 26
- GW04: Finding Limits Using the Limit Rules, Tue Aug 30
- GW05: Limits Involving Infinity for a Function Given by a Graph, Tue Sep 6
- GW06: Limits Involving Infinity for a Rational Function, Tue Sep 13 (GW06 Solutions)
- GW07: Representations of Slopes, Tue Sep 20
- GW08: Finding Derivatives Graphically Using a Ruler, Tue Sep 20
- GW09: Finding Derivatives Graphically Using a Ruler, Tue Sep 20
- GW10: Which is the Function; Which is the Derivative?, Tue Sep 20
- GW11: Rewrite Function, then Find the Derivative., Tue Oct 3
- GW12: The Extreme Value Theorem, Mon Oct 10
- GW13: Finding Absolute Extrema on a Closed Interval, Tue Oct 11
- GW14: Comparing Two Solutions to an Absolute Extrema Problem, Tue Oct 11
- GW15: Analyzing a Polynomial, Tue Oct 18
- GW16: Analyzing a Rational Function with a Vertical Asymptote, Tue Oct 18
- GW17: Analyzing a Rational Function with a Horizontal Asymptote, Tue Oct 18
- GW18: Using Given Information to Sketch a Graph, Wed Oct 19
- GW19: The Idea Behind Newton’s Method, Mon Oct 24
- GW20: Using the Graphing Strategy to Analyze and Graph a Polynomial, Tue Oct 25
- GW21: Newton’s Method, Wed Oct 26
- GW22: Related Rates #1, Tue Nov 1
- GW23: Related Rates #2, Tue Nov 1
- GW24: Related Rates #3, Tue Nov 1
- GW25: Antiderivatives, Tue Nov 8
- GW26: Good and Bad Indefinite Integral Solutions, Tue Nov 15
- GW27: Definite Integrals for a Graph Made up of Geometric Shapes, Tue Nov 15
- GW28: Computing Definite Integrals By Using Geometry, Tue Nov 15
- GW29: Estimating the Area Under a Graph by Using Riemann Sums, Wed Nov 16
- GW30: Computing Riemann Sums, Fri Nov 18
- GW31: The Average Value of a Function on an Interval, Tue Nov 29
- GW32: Position, Velocity, and Acceleration, Tue Nov 29
- GW33: Integration by Substitution, Fri Dec 2

**Full Calendar **displayed on this web page

Lecture Sections 172 and Recitation Sections 173, 174, 175, 176

(taught by Mark Barsamian)

**Book Sections and (Homework Exercises)**

**Section 1.1**An Introduction to Limits (#2, 3, 4, 7-15odd, 21, 23, 27)**Section 1.3**Finding Limits Analytically (#1, 2, 4, 5, 7-18, 19-33odd, 35-38, 39, 43)

**Mon Aug 22 Meeting Topics**

**Piecewise-Defined Functions**Consider this symbol:

$$ f(x) = \left\{ \begin{array}{ll} -2x+10, & \quad x\leq 3 \\ x^2, & \quad x>3 \end{array} \right. $$- What does this symbol mean? How is it spoken?
- What is \( f(2) \)? Why?
- What is \( f(3) \)? Why?
- What is \( f(4) \)? Why?
- What does the graph of \( f(x) \) look like? How can we figure it out?

**The Absolute Value Function**You're all familiar with the behavior of the

*absolute value*when the thing inside is a*number*.- What is \( |5| \)?
- What is \( |-7| \)?
- What is \( |0| \)?

Based on that familiarity, you might feel like you

*understand*the*absolute value*. But do you understand the*absolute value*?**function**- What does the graph of \( |x| \) look like? How can we figure it out?
- Can you express \( |x| \) as a
*piecewise-defined function*? - What does the graph of $$ \frac{|x|}{x} $$ look like? How can we figure it out?
- Can you express $$ \frac{|x|}{x} $$ as a
*piecewise-defined function*?

**The Important and Subtle Issue of Cancelling Terms in Expressions**Consider the following two functions:

$$ f(x) = \frac{x^2-2x-3}{x-3} = \frac{(x-3)(x+1)}{(x-3)} $$ $$ g(x) = x+1 $$- What does the graph of \( g(x) \) look like?
- What does the graph of \( f(x) \) look like?

Are those really two different functions, or are they actually the same function? That is, can we always cancel terms such as \( \frac{(x-3)}{(x-3)}\) in expressions? What do the students have to say about this? If a student has an opinion, can they explain a bit about why they have that opinion?

Conclusions about the functions \(f(x)\) and \(g(x)\)

Functions \(f(x)\) and \(g(x)\) are not the same function, because they do not have the same

. Observe that $$g(3)=4$$ $$f(3)= \ does \ not \ exist$$ The reason that \(f(3)\) does not exist is that $$ f(3) = \frac{((3)-3)((3)+1)}{((3)-3)} = \frac{(0)(4)}{(0)} = \frac{0}{0} \ (undefined)$$ and division by \(0\) is*domain**not defined*. Put another way, one*cannot cancel*\(\frac {0}{0}\).So in general, one

*cannot*replace the expression $$ \frac{x^2-2x-3}{x-3} = \frac{(x-3)(x+1)}{(x-3)} $$ with the simplified expression $$ x+1 $$ This may be contrary to what you have been told in previous courses, and even what you may be asked to do in. That is, you may be told to*MyLab**simplify*the expression $$ \frac{x^2-2x-3}{x-3} = \frac{(x-3)(x+1)}{(x-3)} $$ with the expectation that the*simplified*version is $$ x+1 $$ But this is*incorrect*. That is, without any information about the value of \(x\), one is*not allowed*to do that simplification!!However, if you somehow know (or are told) that \(x\neq 3\), then you

*can*replace the expression $$ \frac{x^2-2x-3}{x-3} = \frac{(x-3)(x+1)}{(x-3)} $$ with the*simplified*expression $$ x+1 $$ That is because when \(x\neq 3\), the value of \( f(x)\) will always be the same as the value of \(g(x)\). But if you do know that \(x\neq 3\), and you use that information to replace the expression $$ \frac{x^2-2x-3}{x-3} = \frac{(x-3)(x+1)}{(x-3)} $$ with the*simplified*expression $$ x+1 $$ then you should explain clearly that you are allowed to do the cancellation because you know that \(x\neq 3\), which tells you that \(x - 3 \neq 0\), which means that you can cancel \(\frac {(x-3)}{(x-3)}\).Fussing over whether one

*can*or*cannot*cancel terms in an expression, and*explaining why*, may seem to be over-thinking what ought to just be simple math. But Calculus is not simple math, and!!.*the issue of whether or not one can cancel terms is one of the most important concepts of the first month of the course*

**Tue Aug 23 Meeting Topics**

A printed Definition of Limit was distributed.

Students worked on Group Work GW01 (* Limits for a Function Given by a Graph*).

Students worked on Group Work GW02 (* Guessing Limits for a Function Given by a Formula by Plugging in Numbers*).

**Wed Aug 24 Meeting Topics**

**Fri Aug 26 Meeting Topics**

A printed Class ActivityClass Activity

We discussed a Student's Solution to Homework Exercise 1.1#11 that they sent to Mark B when asking for help. (Mark B. would like to profoundly thank the student for letting us discuss their solution during class. Their solution provided the inspiration for the entire meeting outline.)

**Book Sections and (Homework Exercises)**

**Section 1.3**Finding Limits Analytically (#1, 2, 4, 5, 7-18, 19-33odd, 35-38, 39, 43)**Section 1.4**One Sided Limits (#1-12, 13-21odd)

**Mon Aug 29 Meeting Topics**

We discussed a Student's Solution to Homework Exercises from Section 1.1 that they sent to Mark B when asking for help. (Mark B. would like to thank the student for letting us discuss their solution during class..)

A printed Worksheet about the Squeeze Theorem was distributed.

**Quiz Q01 on Tue Aug 30 Covering Section 1.1**

**20 Minutes at the end of class****No Calculators**- Similar to GW01 and Suggested Exercises 1.1#7-13 about
*computing \(y\) values*and*estimating limits*for function given by a*formula*.

**Tue Aug 30 Meeting Topics**

Students worked Group Work GW04 (* Finding Limits*).

**Quiz Q01 During Tue Aug 30 Meeting Covering Section 1.1**

**20 Minutes at the end of class****No Calculators**- Similar to GW01 and Suggested Exercises 1.1#7-13 about
*computing \(y\) values*and*estimating limits*for function given by a*formula*.

**Wed Aug 31 Meeting Topics**

**Fri Sep 2 Meeting Topics**

**Class Presentations for Fri Sep 2 **

**Nana Asare CP1: **Present a solution to book exercise 1.3#30.
$$\lim_{x \rightarrow 0} \frac {x^2-7x}{x^2+2x}$$
Note that this is an exercise from Section 1.3, so you should be finding the limit by using the Theorems about Limit Properties that are presented in the book in Section 1.3.

- To prepare for the presentation, study
**Example 1.3.16**in Book Section 1.3 and also study your Class Notes and Recitation Notes. - Prepare your presentation according to the Presentation Guidelines.

**Tyler Boldon CP1: **Present a solution to book exercise 1.3#32
$$\lim_{x \rightarrow -8} \frac {x^2+3x-40}{x^2+13x+40}$$
Note that this is an exercise from Section 1.3, so you should be finding the limit by using the Theorems about Limit Properties that are presented in the book in Section 1.3.

- To prepare for the presentation, study
**Example 1.3.16**in Book Section 1.3 and also study your Class Notes and Recitation Notes. - Prepare your presentation according to the Presentation Guidelines.

**Ellie BowerCP1: **Find the limit using Section 1.3 techniques. That is, find the limit by using the Theorems about Limit Properties that are presented in the book in Section 1.3.
$$\lim_{x \rightarrow 36} \frac {x-36}{\sqrt{x}-6}$$

- To prepare for the presentation, study
**Example 1.3.17**in Book Section 1.3 and also study your Class Notes and Recitation Notes. - Prepare your presentation according to the Presentation Guidelines.

**Andrew Champagne CP1: **Find the limit using Section 1.3 techniques. That is, find the limit by using the Theorems about Limit Properties that are presented in the book in Section 1.3.
$$\text{For } f(x)=-x^2+10x
\\
\text{find } \lim_{h \rightarrow 0} \frac {f(4+h)-f(4)}{h}$$

- To prepare for the presentation, study
**Example 1.3.18**in Book Section 1.3 and also study your Class Notes and Recitation Notes. - Prepare your presentation according to the Presentation Guidelines.

**Drew Conway CP1: **Present a solution to book exercise 1.4#16
$$f(x)=\begin{cases}\cos (x) & \text{if }x \lt \pi \\ \sin (x) & \text{if }x\geq \pi \end{cases} $$

- Find \(lim_{x \rightarrow \pi^-} f(x) \)
- Find \(lim_{x \rightarrow \pi^+} f(x) \)
- Find \(lim_{x \rightarrow \pi} f(x) \)
- Find \(f(\pi) \)
- Illustrate with a graph of \(f(x)\)

- To prepare for the presentation, study the examples in Book Section 1.4 and also study your Class Notes.
- Prepare your presentation according to the Presentation Guidelines.

Handed out Reference R03 The Intermediate Value Theorem

**Book Sections and (Homework Exercises)**

**Section 1.5**Continuity (#1-22, 23-38odd)**Section 1.6**Limits Involving Infinity (#1-14, 19-28)

**Mon Sep 5 is Labor Day: No Class**

**Tue Sep 6 Meeting Topics**

**Class Presentations Involving Continuity for Tue Sep 6 **

**Student #6 CP1: **Present a solution to book exercise 1.5#20
$$f(x)=\begin{cases}x^2 - x^2 & \text{if }x \lt 1 \\ x-2 & \text{if }x\geq 1 \end{cases} $$
Without making a graph of \(f(x)\), answer the following two questions:

- Is \(f\) continuous at \(0\)? Explain.
- Is \(f\) continuous at \(1\)? Explain.

- To prepare for the presentation, study The beginning of Book Section 1.5. The problem that you have to solve has no similar examples in the book. But to solve your problem, you only need to study the
**Definition 1.5.1 of Continuous Functions**and to note that immediately following that definition, there is a procedure that you can follow to establish whether or not a function \(f\) is continuous at \(x=c\). - Prepare your presentation according to the Presentation Guidelines.

**Student #7 CP1: **Present a solution to book exercise 1.5#24. For the function
$$f(x)=\sqrt{x^2-25}$$
give the intervals on which \(f\) is continuous.

- To prepare for the presentation, study Book Section 1.5 through Example 1.5.10. Your problem is similar to Examples 1.5.7 and 1.5.10.
- Prepare your presentation according to the Presentation Guidelines.

**Student #8 CP1: **Present a solution to book exercise 1.5#28. For the function
$$g(t)=\frac{1}{\sqrt{9-t^2}}$$
give the intervals on which \(g\) is continuous.

- To prepare for the presentation, study Book Section 1.5 through Example 1.5.10. Your problem is similar to Examples 1.5.7 and 1.5.10.
- Prepare your presentation according to the Presentation Guidelines.

- In Section 173 (Tue 9:30)
- Amy Evers is Student #6
- Dalana Goddard is Student #7
- Lauren Hartel is Student #8

- In Section 174 (Tue 11:00)
- Carly Doros is Student #6
- Nicole Grant is Student #7
- Tim Jaskiewicz is Student #8

- In Section 175 (Tue 2:00)
- Evan Green is Student #6
- Kierston Harper is Student #7
- Olivia Keener is Student #8

- In Section 176 (Tue 3:30)
- Carlotta Dattilo is Student #6
- Paul Gbadebo is Student #7
- Alan Romero Herrera is Student #8

Discussed Reference R04 (Limits Involving Infinity).

Students worked Group Work GW05 (Limits Involving Infinity for a function given by a graph).

**Quiz Q02 on Wed Sep 7 Covering Sections 1.3, 1.4**

**Wed Sep 7 Meeting Topics**

**Quiz Q02 on Wed Sep 7 Covering Sections 1.3, 1.4**

**Fri Sep 9 Meeting Topics**

**Introduction**

Today, we're discussing *infinite limits*. That, is, limits such as

The *book* gives only a *precise* definition of *infinite limits*, a definition involving *N* and *delta*. This is the same style of definition as the book's initial definition of *regular limits* in Section 1.2, a definition involving *epsilon* and *delta*. There is nothing *wrong* with the book's *precise* definitions, but what is *lacking* in the book's presentation of limits is an *inf
ormal* definition of limits, both *regular limits* and *infinite limits*.

For our course, there are two ** Reference Pages** about limits, posted on the

- R01: Definition of Limit was distributed on Tue Aug 23.
- R04: Limits Involving Infinity was discussed Fri Sep 9.

The *precise* definitions of limits are above the level of MATH 2301, but the *informal* definition of limits (both kinds) are appropriate for a class at the level of MATH 2301. Furthermore, he gave an example of computing limits, both *regular limits* and *infinite limits*, using what could be called an *informal* method.

The *informal method of computing the limit* of an expression involves examining the *trends* in the various terms in the expression, and making a conclusion about the *trend* in the value of the expression.

**[Example 1]** we will revisit the following function, which was studied in Group Work GW02: Guessing Limits for a Function Given by a Formula by Plugging in Numbers on Tue Aug 23.

(a) find \(lim_{x \rightarrow 3^-}f(x)\) by using the *informal method of computing the limit*. That is, by considering the trends in the size of the factors. Use the *Expanded Definition of Limit, Involving Infinity*, if appropriate.

First we note that because \(x \rightarrow 3^-\), we know that \(x \neq 3\), so \(x-3 \neq 0\). Therefore, we can cancel the \(\frac{(x-3)}{(x-3)}\) terms.

$$lim_{x \rightarrow 3^-}f(x)=lim_{x \rightarrow 3^-}\frac{x(x-3)}{(x-2)(x-3)}=lim_{x \rightarrow 3^-}\frac{x}{(x-2)}$$Now, because \(x \rightarrow 3^-\), we know that

- The
*numerator*, \(x\), is getting closer and closer to \(3\). - The
*denominator*, \(x-2\), is getting closer and closer to \(1\). - Therefore, the
*ratio*is getting closer and closer to \(3\).

That is, when the trend in the \(x\) values is that \(x\) is getting closer and closer to \(3\) but less than \(3\), the trend in the values of \(f(x)\) will be that \(f(x)\) is getting closer and closer to \(3\). In limit notation, these trends are denoted as follows.

$$lim_{x \rightarrow 3^-}f(x)=lim_{x \rightarrow 3^-}\frac{x(x-3)}{(x-2)(x-3)}=lim_{x \rightarrow 3^-}\frac{x}{(x-2)}=3$$(b) find \(lim_{x \rightarrow 2^-}f(x)\) by using the *informal method of computing the limit*. That is, by considering the trends in the size of the factors. Use the *Expanded Definition of Limit, Involving Infinity*, if appropriate.

We note that because \(x \rightarrow 2^-\), we know that \(x \neq 3\), so \(x-3 \neq 0\). Therefore, we can cancel the \(\frac{(x-3)}{(x-3)}\) terms.

$$lim_{x \rightarrow 2^-}f(x)=lim_{x \rightarrow 2^-}\frac{x(x-3)}{(x-2)(x-3)}=lim_{x \rightarrow 2^-}\frac{x}{(x-2)}$$Now, because \(x \rightarrow 2^-\), we know that

- The
*numerator*, \(x\), is getting closer and closer to \(2\). - The
*denominator*, \(x-2\), is getting closer and closer to \(0\) but is*negative*. - Therefore, the
*ratio*will be a*huge negative number*. - The closer that \(x\) gets to \(2\), while remaining
*less than*\(2\), the more huge and negative will be the value of the*ratio*.

That is, when the trend in the \(x\) values is that \(x\) is getting closer and closer to \(2\) but less than \(2\), the trend in the values of \(f(x)\) will be that \(f(x)\) is getting huge and negative, without bound. In limit notation, these trends are denoted as follows.

$$lim_{x \rightarrow 2^-}f(x)=lim_{x \rightarrow 2^-}\frac{x(x-3)}{(x-2)(x-3)}=lim_{x \rightarrow 2^-}\frac{x}{(x-2)}=-\infty$$Notice that if we were using the *Original Definition of Limit*, we would said instead that the limit does not exist.

**End of [Example 1]**

**[Example 2]** we will revisit the following function, which was studied in Group Work GW03: Function Values and Limits for a Rational Function on Fri, Aug 26.

(a) find \(lim_{x \rightarrow -3^+}f(x)\) by using the *informal method of computing the limit*. That is, by considering the trends in the size of the factors. Use the *Expanded Definition of Limit, Involving Infinity*, if appropriate.

First we note that because \(x \rightarrow -3^+\), we know that \(x \neq 3\), so \(x+3 \neq 0\). Therefore, we can cancel the \(\frac{(x+3)}{(x+3)}\) terms.

$$lim_{x \rightarrow -3^+}f(x)=lim_{x \rightarrow -3^+}\frac{(x+7)(x+3)}{(x+2)(x+3)}=lim_{x \rightarrow -3^+}\frac{(x+7)}{(x+2)}$$Now, because \(x \rightarrow -3^+\), we know that

- The
*numerator*, \(x+7\), is getting closer and closer to \(4\). - The
*denominator*, \(x+2\), is getting closer and closer to \(-1\). - Therefore, the
*ratio*is getting closer and closer to \(-4\).

That is, when the trend in the \(x\) values is that \(x\) is getting closer and closer to \(-3\) but greater than \(-3\), the trend in the values of \(f(x)\) will be that \(f(x)\) is getting closer and closer to \(-4\). In limit notation, these trends are denoted as follows.

$$lim_{x \rightarrow -3^+}f(x)=lim_{x \rightarrow -3^+}\frac{(x+7)(x+3)}{(x+2)(x+3)}=lim_{x \rightarrow -3^+}\frac{(x+7)}{(x+2)}=-4$$(b) find \(lim_{x \rightarrow -2^+}f(x)\) by using the *informal method of computing the limit*. That is, by considering the trends in the size of the factors. Use the *Expanded Definition of Limit, Involving Infinity*, if appropriate.

We note that because \(x \rightarrow -2^+\), we know that \(x \neq -3\), so \(x+3 \neq 0\). Therefore, we can cancel the \(\frac{(x+3)}{(x+3)}\) terms.

$$lim_{x \rightarrow -3^+}f(x)=lim_{x \rightarrow -3^+}\frac{(x+7)(x+3)}{(x+2)(x+3)}=lim_{x \rightarrow -3^+}\frac{(x+7)}{(x+2)}$$Now, because \(x \rightarrow -2^+\), we know that

- The
*numerator*, \(x+7\), is getting closer and closer to \(5\). - The
*denominator*, \(x+2\), is getting closer and closer to \(0\) but is*positive*. - Therefore, the
*ratio*will be a*huge positive number*. - The closer that \(x\) gets to \(-2\), while remaining
*greater than*\(-2\), the more huge and positive will be the value of the*ratio*.

That is, when the trend in the \(x\) values is that \(x\) is getting closer and closer to \(-2\) but greater than \(2\), the trend in the values of \(f(x)\) will be that \(f(x)\) is getting huge and positive, without bound. In limit notation, these trends are denoted as follows.

$$lim_{x \rightarrow -2^+}f(x)=lim_{x \rightarrow -2^+}\frac{(x+7)(x+3)}{(x+2)(x+3)}=lim_{x \rightarrow -2^+}\frac{(x+7)}{(x+2)}=\infty $$Notice that if we were using the *Original Definition of Limit*, we would said instead that the limit does not exist.

**End of [Example 2]**

**Book Sections and (Homework Exercises)**

**Section 2.1**Instantaneous Rates of Change: The Derivative (1-22, 27-36)

**Mon Sep 12 Meeting Topics**

On Fri Sep 9, Mark discussed *infinite limits*. He presented two ** Reference Pages** about limits, posted on the

- R01: Definition of Limit was distributed on Tue Aug 23.
- R04: Limits Involving Infinity was discussed Fri Sep 9.

Mark also discussed an *informal method of computing the limit* of an expression. This method involves examining the *trends* in the various terms in the expression, and making a conclusion about the *trend* in the value of the expression. The method works equally well for limits that turn out to be *numbers* and limits that turn out to be *infinite*.

See the Calendar entry for Fri Sep 9 to see the discussion of the *informal method of computing the limit* and to see some examples of its use.

We'll start today with three presentations involving using that method to find limits of very similar-looking rational functions.

**Class Presentations Involving the Informal Method for Computing Limits for Mon Sep 12 **

**Ben Oldiges CP1** Let \(f(x)\) be the following function.

(Find \(lim_{x \rightarrow 5^+}f(x)\) by using the *informal method of computing the limit*. That is, by considering the trends in the size of the factors. Use the *Expanded Definition of Limit, Involving Infinity*, if appropriate.

**Austin Kiggins CP1** Let \(g(x)\) be the following function.

Find \(lim_{x \rightarrow 5^+}g(x)\) by using the *informal method of computing the limit*. That is, by considering the trends in the size of the factors. Use the *Expanded Definition of Limit, Involving Infinity*, if appropriate.

**Kelly Koenig CP1** Let \(h(x)\) be the following function.

Find \(lim_{x \rightarrow 5^+}h(x)\) by using the *informal method of computing the limit*. That is, by considering the trends in the size of the factors. Use the *Expanded Definition of Limit, Involving Infinity*, if appropriate.

Observe that all three of the limits in the presentations were of \(\frac{0}{0}\) *indeterminate form*. We see that their limits turned out very differently!

**Mark will Present Conclusions about General Trends** in limits of the form \( \lim_{x\rightarrow c}f(x)\), where \(c\) is a *real number* and \(f(x)\) is a *rational function*.

**Mark will Discuss** limits of the form \( \lim_{x\rightarrow \infty}f(x)\), where \(f(x)\) is a *rational function*.

**Tue Sep 13 Meeting Topics**

**Topic: **Limits Involving Infinity for a Rational Function (Book Section 1.6)

**Group Work: **Students worked onGroup Work GW06 (Limits Involving Infinity for a Rational Function). (GW06 Solutions)

**Exam X1 Covering Chapter 1 on Wed Sep 14**

- Exam is on Wed Sep 13, 2022, for the full duration of the class period, 12:55pm - 1:50pm, in Porter 100
- No Calculators, no Cell Phones, No Books, No Notes
- The Exam is 9 problems, 20 points each.
- Compute limits and function values for a function given by graph. (concepts from Sections 1.1 and 1.6)
- Compute the limit of a function involving trigonometric or logarithmic functions. (concepts from Section 1.3)
- Compute a limit requiring use of the Squeeze Theorem. (concepts from Section 1.3)
- Compute a limit involving factoring or multiplying by the conjugate. (concepts from Section 1.3)
- Compute the limit of a difference quotient. (concepts from Section 1.3)
- Compute the limit of a piecewise-defined function. (concepts from Section 1.4)
- Find the intervals where a function is continuous. (concepts from Section 1.5)
- Compute limits that may or may not be infinite, and find vertical asymptotes for a rational function. (concepts from Section 1.6)
- Compute limits at infinity and find horizontal asymptotes for a rational function. (concepts from Section 1.6)

**Fri Sep 16 Meeting Topics**

**Words:**the derivative of \(f(x)\)**Symbol:**\(f'(x)\)**Meaning:**\(f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\)

**Words:**the line tangent to the graph of \(f(x)\) at \(x=a\)**Meaning:**the line that has these two properties:- The line contains the point \((a,f(a))\).
- The line has slope \(m=f'(a)\).

If a line that has these two properties:

- The line contains the point \((a,b)\).
- the line has slope \((m\).

Since the tangent line that has these two properties:

- The line contains the point \((a,f(a))\).
- the line has slope \(m=f'(a)\).

**Class Presentations Involving Computing Derivatives for Fri Sep 16 **

**Jonah Lewis CP1** Let \(f(x)\) be the following function.

The goal is to find the *Derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, the goal is to find this limit:

- Find \(f(x+h)\).
- Find \(f(x+h)\)-\(f(x)\) and simplify your answer.
- Find \(\frac{f(x+h)-f(x)}{h}\) and simplify your answer, assuming that \(h\neq 0\).
- Finally, find \(f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\).

**Dylan Pohovey CP1** Let \(f(x)\) be the following function.

The goal is to find the *Derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, the goal is to find this limit:

Use as your model **Example 2.1.17** in the book. Show all details and explain key steps.

**Reggie Shaffer CP1** Let \(f(x)\) be the following function.

The goal is to find the *Derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, the goal is to find this limit:

Use as your model **Example 2.1.18** in the book. Show all details and explain key steps.

**Book Sections and (Homework Exercises)**

**Section 2.1**Instantaneous Rates of Change: The Derivative (#1-22, 27-36)**Section 2.2**Interpretations of the Derivative (#1-18)**Section 2.3**Basic Differentiation Rules (#1-38)

**Mon Sep 19 Meeting Topics**

**Words:**the derivative of \(f(x)\)**Symbol:**\(f'(x)\)**Meaning:**\(f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\)

**Words:**the line tangent to the graph of \(f(x)\) at \(x=a\)**Meaning:**the line that has these two properties:- The line contains the point \((a,f(a))\).
- The line has slope \(m=f'(a)\).

If a line that has these two properties:

- The line contains the point \((a,b)\).
- the line has slope \((m\).

Since the tangent line that has these two properties:

- The line contains the point \((a,f(a))\).
- the line has slope \(m=f'(a)\).

**Words:**the line normal to the graph of \(f(x)\) at \(x=a\)**Meaning:**the line that has these two properties:- The line contains the point \((a,f(a))\).
- The line is
*perpendicular*to the line that is tangent to the graph of \(f(x)\) at \(x=a\).

**Remark:**By this definition, realize that if the tangent line has slope \(m=f'(a)\), where \(f'(a) \neq 0\), then the normal line will have slope \(m=-\frac{1}{f'(a)}\). But also realize that if the tangent line has slope \(m=f'(a)\), where \(f'(a) = 0\), then this means that the*tangent line*is*horizontal*. That will mean that the*normal line*will be*vertical*. Remember that*vertical lines*have*undefined slope*. One can still write a*line equation*for a vertical line. But it is in the form \(x=a\). That is, it is not in slope intercept form.

**Class Presentations Involving Tangent Lines and Normal Lines for Mon Sep 19 **

**Colin Sorge CP1** Let \(f(x)\) be the following function.

In a previous presentation, a student found the *derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, they computed the following:

Their result was

$$f'(x)=5$$Using that result,

- Find the
*slope*of the line*tangent*to the graph of \(f(x)\) at \(x=2\). - Find the
*equation*of the line*tangent*to the graph of \(f(x)\) at \(x=2\). - Find the
*slope*of the line*normal*to the graph of \(f(x)\) at \(x=2\). - Find the
*equation*of the line*normal*to the graph of \(f(x)\) at \(x=2\). - Illustrate your results by drawing a graph of \(f(x)\) along with the
*tangent*and*normal*lines.- Make your graph large and neat. (You are welcome to use a graphing utility such as
*Desmos*.) - Label important points (such as
*axis intercepts*and the*point of tangency*) with their \((x,y)\) coordinates. - Label the curve for \(f(x)\) with its equation.
- Label the
*tangent line*with its*line equation*. - Label the
*normal line*with its*line equation*.

- Make your graph large and neat. (You are welcome to use a graphing utility such as

**Bilal Tahir CP1** Let \(f(x)\) be the following function.

In a previous presentation, a student found the *derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, they computed the following:

Their result was

$$f'(x)=10x-11$$Using that result,

- Find the
*slope*of the line*tangent*to the graph of \(f(x)\) at \(x=2\). - Find the
*equation*of the line*tangent*to the graph of \(f(x)\) at \(x=2\). - Find the
*slope*of the line*normal*to the graph of \(f(x)\) at \(x=2\). - Find the
*equation*of the line*normal*to the graph of \(f(x)\) at \(x=2\). - Illustrate your results by drawing a graph of \(f(x)\) along with the
*tangent*and*normal*lines.- Make your graph large and neat. (You are welcome to use a graphing utility such as
*Desmos*.) - Label important points (such as
*axis intercepts*and the*point of tangency*) with their \((x,y)\) coordinates. - Label the curve for \(f(x)\) with its equation.
- Label the
*tangent line*with its*line equation*. - Label the
*normal line*with its*line equation*.

- Make your graph large and neat. (You are welcome to use a graphing utility such as

**Paul Thorp CP1** Let \(f(x)\) be the following function.

In a previous presentation, a student found the *derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, they computed the following:

Their result was

$$f'(x)=-\frac{1}{(x+7)^2}$$Using that result,

- Find the
*slope*of the line*tangent*to the graph of \(f(x)\) at \(x=2\). - Find the
*equation*of the line*tangent*to the graph of \(f(x)\) at \(x=2\). - Find the
*slope*of the line*normal*to the graph of \(f(x)\) at \(x=2\). - Find the
*equation*of the line*normal*to the graph of \(f(x)\) at \(x=2\). - Illustrate your results by drawing a graph of \(f(x)\) along with the
*tangent*and*normal*lines.- Make your graph large and neat. (You are welcome to use a graphing utility such as
*Desmos*.) - Label important points (such as
*axis intercepts*and the*point of tangency*) with their \((x,y)\) coordinates. - Label the curve for \(f(x)\) with its equation.
- Label the
*tangent line*with its*line equation*. - Label the
*normal line*with its*line equation*.

- Make your graph large and neat. (You are welcome to use a graphing utility such as

**Tue Sep 20 Meeting Topics**

**Topic: **Group Works About Interpretations Of The Derivative (Book Section 2.2)

- GW07: Representations of Slopes, Tue Sep 20
- GW08: Finding Derivatives Graphically Using a Ruler, Tue Sep 20
- GW09: Finding Derivatives Graphically Using a Ruler, Tue Sep 20
- GW10: Which is the Function; Which is the Derivative?, Tue Sep 20

**Wed Sep 21 Meeting Topics**

**Class Presentations for Wed Sep 21 **

**Sara Weller CP1** Let \(f(x)\) be the following function.

In book **Example 2.1.19** the the authors computed the *derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, they computed the following:

Their result was

$$f'(x)=\cos(x)$$Using that result,

- Find the
*slope*of the line*tangent*to the graph of \(f(x)\) at \(x=\pi\). - Find the
*equation*of the line*tangent*to the graph of \(f(x)\) at \(x=\pi\). - Find the
*slope*of the line*normal*to the graph of \(f(x)\) at \(x=\pi\). - Find the
*equation*of the line*normal*to the graph of \(f(x)\) at \(x=\pi\). - Illustrate your results by drawing a graph of \(f(x)\) along with the
*tangent*and*normal*lines.- Make your graph large and neat. (You are welcome to use a graphing utility such as
*Desmos*.) - Label important points (such as
*axis intercepts*and the*point of tangency*) with their \((x,y)\) coordinates. - Label the curve for \(f(x)\) with its equation.
- Label the
*tangent line*with its*line equation*. - Label the
*normal line*with its*line equation*.

- Make your graph large and neat. (You are welcome to use a graphing utility such as

**Emily Wilkerson CP1** Let \(f(x)\) be the following function.

In book **Example 2.1.19** the the authors computed the *derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, they computed the following:

Their result was

$$f'(x)=\cos(x)$$Using that result,

- Find the
*slope*of the line*tangent*to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\). - Find the
*equation*of the line*tangent*to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\). - Find the
*slope*of the line*normal*to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\). - Find the
*equation*of the line*normal*to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\). - Illustrate your results by drawing a graph of \(f(x)\) along with the
*tangent*and*normal*lines.- Make your graph large and neat. (You are welcome to use a graphing utility such as
*Desmos*.) - Label important points (such as
*axis intercepts*and the*point of tangency*) with their \((x,y)\) coordinates. - Label the curve for \(f(x)\) with its equation.
- Label the
*tangent line*with its*line equation*. - Label the
*normal line*with its*line equation*.

- Make your graph large and neat. (You are welcome to use a graphing utility such as

**Gavin Wolfe CP1** Let \(f(x)\) be the following function.

In book **Example 2.1.19** the the authors computed the *derivative*, \(f'(x)\), using the *Definition of the Derivative*. That is, they computed the following:

Their result was

$$f'(x)=\cos(x)$$Using that result,

- Find the
*slope*of the line*tangent*to the graph of \(f(x)\) at \(x=\frac{\pi}{4}\). - Find the
*equation*of the line*tangent*to the graph of \(f(x)\) at \(x=\frac{\pi}{4}\). - Find the
*slope*of the line*normal*to the graph of \(f(x)\) at \(x=\frac{\pi}{4}\). - Find the
*equation*of the line*normal*to the graph of \(f(x)\) at \(x=\frac{\pi}{4}\). - Illustrate your results by drawing a graph of \(f(x)\) along with the
*tangent*and*normal*lines.- Make your graph large and neat. (You are welcome to use a graphing utility such as
*Desmos*.) - Label important points (such as
*axis intercepts*and the*point of tangency*) with their \((x,y)\) coordinates. - Label the curve for \(f(x)\) with its equation.
- Label the
*tangent line*with its*line equation*. - Label the
*normal line*with its*line equation*.

- Make your graph large and neat. (You are welcome to use a graphing utility such as

**Quiz Q03 Covering Section 2.1**

**Wed Sep 21 Quiz Q03 Covering Section 2.1**

**Fri Sep 23 Meeting Topics**

**Class Presentations for Fri Sep 23 **

**Nana Asare CP2:** Using **Theorem 2.3.1** and **Theorem 2.3.4**, compute the derivative of the function
$$g(x)=16x^2-4x^3+24x+28$$
Use, as a model, the author's un-numbered example that immediately follows **Theorem 2.3.4**.

**Tyler Boldon CP2:** Using **Theorem 2.3.1** and **Theorem 2.3.4**, compute the derivative of the function
$$g(t)=16t^3+\cos{(t)}+7\sin{(t)}$$
Use, as a model, the author's un-numbered example that immediately follows **Theorem 2.3.4**.

**Ellie Bower CP2:** Using **Theorem 2.3.1** and **Theorem 2.3.4**, compute the derivative of the function
$$f(x)=\ln{(4x^8)}$$
**Hint: ** We don't have a rule for computing the derivative of any sort of *logarithmic* function except the most basic one, \(y=\ln{(x)}\). So you'll have to use *rules of logarithms* to *rewrite* \(f(x)\) into a form involving \(ln{(x)}\).

**Andrew Champagne CP2:** Using **Theorem 2.3.1** and **Theorem 2.3.4**, compute the derivatives of the following two functions

- \( f(x) = \ln{(x)}+e^{(x)}+\cos{(x)} \)
- \( f(x) = \ln{(2)}+e^3+\cos{(\frac{\pi}{3}) }\)

**Drew Conway CP2:** Using **Theorem 2.3.1** and **Definition 2.3.8**, compute the first four derivatives of the function \(g(x)=-5\sin{(x)}\).

**Book Sections and (Homework Exercises)**

**Section 2.4**The Product and Quotient Rules (1-14, 15-47odd)**Section 2.5**The Chain Rule (1-6, 7-39odd, 41, 42)

**Mon Sep 26 Meeting Topics**

**Class Presentations for Mon Sep 26 **

**Mark B presented this example.**

- Using
**Theorem 2.4.1 Product Rule**and theorems from Section 2.3, compute the derivative of the function $$f(x)=x^8\cos{(x)}$$ - Evaluate the derivative at \(x=\pi\). That is, find \(f'(\pi)\).

**Carly Doros Presentation #2**

- Using
**Theorem 2.4.1 Product Rule**and theorems from Section 2.3, compute the derivative of the function $$f(x)=x^3\ln{(x)}$$ Simplify your answer. - Evaluate the derivative at \(x=1\). That is, find \(f'(1)\). Give an exact, simplified answer.
- Evaluate the derivative at \(x=e\). That is, find \(f'(e)\). Give an exact, simplified answer.

**Amy Evers Presentation #2**

- Using
**Theorem 2.4.1 Product Rule**and theorems from Section 2.3, compute the derivative of the function $$f(x)=\left(x^3+5x^2+7x+11\right)e^{(x)}$$ - Evaluate the derivative at \(x=0\). That is, find \(f'(0)\). Give an exact, simplified answer.
- Evaluate the derivative at \(x=1\). That is, find \(f'(1)\). Give an exact, simplified answer.

**Paul Gbadebo Presentation #2**

- Using
**Theorem 2.4.8 Quotient Rule**and theorems from Section 2.3, compute the derivative of the function $$f(x)=\frac{x^2+3x+5}{x+2}$$ - Evaluate the derivative at \(x=1\). That is, find \(f'(1)\). Give an exact, simplified answer.

**Dalana Goddard Presentation #2** Finding a Derivative Two Ways
$$f(x)=\frac{x^2+3x+5}{x}$$

- Using
**Theorem 2.4.8 Quotient Rule**, compute \(f'(x)\). - Start over. This time, start by
*simplifying*\(f(x)\) through division. Rewrite \(f(x)\) in*power function form*. That is, write \(f(x)\) in the form $$f(x)=ax^p+bx^q+cx^r$$ where \(a,b,c\) are constants and \(x^p,x^q,x^r\) are*power functions*. Then find \(f'(x)\) using the simpler derivative rules (the*Sum and Constant Multiple Rule*, the*Power Rule*, and the*Constant Function Rule*).

**Tue Sep 27 Meeting Topics**

Discussed *nested functions*. That is, functions of the form

**Question: ** How to find the *derivative* of such a function?

**Answer: Theorem 2.5.3 The Chain Rule**

**[Example 1]: ** For \(f(x)=\left(\cos{(x)}\right)^2\), find \(f’(x)\).

**[Example 2]: ** For \(f(x)=\cos{\left(x^2\right)}\), find \(f’(x)\).

**[Example 3: ** For \(f(x)=\cos{\left(\ln{(x)}\right)}\), find \(f’(x)\).

**[Example 4]: ** For \(f(x)=\ln\left(\cos{(x)}\right)\), find \(f’(x)\).

Discuss these *Derivative Rules*:

**Constant Function Rule:**If \(f(x)=c\) then \(f’(x)=0\).**Sine Function Rule:**If \(f(x)=\sin{(x)}\) then \(f’(x)=\cos{(x)}\).**Exponential Function Rule:**If \(f(x)=e^{(x)}\) then \(f’(x)=e^{(x)}\).**Logarithm Function Rule:**If \(f(x)=\ln{(x)}\) then \(f’(x)=\frac{1}{x}\).

For each *Derivative Rule*, do an construction similar to those done in Group Works GW08 and GW09, **Finding Derivatives Graphically Using a Ruler**.

- Draw basic graph of \(f(x)\).
- Pin some tangent lines to the graph of \(f(x)\).
- Estimate the slopes of those tangent lines.
- Use the slope date to draw a crude graph of \(f’(x)\).
- Observe that the drawn graph of \(f’(x)\) agrees with the function \(f’(x)\) given by the
*Derivative Rule*.

**Wed Sep 28 Meeting Topics**

**More Examples involving the Chain Rule**

**Class Presentations for Wed Sep 28 **

**Nicole Grant Presentation #2** For the following function
$$f(x)=\left(x^2-4x+3\right)^3$$

- Find \(f'(x)\) using
**Theorem 2.5.3 the Chain Rule**. - Find the \(x\) coordinates of all points on the graph of \(f(x)\) that have
*horizontal tangent lines*.

(**Hint:**Remember that a*horizontal line*has slope \(m=0\). Also remember that the slope of the line tangent to the graph of \(f(x)\) at \(x=a\) is \(m=f'(a)\). So you should look for all values of \(x\) that cause \(f'(x)=0\). In other words, set \(f'(x)=0\) and solve for \(x)\).

**Evan Green Presentation #2** For the following function
$$f(x)=e^{\left(x^2-6x+5\right)}$$

- Find \(f'(x)\) using
**Theorem 2.5.3 the Chain Rule**. - Evaluate the derivative at \(x=0\). That is, find \(f'(0)\).
- Find the
*slope*of the line tangent to the graph of \(f(x)\) at \(x=0\). - Find the \(x\) coordinates of all points on the graph of \(f(x)\) that have
*horizontal tangent lines*.

(**Hint:**Remember that a*horizontal line*has slope \(m=0\). Also remember that the slope of the line tangent to the graph of \(f(x)\) at \(x=a\) is \(m=f'(a)\). So you should look for all values of \(x\) that cause \(f'(x)=0\). In other words, set \(f'(x)=0\) and solve for \(x)\).

Mark Discussed the * Zero Product Property*, and how it is used in solving the equation

**Kierston Harper Presentation #2** For the following function
$$f(x)=\sin{(2x)}$$

- Find \(f'(x)\) using
**Theorem 2.5.3 the Chain Rule**. - Evaluate the derivative at \(x=0\). That is, find \(f'(0)\).
- Find the
*slope*of the line tangent to the graph of \(f(x)\) at \(x=0\). - Find the
*equation*of the line tangent to the graph of \(f(x)\) at \(x=0\).

**Wed Sep 28 Quiz Q04 Covering Sections 2.3 and 2.4**

**20 Minutes at the end of class****No Calculators**- Six Problems
- Compute a derivative using Section 2.3 Derivative Rules (Practice: 2.3 # 11-25 odd)
- Compute a derivative using Section 2.3 Derivative Rules (Practice: 2.3 # 11-25 odd)
- Compute a derivative using Section 2.3 Derivative Rule (Practice: 2.3 # 11-25 odd)
- Compute a derivative using the
*Product Rule*(Section 2.4 Concept) (Practice: 2.4 # 15 - 36 odd) - Compute a derivative using the
*Quotient Rule*(Section 2.4 Concept) (Practice: 2.4 # 15 - 36 odd) - A problem similar to 2.4 # 11 - 14. Given a function \(f(x)\) presented as a
*quotient*,- Use the
*Quotient Rule*to find \(f'(x)\). - Start over. Simplify \(f(x)\) to a form that is
*not*a*quotient*, and then use*simpler derivative rules*to find \(f'(x)\).

- Use the

**Fri Sep 30 Fall Break: Stay Home and Study Math!**

**Book Sections and (Homework Exercises)**

**Section 2.6**Implicit Differentiation (1-4, 5-25odd, 26, 27-41odd)

**Mon Oct 3 Meeting Topics**

Mark B will do an example, and we’ll have two **presentations**.

In our book, the **Power Rule For Derivatives**
$$\frac{d}{dx}x^n=nx^{n-1}$$
is rolled out gradually.

- In
**Book Section 2.3**, the*Power Rule*is presented (without proof) for*positive integer exponents*. That is, \(n \in \mathbb{Z}\) and \( n \gt 0\). - In
**Book Section 2.4**, the authors use the*Quotient Rule*to prove that the*Power Rule*is proved to hold for*all integer exponents*, including*positive*,*negative*and*zero*. That is, \(n \in \mathbb{Z}\). - It is possible to use the
*Chain Rule*(from**Book Section 2.5**) to show that the*Power Rule*actually holds for*all rational number exponents*. That is, \(n \in \mathbb{Q}\). The authors do not do this in the book. - In
**Book Section 2.6**, the authors use*Implicit Differentiation*prove that the*Power Rule*holds for*all rational number exponents*. That is, \(n \in \mathbb{Q}\). Since the method of*Implicit Differentiation*is basically a variation on the*Chain Rule*, the author’s proof in Book Section 2.6 looks similar to what could have been done (but wasn’t done) in Section 2.5. - Finally, in
**Book Section 2.6**, the authors state (but do not prove) the most general result: The*Power Rule*holds for*all real number exponents*. That is, \(n \in \mathbb{R}\).

So far in our *class examples* and *presentations*, we have only found derivatives of power functions that have *positive integer exponents*. We’ll have four **presentations** that involve finding the derivatives of functions that have *more general exponents*.

**Lauren Hartel CP2 (We didn’t get to this last Wednesday.):**

For the following function
$$f(x)=\ln{\left(x^2-4x+3\right)}$$

- Find \(f'(x)\) using
**Theorem 2.5.3 the Chain Rule**. - Find \(f”(x)\).

**Carlotta Dattilo CP2:** For the function
$$f(x)=e^{(x)}+x^e+x^{1.9}+e^{1.9}$$

- Find \(f’(x)\)
- Find \(f'(1)\)

**Tim Jaskiewicz CP2:** For the function

- Find \(f’(x)\). Write your final answer in
*positive exponent form*. That is, eliminate all*negative exponents*. - Find \(f'(8)\)

**Hints for (a): **

- Remember that \(\sqrt[n]{a}=a^{1/n}\).
- Remember that \(a^{-b}=\frac{1}{a^b}\).

**Hint for (b): ** Remember that \(a^{b/c}=a^{(1/c)\cdot b} =\left(a^{(1/c)}\right)^b\).

**Olivia Keener CP2:** For the function

- Find \(f’(x)\) using the
*Chain Rule*. Write your final answer in*positive exponent form*. That is, eliminate all*negative exponents*. - Find \(f'(0)\)
- Find the \(x\) coordinates of all points on the graph of \(f(x)\) that have
*horizontal tangent lines*.

**Hints for (a): **

- For the
*outer function*, use \(outer( \ \ ) = \sqrt{( \ \ )} = ( \ \ )^{1/2}\). - Remember that \(a^{-b}=\frac{1}{a^b}\).

**Hint for (c): **Remember that a *horizontal line* has slope \(m=0\). Also remember that the slope of the line tangent to the graph of \(f(x)\) at \(x=a\) is \(m=f'(a)\). So you should look for all values of \(x\) that cause \(f'(x)=0\). In other words, set \(f'(x)=0\) and solve for \(x)\).

**Austin Kiggins CP2:** For the function
$$f(x)=\frac{5x+7}{\sqrt{x}}$$
the goal is to find \(f’(x)\). This could be done using the *Quotient Rule*, but that would be **really hard**. A better approach is to first *rewrite* \(f(x)\), putting it into a form where simpler derivative rules can be used.

- Rewrite \(f(x)\) in
. That is, rewrite it in the form $$f(x)=ax^b+cx^d$$ This form is a sum of terms where each term is a product of a**power function form***constant*and a*power function*. That’s why it is called.**power function form** - Now find \(f’(x)\). Start by using the simpler derivative rules from
**Section 2.3**. That is, use the**Sum Rule**and the**Constant Multiple Rule**. Then use the**Power Rule**. Write your final answer in*positive exponent form*. That is, eliminate all*negative exponents*.

- Remember that \(\sqrt{a}=a^{1/2}\).
- Remember that \(a^{-b}=\frac{1}{a^b}\).

**Kelly Koenig CP2 (Old Stuff):** Show how the **Theorem 2.4.8 the Quotient Rule** can be used to find the derivative of \(f(x)=\tan{(x)}\).

**Tue Oct 4 Meeting Topics**

(Used for finding \(\frac{dy}{dx}\) when \(x\) and \(y\) are related by an equation that is not solved for \(y\).)

**Starting with: **An equation involving \(x\) and \(y\).

**Step 1: **Replace all \(y\) with the symbol \(y(x)\), indicating that \(y\) is a function of \(x\) . Add parentheses, if necessary, to clarify notation and order of operations.
The result will be a new equation involving \(x\) and \(y(x)\).

**Step 2: **Take derivative of left and right sides of the equation from **Step 1** with respect to \(x\) . That is, take \(\frac{d}{dx}\) of both sides. This will require the **Chain Rule**. Note that when the inner function is \(inner(x)=y(x)\), you should just leave the \(inner'(x)\) in the form \(inner’(x)=\frac{dy(x)}{dx}\). The result will be a new equation involving \(x\) and \(y(x)\) and \(\frac{dy(x)}{dx}\).

**Step 3: **In the equation from **Step 2**, replace all \(y(x)\) with just \(y\). The result will be a new equation involving \(x\) and \(y\) and \(\frac{dy}{dx}\).

**Step 4: **Solve the equation from **Step 3** for \(\frac{dy}{dx}\). The result will be a new equation of the form
$$\frac{dy}{dx}=\text{expression involving }x\text{ and }y$$

Mark B will present two examples involving the method of *Implicit Differentiation*.

Two students will do **presentations** involving .

**Student #1 Presentation CP2**

Suppose that \(3x^2+5xy+7y^2=11\). Use *Implicit Differentiation* to find \(\frac{dy}{dx}\).

**Student #2 Presentation CP2**

Suppose that \(x^2+y^2=1\).

- Use
*Implicit Differentiation*to find \(\frac{dy}{dx}\). - Find the
*slope*of the line tangent to the graph of \(x^2+y^2=1\) at the point \((x,y)=(-\frac{\sqrt3}{2},\frac{1}{2})\). - Find the
*equation*of the line tangent to the graph of \(x^2+y^2=1\) at the point \((x,y)=(-\frac{\sqrt3}{2},\frac{1}{2})\). - Graph the equation \(x^2+y^2=1\), along with the tangent line that you found in (c). Make your graph large and neat, and put \((x,y)\) coordinates on all important locations.

- In Section 173 (Tue 9:30)
- Ben Oldiges is Student #1
- Sara Weller is Student #2

- In Section 174 (Tue 11:00)
- Bilal Tahir is Student #1
- Gavin Wolfe is Student #2

- In Section 175 (Tue 2:00)
- Jonah Lewis is Student #1
- Dylan Pohovey is Student #2

- In Section 176 (Tue 3:30)
- Alan Romero Herrera is Student #1
- Reggie Shaffer is Student #2

**Wed Oct 5 Meeting Topics**

We know how to find derivatives of two kinds of functions that contain exponents.

- \(x^n\) is a
*power function*, so we use the*Power Rule*to find its derivative. That is, \(\frac{d}{dx}x^n=nx^{n-1}\). - \(a^x\) is an
*exponential function*, so we use the*General Exponential Function Rule*to find its derivative. That is, \(\frac{d}{dx}a^n=a^\cdot \ln{(a)}\).

What about the function \(x^x\)? It is neither a power function nor an exponential function. How do we find its derivative?

Mark B will explain the technique of **Logarithmic Differentiation**.

**Colin Sorge CP2:** For the function \(f(x)=e^{(x)}\),

- Find \(f(0)\).
- Use a calculator or computer to get a decimal approximation for the value of \(f(0.1)\), rounded to 5 decimal places.
- Find the
*slope*of the*line tangent to the graph of \(f(x)\) at \(x=0\)*. - Find the
*equation*of the*line tangent to the graph of \(f(x)\) at \(x=0\)*. Present your result in*slope intercept form*. - Without using a calculator or computer, find the \(y\) value on the
*tangent line*from (d) at \(x=0.1\). - Compare your results of (b) and (e).

**Paul Thorp CP2:** (Use units of *radians* in this problem.) For the function \(f(x)=\sin{(x)}\),

- Find \(f(0)\).
- Use a calculator or computer to get a decimal approximation for the value of \(f(0.1)\), rounded to 5 decimal places.
- Find the
*slope*of the*line tangent to the graph of \(f(x)\) at \(x=0\)*. - Find the
*equation*of the*line tangent to the graph of \(f(x)\) at \(x=0\)*. Present your result in*slope intercept form*. - Without using a calculator or computer, find the \(y\) value on the
*tangent line*from (d) at \(x=0.1\). - Compare your results of (b) and (e).

**Emily Wilkerson CP2:** For the function \(f(x)=\sqrt x\),

- Find \(f(4)\).
- Use a calculator or computer to get a decimal approximation for the value of \(f(4.1)\), rounded to 5 decimal places.
- Find the
*slope*of the*line tangent to the graph of \(f(x)\) at \(x=4\)*. - Find the
*equation*of the*line tangent to the graph of \(f(x)\) at \(x=4\)*. Present your result in*slope intercept form*. - Without using a calculator or computer, find the \(y\) value on the
*tangent line*from (d) at \(x=4.1\). - Compare your results of (b) and (e).

**Fri Oct 7 Exam X2 Covering Chapter 2**

**The full duration of the class meeting****No Calculators**- 8 problems, typeset on 4 pages, printed on front & back of 2 sheets of paper.
- Find one derivative using the
*Definition of the Derivative*(**not**the*Derivative Rules*). Study 2.1#7-14 - Compute some derivatives using the
*Derivative Rules*(**not**the*Definition of the Derivative*). Problems of this sort are found in Sections 2.3, 2.4, 2.5, 2.6. - Compute some derivatives using the
*Derivative Rules*(**not**the*Definition of the Derivative*). Problems of this sort are found in Sections 2.3, 2.4, 2.5, 2.6. - Compute some derivatives using the
*Derivative Rules*(**not**the*Definition of the Derivative*). Problems of this sort are found in Sections 2.3, 2.4, 2.5, 2.6. - A problem like the problem on the 2nd page of Group Work GW11. Study that GW.
- A problem about
*slope*of the*tangent line*and/or*equation*of the*tangent line*(but not involving*Implicit Differentiation*). Problems of this sort are found in Sections 2.3, 2.4, 2.5. - A problem involving
*Implicit Differentiation*. Problems of this sort are found in Section 2.6. - A problem about
*approximating*using the*tangent line*. (Problems of this sort are found in Sections 2.2 and 2.3 and in the Presentations for Wed Oct 5.)

- Find one derivative using the

**Book Sections and (Homework Exercises)**

**Section 3.1**Extreme Values (1-6, 7-25odd)**Section 3.2**The Mean Value Theorem (1, 2, 3-20odd)**Section 3.3**Increasing and Decreasing Functions (1-6, 7-23odd)

**Mon Oct 10 Meeting Topics**

**Book Sections and (Homework Exercises):** Section 3.1 Extreme Values (1-6, 7-25odd)

Discussed these topics from Section 3.1:

- Absolute Extrema (Absolute Max and Absolute Min)
- The Extreme Value Theorem
- Relative Extrema (Relative Max and Relative Min)

Students did Group Work GW12: The Extreme Value Theorem.

**Tue Oct 11 Meeting Topics**

**Book Sections and (Homework Exercises):** Section 3.1 Extreme Values (1-6, 7-25odd)

Discuss these topics from Section 3.1:

**Definition 3.1.11 Critical Numbers and Critical Points****Theorem 3.1.12 Relative Extrema and Critical Points****Key Idea 3.1.14 Finding Extrema on a Closed Interval**(**presentations**are about this)

**Student #1 Presentation CP3**

- Find the extrema of \(f(x)=x \cdot e^{(x)}\) on the interval \([-2,2]\).
- Illustrate with a graph of \(f(x)\).

**Student #2 Presentation CP3**

- Find the extrema of \(f(x)=x^3-3x^2-9x-1\) on the interval \([-2,5]\).
- Illustrate with a graph of \(f(x)\).

- In Section 173 (Tue 9:30)
- Nana Asare is Student #1
- Tyler Boldon is Student #2

- In Section 174 (Tue 11:00)
- Andrew Champgagne is Student #1
- Drew Conway is Student #2

- In Section 175 (Tue 2:00)
- Evan Green is Student #1
- Kierston Harper is Student #2

- In Section 176 (Tue 3:30)
- Carlotta Dattilo is Student #1
- Paul Gbadebo is Student #2

Discussed more examples of **Finding Extrema on a Closed Interval**.

Students did Group Work GW13: Finding Absolute Extrema on a Closed Interval.

Students did Group Work GW14: Comparing Two Solutions to an Absolute Extrema Problem.

**Wed Oct 12 Meeting Topics**

**Book Sections and (Homework Exercises):** 3.2 The Mean Value Theorem (1, 2, 3-20odd)

**Reviewed these old concepts (from book Section 2.1)**

- The
**Average Rate of Change**, \(m=\frac{f(b)-f(a)}{b-a}\), which is the slope of a*secant line*on the graph of the function - The
**Instantaneous Rate of Change**, \(m=\lim_{h\rightarrow 0}\frac{f(c+h)-f(C)}{h}\), which is the slope of a*secant line*on the graph of the function

**Discussed the Mean Value Theorem (from book Section 3.2)**

**Theorem 3.2.3 The Mean Value Theorem**

Let \(f(x)\) be a continuous function on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\). There exists an \(x\)value, \(x=c\), such that
$$f’(c)=\frac{f(b)-f(a)}{b-a}$$
That is, there is at least one \(x\) value, \(x=c\), in the interval \((a,b)\) such that the *instantaneous rate of change of \(f(x)\) at \(x=c\)* is equal to the *average rate of change of \(f(x)\) on the interval \([a,b]\)*.

In terms of the graph, this meant that there is at least one \(x\) value, \(x=c\), in the interval \((a,b)\) such that the line *tangent* to the graph of \(f(x)\) at \(x=c\) has the same slope as the *secant line* that touches the graph of \(f(x)\) at \(x=a\) and \(x=b\).

**Saw Four Examples Involving the Mean Value Theorem**

**Tim Jaskiewicz CP3:**(book exercise 3.2#12) Can the Mean Value Theorem be applied to the function \(f(x)=5x^x-6x+8\) on the interval \([0,5]\)? If so, find a number \(c\) in the interval \((0,5)\) that is guaranteed by the Mean Value Theorem.**Olivia Keener CP3:**(book exercise 3.2#14) Can the Mean Value Theorem be applied to the function \(f(x)=\sqrt{25-x}\) on the interval \([0,9]\)? If so, find a number \(c\) in the interval \((0,9)\) that is guaranteed by the Mean Value Theorem.**Paul Thorp CP3:**(book exercise 3.2#16) Can the Mean Value Theorem be applied to the function \(f(x)=\ln{(x)}\) on the interval \([1,5]\)? If so, find a number \(c\) in the interval \((1,5)\) that is guaranteed by the Mean Value Theorem.**Mark B:**For the function \(f(x)=\frac{1}{x}\) on the interval \([-1,1]\), it does not seem possible to find a number \(c\) in the interval \((-1,1)\) such that $$f’(c)=\frac{f(1)-f(-1)}{1-(-1)}$$ Why not?!?

**Fri Oct 14 Meeting Topics**

**Book Sections and (Homework Exercises):**
Section 3.3 Increasing and Decreasing Functions (1-6, 7-23odd)

Discussed these topics from Section 3.3:

**Definition 3.3.2 Definition of Increasing and Decreasing Functions****Theorem 3.3.4 Test for Increasing/Decreasing**

Discussed method of making a **Sign Chart** to determine the *sign behavior* of a function.

**[Increasing/Decreasing Example] ** For the function
$$f(x)=x^4-6x^2+5$$
find the intervals where \(f(x)\) is **increasing** and the intervals where \(f(x)\) is **decreasing**.

Note that this was the same function that was studied on Tue Oct 11, in Group Work GW13: Finding Absolute Extrema on a Closed Interval. In that Group Work, students found that $$f’(x)=4x^3-6x=4x(x^2-3)=4x(x+\sqrt{3})(x-\sqrt{3})$$ The critical numbers for \(f(x)\) are \(x=-\sqrt{3},0,\sqrt{3}\).

To solve the problem in the current **[Example]**, a **sign chart** was constructed for \(f’(x)\) to determine its sign behavior. Then **Theorem 3.3.4 Test for Increasing/Decreasing** was used to make the following conclusions:

- \(f(x)\) is
**increasing**on the intervals \([-\sqrt{3},0]\) and \([\sqrt{3},\infty)\) because \(f’(x)\) is**positive**there. - \(f(x)\) is
**decreasing**on the intervals \((-\infty,-\sqrt{3}]\) and \([0,\sqrt{3}]\) because \(f’(x)\) is**negative**there.

**Quiz Q5 on Friday Oct 14 covering Sections 3.1 and 3.2**

**Book Sections and (Homework Exercises)**

**Section 3.3**Increasing and Decreasing Functions (1-6, 7-23odd)**Section 3.4**Concavity and the Second Derivative (1-4, 5-56odd)**Section 3.5**Curve Sketching (1-5, 6-25odd, 26-28)

**Mon Oct 17 Meeting Topics**

**Book Sections and (Homework Exercises):** Section 3.3 Increasing and Decreasing Functions (1-6, 7-23odd)

Reviewed past topics from Section 3.1 and 3.3 that were discussed last week

**Definition 3.1.1 Extreme Values (absolute max and absolute min)****Theorem 3.1.3 The Extreme Value Theorem**If \(f(x)\) is a*continuous*function defined on a*closed interval*\(I=[a,b]\), then \(f(x)\) has both an*absolute max*and an*absolute min*on the interval \(I\).**Definition 3.1.6 Relative Max and Relative Min****Definition 3.1.11 Critical Numbers and Critical Points****Theorem 3.1.11 Critical Numbers and Critical Points**Relative extrema can only occur at \(x\) values that are**critical numbers**.**Fact**Absolute extrema can only occur at \(x\) values that are**endpoint of the domain**or**critical numbers**.**Key Idea 3.1.4**Method for finding**absolute extrema**for a function \(f(x)\) that is*continuous*on a*closed interval*.**Definition 3.3.2 Definition of Increasing and Decreasing Functions****Theorem 3.3.4 Test for Increasing/Decreasing**

Discussed new topic from Section 3.3:

**Theorem 3.3.10 First Derivative Test**

**Amy Evers Presentation CP3:** Find the **critical numbers** for the function \(f(x)=xe^{(-x)}\).

**[Example 1]** (done by Mark B) Find the **relative extrema** for the function \(f(x)=xe^{(-x)}\).

**Ben Oldiges Presentation CP3:** Find the **critical numbers** for the function \(f(x)=\frac{1}{x^2}\).

**[Example 2]** (done by Mark B) Find the **relative extrema** for the function \(f(x)=\frac{1}{x^2}\).

**Tue Oct 18 Meeting Topics**

**Book Sections and (Homework Exercises):** Section 3.3 Increasing and Decreasing Functions (1-6, 7-23odd)

A **partition number** for a function \(g(x)\) is an \(x\) value where \(g(x)=0\) or \(g\) is *discontinuous*.
**Remark: **A function \(g(x)\) can only *change sign* at its *partition numbers*.

A **critical number** for a function \(f(x)\) is an \(x\) value \(x=c\) that has these two properties:

- \(x=c\) is a
*partition number for*\(f'(x)\). That is, \(f'(c)=0\) or \(f'\) is*discontinuous*at \(x=c\). - \(f\) is
*continuous*at \(x=c\).

**Student #1 Presentation CP3** Find the **critical numbers** for the function
$$f(x)=x+\frac{4}{x}$$

**Student #2 Presentation CP3** Find the **critical numbers** for the function
$$f(x)=\frac{x}{x^2+9}$$

- In Section 173 (Tue 9:30)
- Sara Weller is Student #1
- Ellie Bower is Student #2

- In Section 174 (Tue 11:00)
- Nicole Grant is Student #1
- Bilal Tahir is Student #2

- In Section 175 (Tue 2:00)
- Austin Kiggins is Student #1
- Jonah Lewis is Student #2

- In Section 176 (Tue 3:30)
- Carlotta Dattilo is Student #1
- Paul Gbadebo is Student #2

- GW15: Analyzing a Polynomial, Tue Oct 18
- GW16: Analyzing a Rational Function with a Vertical Asymptote, Tue Oct 18
- GW17: Analyzing a Rational Function with a Horizontal Asymptote, Tue Oct 18

**Wed Oct 19 Meeting Topics**

**Book Sections and (Homework Exercises)**

**Section 3.4**Concavity and the Second Derivative (1-4, 5-56odd)**Section 3.5**Curve Sketching (1-5, 6-25odd, 26-28)

- GW18: Using Given Information to Sketch a Graph, Wed Oct 19

**Fri Oct 21 Meeting Topics**

**Book Sections and (Homework Exercises):** Section 3.5 Curve Sketching (1-5, 6-25odd, 26-28)

**Dalana Goddard Presentation CP3: **Given the following in formation about \(f,f’,f”\):

- \(f(x)\)is continuous for all x
- \(f(7)=10\)
- \(f’(x)\) is positive on the interval \((-\infty,2)\).
- \(f’(2)=0\)
- \(f’(x)\) is negative on the interval \((2,12)\).
- \(f’(12)=0\)
- \(f’(x)\) is positive on the interval \((12,\infty)\).
- \(f’’(x)\) is negative on the interval \((-\infty,7)\).
- \(f’’(7)=0\)
- \(f’’(x)\) is positive on the interval \((7,\infty)\).

**Lauren Hartel Presentation CP3** The graph of a function \(f(x)\) is given without gridlines and without coordinate axes.

The formula for \(f(x)\) is not given, but the formulas for its first and second derivatives are

$$\begin{eqnarray} f’(x) &=& \frac{x}{x^2+1} \\ f”(x) &=& -\frac{2(x^2-1)}{(x^2+1)^2}=-\frac{2(x+1)(x-1)}{(x^2+1)^2} \end{eqnarray}$$- It looks like there is a
*relative min*on the graph of \(f(x)\). What is the \(x\) coordinate of that point? Explain how you know. - It looks like there is are two
*inflection points*on the graph of \(f(x)\). What are the \(x\) coordinates of those point? Explain how you know.

**Quiz Q6 on Friday Oct 21 covering Section 3.3**

**Book Sections and (Homework Exercises)**

**Section 4.1**Newton's Method (3, 5, 7, 17)**Section 4.2**Related Rates (3-15odd)

**Mon Oct 24 Meeting Topics**

**Carly Doros Presentation CP3: **The derivative of \(f(x) = e^{(-x^2)}\) is
$$f'(x)= -2xe^{(-x^2)}$$

- Show how \(f'(x)\) is obtained.
- Make a sign chart for \(f'(x)\).
- Using your sign chart for \(f'(x)\), determine the intervals where \(f(x)\) is
*increasing*and*decreasing*. Present your answer in*interval notation*. - Determine the \(x\) coordinates of all
*relative extrema*of \(f(x)\). - Find the corresponding \(y\) coordinates of the
*relative extrema*.

**Dylan Pohovey Presentation CP3: **The second derivative of \(f(x) = e^{(-x^2)}\) is
$$f''(x)= (4x^2-2)e^{(-x^2)}$$

- Show how \(f''(x)\) is obtained.
- Make a sign chart for \(f''(x)\). It will be helpful to note that \(f''(x)\) can be factored as $$f''(x)= (4x^2-2)e^{(-x^2)} = 4(x^2-\frac{1}{2})e^{(-x^2)} = 4(x+\frac{1}{\sqrt{2}})(x-\frac{1}{\sqrt{2}})e^{(-x^2)}$$
- Using your sign chart for \(f'(x)\), determine the intervals where \(f(x)\) is
*concave up*and*concave down*. Present your answer in*interval notation*. - Determine the \(x\) coordinates of all
*inflection points*of \(f(x)\). - Find the corresponding \(y\) coordinates of the
*inflection points*.

Students worked on Group Work GW19: The Idea Behind Newton’s Method

Then Mark discussed Newton’s Method

**Tue Oct 25 Meeting Topics**

We will be discussing the following function:

$$f(x)=-x^4+4x^3$$Refer to the Reference R05: Graphing Strategy, handed out in class on Wed Oct 19.

**First Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Nana Asare CP4** - In Section 174 (Tue 11:00),
**Gavin Wolfe CP3** - In Section 175 (Tue 2:00),
**Kierston Harper CP4** - In Section 176 (Tue 3:30),
**Kelly Koenig CP3**

**Second Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Tyler Boldon CP4** - In Section 174 (Tue 11:00),
**Drew Conway CP4** - In Section 175 (Tue 2:00),
**Evan Green CP4** - In Section 176 (Tue 3:30),
**Reggie Shaffer CP3**

**Groups: **Complete GW20: Using the Graphing Strategy to Analyze and Graph a Polynomial

**Wed Oct 26 Meeting Topics**

Students finished the Group Work that was begun on Monday: GW19: The Idea Behind Newton’s Method.

Mark Discussed Newton’s Method.

Students worked on Group Work GW21: Newton’s Method

**Ellie Bower CP4: ** In the Group Work GW21: Newton’s Method, students used *Newton’s Method* to find an *approximate* value for the *root* of the function
$$f(x)=x^3-x^2-1$$
In the group work, they were given an initial approximation \(x_0=1\), and they used *Newton’s Method* to find \(x_1\) and \(x_2\).

Your job is to find a web site that has a *Newton’s Method Calculator*. There are lots of them. Using the function \(f(x)=x^3-x^2-1\) and the initial approximation \(x_0=1\), show the web site calculator results for the first \(10\) or so steps. That is, display \(x_0\) through roughly \(x_{10}\). Comment on whether the results that the calculator displays for \(x_1\) and \(x_2\) match the values obtained by your group.

**Andrew Champagne CP4: ** In the Group Work GW21: Newton’s Method, students used *Newton’s Method* to find an *approximate* value for the *root* of the function
$$f(x)=x^3-x^2-1$$
Show what happens when you ask *Wolfram Alpha* to find a root of \(f(x)\). That is, ask Wolfram Alpha to solve the equation
$$f(x)=0$$
Show the result in both the *exact form* and the *decimal approximation* (the *approximate form*).

**Fri Oct 28 Meeting Topic: **Section 4.2 Related Rates (3-15odd)

The Quiz will be over Newton’s Method (from Section 4.1). The quiz will be one problem, with three questions that are similar to the questions in Group Work GW21: Newton’s Method. That is, a problem like this:

The goal is to use Newton’s Method to find an approximate value for a root of the function \(f(x)=\text{some function}\), using the initial approximation \(x_0=4\).

- Compute \(f’(x)\).
- Fill in a table that is a worksheet for finding the values of \(x_1\), \(x_2\), and \(x_3\). (similar to the table in GW21)
- A graph of \(f(x)\) is shown. Illustrate your results on this graph. (illustrations similar to the illustration in GW21)

To prepare for the quiz:

- Read your MATH 2301 class notes from Wed Oct 26 meeting.
- Review the Group Work GW21: Newton’s Method.
- Work the suggested exercises 4.1#3,5,7. These book exercises are written with calculators in mind. Go ahead and use your calculator for your studying. The quiz problem will be a simple polynomial function that you can analyze without a calculator.

**Book Sections and (Homework Exercises)**

**Section 4.3**Optimization (8, 9, 11, 12, 13, 15, 18)**Section 4.4**Differentials (1-6, 7-13odd, 17-39odd)**Section 5.1**Antiderivatives and Indefinite Integration (9-27odd, 28, 29, 31-39odd)

**First Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Amy Evers CP4** - In Section 174 (Tue 11:00),
**Carly Doros CP4** - In Section 175 (Tue 2:00),
**Austin Kiggins CP4** - In Section 176 (Tue 3:30),
**Paul Gbadebo CP4**

**Second Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Dalana Goddard CP4** - In Section 174 (Tue 11:00),
**Nicole Grant CP4** - In Section 175 (Tue 2:00),
**Dylan Pohovey CP4** - In Section 176 (Tue 3:30),
**Alan Romerero Herrera CP3**

**Third Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Lauren Hartel CP4** - In Section 174 (Tue 11:00),
**Tim Jaskiewicz CP4** - In Section 175 (Tue 2:00),
**Colin Sorge CP3** - In Section 176 (Tue 3:30),
**Emily Wilkerson CP3**

**Groups: **Complete the three group works that were started by your classmates

**Book Sections and (Homework Exercises)**

**Section 5.1**Antiderivatives and Indefinite Integration (9-27odd, 28, 29, 31-39odd)**Section 5.2**The Definite Integral (5-17odd, 19-22)

- A problem about Increasing & Decreasing Functions and Relative Extrema (Section 3.3 Concepts)
- A problem about Concavity and the Second Derivative (Section 3.4 Concepts)
- A problem about Curve Sketching, making use of results from problems 1,2 (Section 3.5 Concepts)
- A problem about Related Rates (Section 4.2 Concepts)
- A problem about Optimization (Section 4.3 Concepts)
- A problem about Differentials (Section 4.4 Concepts)

**Book Sections and (Homework Exercises)**

**Section 5.1**Antiderivatives and Indefinite Integration (9-27odd, 28, 29, 31-39odd)

In book Section 5.1, in **Definition 5.1.1** at the beginning of the section, the authors present the definitions of *Antiderivatives and Indefinite Integral* together. I think it is useful to spend some time learning about antiderivatives *before* learning about indefinite integrals. In today’s meeting, you’ll learn about, and discuss, antiderivatives.

**Words:**\(F\)*is an antiderivative of*\(f\). (Note the uppercase and lower case letters!)**Meaning:**\(f\)*is the derivative of*\(F\). That is, \(f=F’\).**Arrow diagram:**\(F \xrightarrow[\text{take derivative}] {}f\)

In coming days, you will be learning some techniques for *finding* antiderivatives. We’ll start discussing those techniques on Wed Nov 9. But in today’s meeting, you will focus on the following kind of problem:

**Question:**Is one given function \(f(x)\) and*antiderivative*of another given function \(g(x)\).**Solution Strategy:**Find \(f’(x)\) and see if it equals \(g(x)\).

**[Example 1] Question: **
$$\text{Is }F(x)=\frac{x^3}{3}\text{ an antiderivative of }f(x)=x^2?$$
Explain why or why not. (Show the steps!)

**Solution: ** Strategy: Find \(F’(x)\) and see if it equals \(f(x)\).

First, rewrite $$F(x)=\frac{x^3}{3}=\left(\frac{1}{3}\right)x^3$$

Now, find the derivative. $$F’(x)=\frac{d}{dx}\left(\frac{1}{3}\right)x^3=\left(\frac{1}{3}\right)\frac{d}{dx}x^3=\left(\frac{1}{3}\right)\left(3x^{3-1}\right)=x^2=f(x)$$

The answer is **yes**, \(F(x)=\frac{x^3}{3}\) an antiderivative of \(f(x)=x^2\), because \(F’(x)=f(x)\)

**End of [Example 1]**

**First Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Ben Oldiges CP4** - In Section 174 (Tue 11:00),
**Bilal Tahir CP4** - In Section 175 (Tue 2:00),
**Olivia Keener CP4** - In Section 176 (Tue 3:30),
**Carlotta Datillo CP4**

(

**Second Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Sarah Weller CP4** - In Section 174 (Tue 11:00),
**Andrew Champagne CP5** - In Section 175 (Tue 2:00),
**Jonah Lewis CP4** - In Section 176 (Tue 3:30),
**Kelly Koenig CP4**

(

**Third Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Nana Asare CP5** - In Section 174 (Tue 11:00),
**Drew Conway CP5** - In Section 175 (Tue 2:00),
**Colin Sorge CP4** - In Section 176 (Tue 3:30),
**Alan Romero Herrera CP4**

(

**Fourth Presentation**, presented by these students:

- In Section 173 (Tue 9:30),
**Tyler Boldon CP5** - In Section 174 (Tue 11:00),
**Carly Doros CP5** - In Section 175 (Tue 2:00),
**Evan Green CP5** - In Section 176 (Tue 3:30),
**Reggie Shaffer CP4**

(

**Groups: **Work on Group Work GW25: Antiderivatives.

Exercises: Section 5.1# 9-27odd, 28, 29, 31-39odd

Mark will do some Examples

Exercises: Section 5.2# 5-17odd, 19-22

Mark Discussed the *informal* definition of the **Definite Integral** presented in the book:

**Symbol:**$$\int_{x=a}^{x=b}f(x)dx$$**Spoken:**The definite integral of \(f(x)\) from \(x=a\) to \(x=b\).**Usage:**The function \(f(x)\) is*continuous*on the interval \([a,b]\).**Informal Definition:**the signed area of the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\).**Remark:**This is an*informal*definition because we have only have a notion of area for certain basic geometric shapes. For now, this definition of definite integral can only be used in situations where the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is made up of basic geometric shapes. In those situations, the value of the definite integral can be found by using familiar geometric formulas to compute the areas of the shapes that make up the region. (Note, however, that the book does not call this an*informal*definition. Rather, the book just presents this as the*definition*of the*Definite Integral*. That is a significant mistake in the book.)

Mark will do some Examples of Definite Integrals Gound Using Geometry

Students worked on Group Works involving concepts from these two sections:

**Section 5.1 Indefinite Integrals**(Exercises: 5.1# 9-27odd, 28, 29, 31-39odd)**Section 5.2 Definite Integrals**(Exercises: 5.2# 5-17odd, 19-22)

The Group Works:

- GW26: Good and Bad Indefinite Integral Solutions (Section 5.1 Concepts)
- GW27: Definite Integrals for a Graph Made up of Geometric Shapes (Section 5.2 Concepts)
- GW28: Computing Definite Integrals By Using Geometry (Section 5.2Concepts)

Sections for Today

- Section 5.3: Riemann Sums (Exercises: 17-39odd)

Mark Discussed the *informal* definition of the **Definite Integral** presented in the book:

**Symbol:**$$\int_{x=a}^{x=b}f(x)dx$$**Spoken:**The definite integral of \(f(x)\) from \(x=a\) to \(x=b\).**Usage:**The function \(f(x)\) is*continuous*on the interval \([a,b]\).**Informal Definition:**the signed area of the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\).**Remark:**This is an*informal*definition because we have only have a notion of area for certain basic geometric shapes. For now, this definition of definite integral can only be used in situations where the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is made up of basic geometric shapes. In those situations, the value of the definite integral can be found by using familiar geometric formulas to compute the areas of the shapes that make up the region. (Note, however, that the book does not call this an*informal*definition. Rather, the book just presents this as the*definition*of the*Definite Integral*. That is a significant mistake in the book.)

**Natural Question: **What do we do when the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is **not** made up of basic geometric shapes?

That is the essence of ** The Area Problem**.

**The Area Problem**

When the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is **not** made up of basic geometric shapes,

- What does the
*area of the region*even mean? - How do we compute a
*value*for the area of the region?

We start by thinking about a short *wish list* for *Area*, things that we *expect* to be true for Area. We don’t yet have a definition of what Area *calculate the area*, but however Area is eventually defined, we expect the definition to satisfy these three requirements. This short list could be called *Area Axioms*.

**Area Axioms**

- The area of a region should be \(\geq 0\), and should \(=0\) only when the region does not contain any discs. (That’s a cheesy, but actually mathematically fine, way of waying that the region does not enclose any space.)
- The area of a region should equal the sum of the areas of the sub-regions that make up the larger region.
- (This is a consequence of axiom 2) If one region is entirely contained in some larger region, then the area of the smaller, inner region should be less than the area of the larger region.

With that third axiom in mind, we can think of a way of getting an *underestimate* and an *overestimate* for the unknown area of some curvy region by constructing a sort of *sandwich* involving the curvy region and two simple geometric regions. A region made up of rectangles is built in a way that it is entirely contained in the curvy region. A second region made up rectangles is built in a way that the curvy region is entirely contained in the second region.

This led to the introduction of the *Left Riemann Sum with \(n\) subintervals*, denoted by \(L_n\), and the *Right Riemann Sum with \(n\) subintervals*, denoted by \(R_n\). These sums are the area of regions made up of *left rectangles* and *right rectangles*.

Students worked on this Group Work involving calculating *Left and Right Riemann Sums* for a function given by a *graph*.

Then Mark B worked on an analogous problem involving a function given by a *formula* not a graph. That problem was framed as a ** Quest**.

For the function \(f(x)\) given by the formula
$$f(x)=5+\frac{x^2}{10}$$
the region between the graph of \(f(x)\) and the \(x\) axis from \(x=2\) to \(x=12\) is a *curvy* region, not made up of simple geometric shapes. We would like to find the *area* of that region.

We don’t have a method for calculating the area of that region. We don’t even have a definition of what area even *means* for a region like that. But it is possible to find an *underestimate* and an *overestimate* using *Riemann Sums*. Mark did this with a *Left Sum with \(5\) subintervals* and a *Right Sum with \(5\) subintervals*. The result was
$$L_5=94 \lt \text{ unknown area } \lt R_5=122$$

Sections for Today

- Section 5.3: Riemann Sums (Exercises: 17-39odd)
- Useful video
**Approximating areas with Sums**

Using Reference R06: Steps for Computing Riemann Sums as a guide, students worked on Group Work GW30: Computing Riemann Sums.

Recall our *Quest* from Wednesday’s meeting:

For the function \(f(x)\) given by the formula
$$f(x)=5+\frac{x^2}{10}$$
the region between the graph of \(f(x)\) and the \(x\) axis from \(x=2\) to \(x=12\) is a *curvy* region, not made up of simple geometric shapes. We would like to find the *area* of that region.

We don’t have a method for calculating the area of that region. We don’t even have a definition of what area even *means* for a region like that. But it is possible to find an *underestimate* and an *overestimate* using *Riemann Sums*. Mark did this with a *Left Sum with \(5\) subintervals* and a *Right Sum with \(5\) subintervals*. The result was
$$L_5=94 \lt \text{ unknown area } \lt R_5=122$$

In this part of the meeting, we will try to improve the underestimate and overestimate by using more subintervals. We use a computer tool to do the repetitive work.

Here is a handy Riemann Sum Calculator.

Rather than lecturing, Mark B will be summarizing the video **Approximating areas with Sums**

The Quiz will be over **Indefinite Integrals** (concepts from Section 5.1).

**Book Sections and (Homework Exercises)**

**Section 5.4**The Fundamental Theorem of Calculus (5-29odd, 35-57odd)**Video that may be helpful:**(video) (notes)

**Book Sections and (Homework Exercises)**

**Section 5.4**The Fundamental Theorem of Calculus (5-29odd, 35-57odd)**Video that may be helpful:**(video) (notes)

**Book Sections and (Homework Exercises)**

**Section 5.4**The Fundamental Theorem of Calculus (5-29odd, 35-57odd)**Video that may be helpful:**(video) (notes)

**Tue Nov 22: **

- Student presentations involve showing how to find a definite integral using the Fundamental Theorem of Calculus.
- All students will have presentations
- Extra Credit Presentations (for 10 points):
- If a student is absent, another student can do their presentation for extra credit.
- Also, notice that there are 10 presentations. But no section has that many students. If there is a presentation with no student assigned to it, a student do that presentation for extra credit.
- Each student can do at most one extra credit presentation.
- The substitute student must have prepared the presentation.
- Therefore, students hoping to earn extra credit points by doing an extra presentation should prepare by studying
*all*of the presentations and knowing how to do them.

**The Fundamental Theorem of Calculus: **
$$\int_{x=a}^{x=b}f(x)dx\underset{\text{FTC}}{=}\left. \left(\int f(x)dx\right)\right\vert_{x=a}^{x=b}$$
Mark B introduced this theorem in lecture on Mon Nov 21. He also discusses it at length in this video and its accompanying notes. And of course the Theorem is discussed in the online Apex Calculus book, in Section 5.4.

**The Presentations:** (Student Numbers are shown below.)

- Show why $$\int_0^6(x+7)^2dx=618$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_0^\frac{\pi}{2}\cos{(x)}dx=1$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_{1}^2e^{(x)}dx=e^2-e$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_{1}^{2}3^{(x)}dx=\frac{6}{\ln{(3)}}$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_{16}^{49}\sqrt{x}dx=186$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_{16}^{49}\frac{1}{\sqrt{x}}dx=6$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_0^{\pi}(3\cos{(x)}-2\sin{(x)})dx=-4$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_{75}^{100}dx=25$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_1^{e^2}\frac{1}{x}dx=2$$ That is, clearly show the steps that lead to this result.
- Show why $$\int_1^{2}\frac{1}{x}dx=\ln{(2)}$$ That is, clearly show the steps that lead to this result.

**Student Numbers for Section 173 (Tue 9:30)**

- Nana Asare
- Tyler Boldon
- Ellie Bower
- Amy Evers
- Dalana Goddard
- Lauren Hartel
- Sara Weller

**Student Numbers for Section 174 (Tue 11:00)**

- Andrew Champagne
- Drew Conway
- Carly Doros
- Nicole Grant
- Tim Jaskiewicz
- Bilal Tahir
- Gavin Wolfe

**Student Numbers for Section 175 (Tue 2:00**

- Evan Green
- Kierston Harper
- Olivia Keener
- Austin Kiggins
- Jonah Lewis
- Dylan Pohovey
- Colin Sorge

**Student Numbers for Section 176 (Tue 3:30)**

- Carlotta Dattilo
- Paul Gbadebo
- Kelly Koenig
- Alan Romero Herrera
- Reggie Shaffer
- Paul Thorp
- Emily Wilkerson

**Book Sections and (Homework Exercises)**

**Section 5.4**The Fundamental Theorem of Calculus (5-29odd, 35-57odd)**Section 6.1**Substitution (3-85odd)

- Discuss
**The Average Value of a Function on an Interval**(Concept from APEX Section 5.4) - Discuss
**Position, Velocity, and Acceleration**(Concepts from APEX Sections 2.2 and 5.4)

- Group Work GW31: The Average Value of a Function on an Interval
- Group Work GW32: Position, Velocity, and Acceleration

- Discuss the
**Area So Far Function**(Concept from APEX Section 5.4) - Discuss
**Integration by Substitution**(Concept from APEX Section 6.1) - Refer to Reference R07: The Method of Integration by Substitution
**Quiz Q09**on Section 5.4

- Discuss
**Integration by Substitution**(Concept from APEX Section 6.1) - Refer to Reference R07: The Method of Integration by Substitution
- Group Work GW33: Integration by Substitution

**Final Exam, Mon Dec 5, 2:30pm - 4:30pm
**

- Students of Shadik, Eisworth and Barsamian in Morton 235
- Students of Wu and Regan in Morton 237

**Brief Calendar** on a printable PDF document.

page maintained by Mark Barsamian, last updated Dec 4, 2022