**Campus: **Ohio University, Athens Campus

**Department: **Mathematics

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

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

**Meeting Times and Locations:**

- Lecture Section 100 meets Mon Wed, Fri 8:35am – 9:30m in Morton Hall 127.

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

- Lecture Section 110 meets Mon Wed, Fri 11:50am – 12:45pm in Morton Hall 127.

- Recitation Section 111 meets Tue 9:30am – 10:25am in Ellis 108
- Recitation Section 112 meets Tue 11:00am – 11:55am in Ellis 108
- Recitation Section 113 meets Tue 2:00pm – 2:55pm in Morton 122
- Recitation Section 114 meets Tue 3:30pm – 4:25pm in Morton 318

- Lecture Section 120 meets Mon Wed, Fri 2:00pm – 2:55pm in Morton Hall 127.

- 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

**Information about the Instructors: **

**Instructor for Lecture Sections 100 and 110:** Daniel Ntiamoah

**Office Location:**Morton 558**Office Hours:**Mon, Fri 9:30am - 10:30am**Office Phone:**740-593-9806**Email:**ntiamoah@ohio.edu

**Instructor for Recitation Sections 101, 102, 103, 104:** Kingsley Osae

**Office Location:**Morton 564**Office Hours:**- Tue 10:30am 11:30am
- Thu 11:00am noon

**Office Phone:**740-593-1286**Email:**ko313720@ohio.edu

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

**Office Location:**Morton 203**Office Hours:**- Mon 1:00pm 12:30pm
- Thu 2:30pm 4:00pm
- Also by appointment if both time frames don’t work

**Office Phone:**740-593-1245**Email:**ks698620@ohio.edu

**Instructor for Lecture Section 120:** Mark Barsamian

**Office Location:**Morton 521**Office Hours:**See Mark Barsamian's Web Page for his office hours this semester.**Office Phone:**740-593-1273**Email:**barsamia@ohio.edu

**Instructor for Recitation Sections 121, 122, 123, 124:** Isaac Agyei

**Office Location:**Morton 532**Office Hours:**- Tuesday 12:30pm – 1:30pm (no appointment necessary)(This office hour is held in Ellis Hall. Ask Isaac about the room.)
- Additional hours available by appointment

**Office Phone:**740-593-1271**Email:**ia520320@ohio.edu

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

**Final Exam Date: **All Athens Campus Sections of MATH 2301 have a Common Final Exam on Thu May 2, 2024, from 4:40pm – 6:40pm in various Morton Hall rooms. (Room assignments will be made later.)

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

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

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

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

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

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

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

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

For MATH 2301, this means that if you will be missing any Spring 2024 Quizzes or Exams for religious reasons, and if you want to have a Make-Up Quiz/Exam, **you will need to notify your Professor no later than Tuesday, January 30, 2023**. You and your Professor will work out the dates/times of your Make-Up Quiz/Exam. (In general, if you are going to miss a Friday Quiz/Exam, your Professor will schedule you for a Make-Up on the following Monday or Tuesday.)

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

Policy for Electronic communication between MATH 2301 Students and Instructors

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

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

Jeff Bezos

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

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

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

**Textbook and WebAssign Information: **

**Required Online Course Materials: ** Through a program called *Inclusive Access*, the University has negotiated with the publisher a special price for this course's *Required Online Course Materials*. On the first day of class, you will receive access to an an online system called *WebAssign*. The *WebAssign* system includes an *eText* version of the textbook and an *online homework system*. The cost of the Online Course Materials is a discounted Inclusive Access Price of $45 plus 7% Ohio sales tax, for a total of about $48.15. That cost will be automatically billed to your Ohio University Student Account. If you drop the course before the drop deadline (Fri, Jan 26), your student account will be credited for any amount billed. After you register, you will receive more information about the Inclusive Access program, including an option to "Opt Out" of participation in the program. To "Opt Out" means that your payment for the Online Course Materials is not handled by the Inclusive Access program. If you do that, you can still use the Online Course Materials, but in order to access them, you will be asked to make a credit card payment for the Retail Price of the materials. (Note that the Retail Price is $111 plus 7% ohio tax, for a total of about $118.77. That is significantly higher than the Inclusive Access Price.)

**Optional Print Copy of the Textbook: ** Many students (and instructors) prefer reading printed textbooks rather than eTexts. Students in Ohio University MATH 2301 Sections 100, 110, and 120 can purchase a print copy of the book at the
**College Bookstore**
(at the corner of Court Street and Union Street in Athens) for the discounted price of $46.75 + 7% Ohio sales tax, for a total of around $50.02. This is an extraordinarily low price for a print textbook, and you are strongly encouraged to buy the print copy. Note that your purchase of the print copy will be **in addition to** the *Online Course Materials* that you receive as part of the *Inclusive Access* program, described above. So if you do buy the print copy, your total expenditures will be $48.15 (for the *Online Course Materials* purchased through the *Inclusive Access* program) plus $50.02 (for the print copy of the textbook, purchased at the College Bookstore) for a total of $98.17. That is still an excellent price for course materials. The print copy is a loose-leaf book; its full description is:

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

**Exercises: **

(from Stewart Essential Calculus Early Transcendentals 2nd Edition)

Your goal should be to write solutions to all 362 exercises in this list.

Printable PDF of the Exercise List

(__Underlined__ exercises are in the textbook but are __not__ in *WebAssign*. Do not overlook them: they are on this list because they are __important__.)

- 1.3 The Limit of a Function: 1, 5, 7, 10, 11, 12,
__13__, 15, 16 - 1.4 Calculating Limits: 5, 7, 10, 11, 17, 21, 23, 25, 27, 31, 33, 35, 38, 42, 49, 51, 55
- 1.5 Continuity: 3, 5, 7,
__17__, 19, 27, 33, 39, 43, 47 - 1.6 Limits Involving Infinity: 1, 5, 7, 9, 10, 13, 19, 21, 25, 29, 33, 35, 40, 41, 45, 49
- 2.1 Derivatives & Rates of Change: 1, 5, 7,
__9__, 11, 15, 16, 18, 25, 27, 29, 31, 33, 35, 43, 47 - 2.2 The Derivative as a Function: 1, 3, 5, 9, 11, 13, 19, 20, 22, 23, 25, 33, 35, 39
- 2.3 Basic Differentiation Formulas: 1, 7, 9, 11, 13, 19,
__27__, 29, 31, 33, 35, 37, 39, 45, 50, 57, 69 - 2.4 The Product & Quotient Rules: 3, 5, 7, 13, 16, 17, 19, 21, 26, 27, 31, 34, 37, 41, 51,
__55__ - 2.5 The Chain Rule: 1, 7, 13, 14, 17, 21, 25, 35, 43, 47, 51, 55, 63, 64
- 2.6 Implicit Differentiation: 5, 7, 9, 11, 13, 19, 21
- 2.7 Related Rates: 4, 5, 11, 13, 15, 20, 23, 25, 27, 28, 31
- 2.8 Linear Approx & Differentials: 1, 5, 6, 11, 13, 17, 19, 21, 23
- 3.1 Exponential Functions: 1, 5, 7, 9, 13, 15, 16, 17,
__23__,__24__,__25__, 27, 29, 30 - 3.2 Inverse Functions, Logarithms: 5, 7, 9, 11, 15, 17, 18, 22, 23, 25, 35, 36,
__39__, 67, 71, 76 - 3.3 Derivs of Log. & Exp. Functs.: 1, 3, 4, 6, 13, 20, 26, 31, 35, 41, 45, 55, 57
- 3.4 Exponential Growth & Decay: 1, 2, 3, 9, 13, 16
- 3.5 Inverse Trig Functions: 1, 2, 3, 5, 6, 9, 17, 19, 21
- 3.7 L'Hospital's Rule: 1, 2, 3, 4, 18, 21, 25, 26, 31, 35
- 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, 8,
__10__, 13, 15, 19, 23, 27, 35, 37, 45 - 4.4 Curve Sketching: 1, 9, 11, 13, 15, 19, 31, 33,
__39__ - 4.5 Optimization Problems: 2, 7,
__9__, 11, 12, 15, 17, 22, 25, 26, 28, 30, 37, 39 - 4.6 Newton's Method: 4, 7, 9, 11, 13
- 4.7 Antiderivatives: 1, 2, 7, 12, 13, 15, 20, 27, 38, 40, 43, 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,
__31__, 33, 35, 39, 40, 44 - 5.3 Evaluating Definite Integrals: 3, 7, 11, 18, 26, 29, 35, 51, 56, 59, 61, 65, 69
- 5.4 The Fund. Thm. of Calculus: 1, 3, 5, 6, 10, 11, 15, 19, 25, 27
- 5.5 The Substitution Rule: 7, 11, 13, 17, 19, 23, 26, 27, 33, 37, 39, 44, 50, 53, 55, 61

**A Suggestion for Studying: **Even though *WebAssign* does not require that you write stuff down, you will learn a lot by focusing on your writing. Furthermore, having good writing skills will really help when working on a written Quiz or Exam. Therefore, you should write down a complete solution to each problem *before* you type the answer into the answer box in *WebAssign*. Focus on the clarity and correctness of your written solution. Keep your written work organized in a notebook. Compare your written solutions to your Instructors' written solutions in Lectures and Recitations. Find another student, or a tutor, or the Recitation Instructor, or your Professor 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:**Best 8 of 9 Quizzes @ 30 points each = 240 points possible**Exams:**Best 2 of 3 Exams @ 220 points each = 440 points possible**Final Exam:**250 points possible**WebAssign:**30 Assignments @ 1 point each = 30 points possible (Extra Credit points)

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

Observe that the **Total Possible Points** is \(1030\), but your points total is divided by \(1000\) to get the percentage that is used in computing your course grade. This is because the \(30\) points that can be earned by doing **WebAssign Homework** are considered **Extra Credit Points**.

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

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

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

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

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

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

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

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

Use this to calculate your ** Current Letter Grade** throughout the semester: Grade Calculation Worksheet

**Calendar: **

Items in **red** are graded.

**Mon Jan 15: **Holiday: No Class

**Tue Jan 16 **Recitation **R01**: Course Intro and Section 1.3: The Limit of a Function

**Wed Jan 17: **
Section 1.3: The Limit of a Function
(Class Drill on Limits)

**Fri Jan 19: **
Section 1.4: Calculating Limits
(Handout of Information about Limits)

**Mon Jan 22: **
Section 1.5: Continuity

**Tue Jan 23: **Recitation **R02**

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

$$\lim_{h\rightarrow 0}\frac{(-5+h)^2-25}{h}$$(This problem is Exercise 1.4#23, similar to Book Section 1.4 Example 5.)

Find the limit

**[3]** (This problem is related to Exercise 1.4#35, but is simpler.)
Show that
$$\lim_{x\rightarrow 0}\left[x^2\cos{(20\pi x)}\right]=0$$
Hint: you do not need to use the ** Squeeze Theorem** for this one!

**[4]** (This problem is related to Exercise 1.4#35, but is harder and is similar to Book Section 1.4 Example 9)
Use the ** Squeeze Theorem** to show that
$$\lim_{x\rightarrow 0}\left[x^2\cos{\left(\frac{20\pi}{x}\right)}\right]=0$$

**[5]**
(This problem is Exercise 1.4#41, similar to Book Section 1.4 Example 10)
Find the limit

**[6]**
(This problem is Exercise 1.4#51, similar to Book Section 1.4 Example 10)
Find the limit

(This problem is Exercise 1.5#40, similar to 1.5#39)
Use the ** Intermediate Value Theorem** to show that there is a root of the equation
$$\sqrt[3]{x}=1-x$$
on the interval \((0,1)\)

**Wed Jan 24: **
Section 1.6: Limits Involving Infinity

**Fri Jan 26: **
Section 1.6: Limits Involving Infinity
(Last Day to Drop Without a W)
(Quiz **Q1**)

**Mon Jan 29: **
Section 2.1: Derivatives and Rates of Change
(Handout on Rates of Change)

**Tue Jan 30: **Recitation **R03**

Groups of 2 students will work together to solve a problem. When they are done, they will write a solution to the problme on the whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear! (If there is one student left without a partner, they may join a group of 2 students to make one group of 3 students.) Then Isaac will discuss all the solutions of that problem. Then you will all move on to the next problem.

**Similar Problem from Exercise List:**Exercise 1.4#42 is similar to one of the limits in part (a) and (b)**Similar Book Example:**Section 1.6 Example 1 is similar to one of the limits in part (b)**Similar Class Example:**None

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

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

**Similar Problem from Exercise List:**1.6 # 25**Similar Book Example:**Section 1.6 Example 6**Similar Class Example:**None

- Find the limit of the function using the methods of
**Section 1.6 Example 6**. (You'll notice that this problem is harder than that example.) Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty} \left( \sqrt{49x^2+x}-7x\right)$$ - What does the result of (a) tell you about the graph of the function?

**Similar Problem from Exercise List:**2.1#16**Similar Book Example:**none**Similar Class Example:**none

Suppose that a function \(g(x)\) is known to have these properties:

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

**Similar Problem from Exercise List:**2.1#18**Similar Book Example:**none**Similar Class Example:**none

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 \(f'(4)\)

**Similar Problem from Exercise List:**2.1#7**Similar Book Example:**none**Similar Class Example:**none

For the function

$$f(t)=\frac{2t+1}{t+3}$$- Find \(f'(2)\). That is, find
$$f'(2)=\lim_{h\rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use
that you may have learned in previous courses.*Derivative Rules* - What is the slope of the line tangent to the graph of \(f(t)\) at \(t=2\)? Explain.

**Similar Problem from Exercise List:**1.6 # 35**Similar Book Example:**none**Similar Class Example:**none

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

**Similar Problem from Exercise List:**1.6 # 40**Similar Book Example:**none**Similar Class Example:**none

Find a formula for a function that has vertical asymptotes at \(x=2\) and \(x=5\) and horizontal asymptote \(y=3\). Explain how you determined your function.

**Wed Jan 31: **
Section 2.2: The Derivative as a Function
(Class Activity: Find the Derivative of a Function Given by a Graph)

**Fri Feb 2: **
Section 2.2: The Derivative as a Function
(Quiz **Q2**)

**Mon Feb 5: **
Section 2.3: Basic Differentiation Formulas
(Class Drill on Rewriting Function Before Differentiating)

**Tue Feb 6 **Recitation **R04**:

**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.)**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$

**[1]** (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. (*Definition of the Derivative***Do not use the**!!) Show all steps clearly and explain key steps.*Derivative Rules* - Start over. Find \(g'(t)\) again, but this time use the
. (You should get the same result that you got in (a).*Derivative Rules*

**[2]** (2.3#2) Use the ** Derivative Rules** to find the derivative of the function
$$f(x) = \pi^2$$
Show all details clearly and use correct notation.

**[3]** (2.3#7)

- NO CALCULATORS!!
- Give exact answers in symbols, not decimal approximations.
- You may have to review the \((x,y)\) coordinates of the points where the rays of the famous angles intersect the
*Unit Circle*.

- Find \(f(\pi)\).
- Find \(f'(x)\) using the
*Derivative Rules* - Find \(f'(\pi)\)
- Find the
*height*of the graph of \(f(x)\) at \(x=\pi\). - Find the
*slope*of the graph of \(f(x)\) at \(x=\pi\).

**[4]** (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 \(f'(x)\) using the
*Derivative Rules*

**[5]** (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 \(v'(t)\) using the
*Derivative Rules*

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*.

**[6]** (2.3#27)

- NO CALCULATORS!!
- Give exact answers in symbols, not decimal approximations.
- You may have to review the \((x,y)\) coordinates of the points where the rays of the famous angles intersect the
*Unit Circle*.

- Find the equation for the line
*tangent to the graph of \(f(x)\) at \(x=\frac{2\pi}{3}\)*. - Find the equation for the line
*normal to the graph of \(f(x)\) at \(x=\frac{2\pi}{3}\)*. - Draw the graph and draw your tangent line and normal line. Label important stuff.

**[7]** 2.3#29) For the function
$$f(x)=-x^2+8x=-x(x-8)$$

- Find the equation for the line
*tangent to the graph of \(f(x)\) at \(x=3\)*. - Find the equation for the line
*normal to the graph of \(f(x)\) at \(x=3\)*. - Draw the graph and draw your tangent line and normal line. Label important stuff.

**[8]** (2.2#22) For the function
$$g(t)=\frac{1}{\sqrt{t}}$$

- (Hard problem.) 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. (*Definition of the Derivative***Do not use the**!!) Show all steps clearly and explain key steps.*Derivative Rules* - Start over. Find \(g'(t)\) again, but this time use the
. (You should get the same result that you got in (a).*Derivative Rules*

**Wed Feb 7: **
Section 2.3: Basic Differentiation Formulas

**Thu Feb 8: **
Review Session run by Isaac Agyei, At noon in Morton 219 (**Note that Isaac does not know what problems are on Exam X1. ** He will review questions that he feels are important, and he will review questions that students request.)

**Fri Feb 9: Exam ****X1** **Covering Through Section 2.3**

**No books, notes, calculators, phones, or smart watches**- The Exam will last the full duration of the class period.
- Ten problems, printed on front & back of three sheets of paper (but not quite as long as three quizzes).
- A problem about
,*limits*,*calculating limits*, or*infinite limits*, based on suggested exercises from Section 1.3, 1.4, 1.6*infinite limits* - Another problem about
,*limits*,*calculating limits*, or*infinite limits*, based on suggested exercises from Section 1.3, 1.4, 1.6*infinite limits* - Another problem about
,*limits*,*calculating limits*, or*infinite limits*, based on suggested exercises from Section 1.3, 1.4, 1.6*infinite limits* - Another problem about
,*limits*,*calculating limits*, or*infinite limits*, based on suggested exercises from Section 1.3, 1.4, 1.6*infinite limits* - A problem using the concept of
, based on suggested exercises from Section 1.5*continuity* - A problem about calculating a derivative using the
, based on suggested exercises from Section 2.2*Definition of the Derivative* - A problem involving calculating a derivative using the
, based on suggested exercises from Section 2.3*Derivative Rules* - Another problem involving calculating a derivative using the
, based on suggested exercises from Section 2.3*Derivative Rules* - A problem about
,*secant lines*or*tangent lines*, based on suggested exercises from Sections 2.1, 2.2, 2.3*normal lines* - A problem about
or*rates of change*, based on suggested exercises from Section 2.1, 2.2, 2.3*position & velocity* **Note that neither the Recitation Instructor, Isaac Agyei, nor the SI leader, Ben Oldiges, know what problems are on Exam X1. The coverage of problems in Review Sessions and SI Sessions is not an indication of what will or will not be on the Exam.**

**Mon Feb 12: **
Section 2.4: The Product and Quotient Rules

**Tue Feb 13: **Recitation **R05**

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

**Derivatives of Trig Functions**
$$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$
$$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$

**[1] ** (2.4#3) For the function \(f(t)=t^3\cos t \), find \(f'(\pi/4)\). Give an exact, simplified answer, not a decimal approximation.

**[2] ** For the function \(f(x)=\sec x \),

- Find \(f'(x)\). Give an exact, simplified answer.
- Find the equation for the line tangent to the graph of \(f(x)\) at \(x=\pi/3\). Give an exact, simplified answer.
- Find \(f''(x)\). Give an exact, simplified answer.
- Find \(f''(\pi/4)\). Give an exact, simplified answer.

**[3] ** For the function $$f(\theta)=\frac{\sec \theta}{1+\sec \theta} $$
find \(f'(\theta)\). Give an exact, simplified answer. **Hint: **Simplify the function before differentiating!

**[4] ** For the function \(f(x)= x^2\sin x \tan x\), find \(f'(x)\). Give an exact, simplified answer.

**[5] ** Suppose that \(f(5)=1\), \(f'(5)=6\), \(g(5)=-3\), and \(g'(5)=2\). Find the following values.

- \((fg)'(5)\)
- \((f/g)'(5)\)
- \((g/f)'(5)\)

**[6] ** An object attach moves left and right on a smooth level surface with *position function* \(s(t) = 8\sin t\), where \(t\) is the time in seconds and \(s(t)\) is the *position* at time \(t\), in centimeters. (A *positive* position means that the object is to the *right* of the zero position; *negative* position means that the object is to the *left* of the zero position.)

- Find the
*velocity*at time \(t\). - Find the
*acceleration*at time \(t\). - Find the
*position*,*velocity*, and*acceleration*of the object at time \(t=2\pi/3\) seconds. - In what direction (left or right) is the object moving at that time?
- Is the object speeding up or slowing down at that time?

**Wed Feb 14: **
Section 2.5: The Chain Rule

**Fri Feb 16: **
Section 2.6: Implicit Differentiation
(Handout on Implicit Differentiation)
(Quiz **Q3**)

**Mon Feb 19: **
Section 2.7: Related Rates
(Handout on Implicit Differentiation and related Rates)

**Tue Feb 20: **Recitation **R06**

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

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

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

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

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

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

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

**[1] ** (2.6#9) Suppose that \(x\) and \(y\) are related by the equation
$$4\cos x \sin y=1$$
Find \(dy/dx\) by *implicit differentiation*

**[2] ** (2.6#13) Suppose that \(x\) and \(y\) are related by the equation
$$4\sqrt{xy}=1+x^2y$$
Find \(dy/dx\) by *implicit differentiation*

**[3] ** (2.6#19) Suppose that \(x\) and \(y\) are related by the equation
$$x^2+xy+y^2=3$$

- Find \(dy/dx\) by
*implicit differentiation* - Find the equation of the line tangent to the curve at the point \((1,1)\)

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

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

**[2] ** (2.7#25) A trough is \(10\) ft long and its ends have the shape of isosceles triangles that are \(3\) ft across the top and have a height of \(1\) ft. The trough is being filled with water at a rate of \(12\) ft^{3}/min. How fast is the water level rising when the water is 6 inches deep?
Make a good drawing and use correct units in your answer.

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

**[1] ** (2.7#15) Two cars start moving from the same point. One travels south at \(60\) mi/h and the other travels west at \(25\) mi/h. At what rate is the distance between the cars increasing two hours later?
Make a good drawing and use correct units in your answer.

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

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

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

**[3] ** (2.7#28) A kite \(100\) ft above the ground moves horizontally at a speed of \(8\) ft/s. At what rate is the angle between the string and the horizontal decreasing when \(200\) ft of string have been let out? (angles in radians)
Make a good drawing and use correct units in your answer.

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

**Wed Feb 21: **
Section 2.8: Linear Approximations and Differentials
(Handout on Linearizations, Linear Approximations, and Differentials)

**Fri Feb 23: **
Section 3.1 Exponential Functions
(Quiz **Q4**)

**Mon Feb 26: **
Section 3.2: Inverse Functions and Logarithms (Remarks on Cancelling Before Multiplying)

**Tue Feb 27: **Recitation **R07**

**[1]** (Similar to Exercise 2.8#11) The goal is to use a *Linear Approximation* to estimate the number \(3.1^2\). Answer questions (a) - (f) below.

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

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

- What is the related
*function*, \(f(x)\)? - What is the
*inconvenient \(x\) value*, \(\hat{x}\)? - What is a
*convenient nearby \(x\) value*, \(a\)? - Build the
*Linearization of \(f\) at \(a\)*. That is, build the function $$L(x)=f(a)+f'(a)\cdot(x-a)$$ .
- Use your
*linearization*to find \(L(\hat{x})\). That is, find the value $$L(\hat{x})=f(a)+f'(a)\cdot(\hat{x}-a)$$ This is the estimate that was the goal of the problem. - In problem [1], you can find an exact value for \(f(\hat{x})\) by just multiplying \(3.1^2 = 3.1 \cdot 3.1\) by hand. In problems [2], [3], while it might not be possible to write an exact value for \(f(\hat{x})\), you can use a calculator to get a very precise (but not exact) decimal value for \(f(\hat{x})\). Do that, and see how it compares to your
*estimate*from part (e).

**[1]** (Similar to Exercise 2.8#19) Let \(y=\tan x\).

- Write the expression for the exact change \(\Delta y\) if \(x_1=\pi/4\) and \(\Delta x = 0.1\).
- Using a calculator, find a very precise approximate value of \(\Delta y\).
- Without a calculator, evaluate the approximate change \(dy\) if \(x_1=\pi/4\) and \(\Delta x = 0.1\).

**[2]** (Similar to Exercise 2.8#21) A cube has edges of length 10cm, using a ruler that has a possible error in measurement of \(\pm 0.05 cm\).

- Calculate the volume using an edge length of 10cm. Use differentials to estimate the possible error in the calculated volume.
- Calculate the surface area using an edge length of 10cm. Use differentials to estimate the possible error in the calculated surface area.

**[1]** (Similar to Exercise 3.1#27) Let \(y=\tan x\).
$$\text{(a) Find } \lim_{x\rightarrow 2^+}e^{3/(2-x)} \\
\text{(b) Find } \lim_{x\rightarrow 2^-}e^{3/(2-x)}$$

**[1]** (Similar to Exercise 3.2#23) Let \(f(x)=e^{2x-3}\).

- What are the domain and range of \(f(x)\)?
- Find a formula for \(f^{-1}(y)\)
- What are the domain and range of \(f^{-1}(y)\)?

**Wed Feb 28: **
Section 3.3: Derivatives of Logarithmic and Exponential Functions

**Fri Mar 1: **
Section 3.4: Exponential Growth & Decay
(Quiz **Q5** on Sections 3.1 and 3.2)

**Mon Mar 4: **
Section 3.5: Inverse Trig Functions

**Tue Mar 5: **Recitation **R08**

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

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

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

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

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

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

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

**Section 3.2 Theorem [7]**(on page 156 of the printed book): If \(f\) is a one-to-one differntiable function with inverse function \(f^{-1}\) and \(f'(f^{-1}(a))\neq 0\), then the inverse function \(f^{-1}\) is differentaible at \(a\), with the value of the derivative given by the equation
$$(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}$$

**The Natural Logarithm Rule**
$$\frac{d}{dx}\left( \ln (x) \right)=\frac{1}{x} \ \ \text{ valid on the domain }x \gt 0$$

**The General Logarithm Rule**
$$\frac{d}{dx}\left( \log_b (x) \right)=\frac{1}{x\ln b} \ \ \text{ valid on the domain }x \gt 0$$

**The Natural Exponential Rule**
$$\frac{d}{dx}\left( e^x \right)= e^x$$

**The General Exponential Rule**
$$\frac{d}{dx}\left( b^x \right)= b^x \ln b$$

**Steps in Logarithmic Differentiation**

- Starting with an equation \(y=f(x)\). This will be an equation involving \(x\) and \(y\)
- Take natural logarithms of both sides of the equation \(y=f(x)\) to obtain a new equation of the form \(\ln(y)=\ln(f(x))\), and use the Laws of Logarithms to simplify the right side. The result will be an equation involving \(x\) and \(\ln(y)\)
- Find \(\frac{d}{dx}\) of both sides of the new equation, using the technique of
. The result will be a new equation involving \(x\) and \(y\) and \(y'\).*implicit differentiation* - Solve this new equation for \(y'\). The result will be an equation of the form $$y'= \ \text{expression involving } \ x \ \text{and} \ y$$
- Replace the \(y\) in this new expresison with the original exppression for \(y\) from step (1). The result will be an equation of the form $$y'= \ \text{expression involving } \ x$$

**Derivatives of Inverse Trig Functions**
$$\frac{d}{dx}\sin^{-1}{(x)}=\frac{1}{\sqrt{1-x^2}}$$
$$\frac{d}{dx}\cos^{-1}{(x)}=\frac{1}{\sqrt{1-x^2}}$$
$$\frac{d}{dx}\tan^{-1}{(x)}=\frac{1}{1+x^2}$$

**Derivatives of Inverse Trig Functions, using Prime Notation**
$$\left(\sin^{-1}\right)'=\frac{1}{\sqrt{1-x^2}}$$
$$\left(\cos^{-1}\right)'=\frac{1}{\sqrt{1-x^2}}$$
$$\left(\tan^{-1}\right)'=\frac{1}{1+x^2}$$

**Derivatives of Inverse Trig Functions, Empty Versions**
$$\left(\sin^{-1}\right)'{( \ )}=\frac{1}{\sqrt{1-( \ )^2}}$$
$$\left(\cos^{-1}\right)'{( \ )}=\frac{1}{\sqrt{1-( \ )^2}}$$
$$\left(\tan^{-1}\right)'{( \ )}=\frac{1}{1+( \ )^2}$$

**[1]** (Similar to Exercise 3.2#17 and #35,36) Let \(g(x)=3+x+e^x\). Then \(g\) is known to be one-to-one, so it has an inverse function, \(g^{-1}\).

- Find \(g^{-1}(4)\)
- Find \((g^{-1})'(4)\)

**[2]** (similar to 3.2#76) Find the limits
$$
(a) \lim_{x\rightarrow \infty}\left[\ln (2+7x) - \ln(5+3x)\right] \\
(b) \lim_{x\rightarrow \infty}\left[\ln (2+3x) - \ln(5+3x)\right] \\
(c) \lim_{x\rightarrow \infty}\left[\ln (2+3x^2) - \ln(5+3x)\right] \\
(d) \lim_{x\rightarrow \infty}\left[\ln (2+3x) - \ln(5+3x^2)\right]
$$

(See the derivative rules for Logarithmic and Exponential Functions in the **List of Derivative Formulas** above.)

**[3]** (Similar to Exercise 3.3#41) Let \(y=e^{(-2x)}\cos{(3x)}\). Find \(y'\).

**[4]** (Similar to Exercise 3.3#13) Find the derivative of \(G(x)=\ln \frac{(2x+1)^5}{\sqrt{x^2+1}}\).
**Extremely Important Hint: Simplify the function before differentiating**! Do this as follows

- First, use the rule of logarithms $$\ln \frac{a}{b} = \ln a - \ln b$$ to rewrite the logarithm of a quotient as a difference of two logarithms.
- Then use that result, along with another rule of logarithms $$\ln q^p = p\ln q$$ to rewrite the expressions further.
- After you have simplified the function completely, you can then move on to finding the derivative.

**[5]** (Similar to Exercise 3.3#35) Find the derivative of \(y=2x\log_{10}{\sqrt{x}}\).
**Extremely Important Hint: Simplify the function before differentiating**! Do this as follows

- First rewrite\(\sqrt{x}\) as a
.*power function* - Then use that result, along with a rule of logarithms, to simplify the expression \(\log_{10}{\sqrt{x}}\).
- Then use that result to simplify the expression \(2x\log_{10}{\sqrt{x}}\)
- After you have simplified the function completely, you can then move on to finding the derivative.

(See the steps for Logarithmic Differentiation in the **List of Derivative Formulas** above.)

**[6]** (Similar to Exercise 3.3#56,57)

- Use logarithmic differentiation to find the derivative of \(y=x^{\cos x}\).
- Use logarithmic differentiation to find the derivative of \(y=\left(\cos x \right)^x\).

**[7]** (Similar to Exercise 3.4#2) The *E. coli* bacterium has the property that each cell divides into two cells every 20 minutes. (This means that the number of cells in a population of *E. coli* will double every 20 minutes.) Suppose that a particular population of *E. coli* has 60 cells initially.

- Find the
*relative growth rate*. - Find an expression for the number of cells after \(t\) hours.
- Find the number of cells after 8 hours.
- Find the rate of growth after 8 hours.
- At what time will the population reach 20,000 cells?

(See the derivative rules for Inverse Trig Functions in the **List of Derivative Formulas** above.)

**[8]** (Similar to Exercise 3.5#16,17) Find the derivative of each function

- \(y=\tan^{-1}(x^2)\)
- \(y=\left(\tan^{-1}(x)\right)^2\)

**[9]** Let \(f(t)=\cos^{-1}\left(\sin(t)\right)\)

- Find \(f'(t)\) by using the chain rule, using \(inner(t)=\sin(t)\) and \(outer( \ ) = \left(\cos^{-1}\right)'{( \ )}\).
- Start over. This time, simplify the expression for \(f(t)\) before differentiating. Then find the derivative.

**Wed Mar 6: **
Section 3.7: L'Hospital's Rule

**Fri Mar 8: Exam ****X2** **Covering Section 2.4 through Section 3.5**

**The Exam will last the full duration of the class period.****No books, notes, calculators, or phones**- Ten problems, 20 points each.
- Best score gets counted twice.
- Printed on front & back of three sheets of paper.
- All problems are based on suggested exercises.
- Five problems about finding derivatives using various methods that we have studied (in sections 2.4, 2.5, 2.6, 3.2, 3.3, 3.5)
- Five problems about using derivatives to find things.
- Related rates (Section 2.7)
- Linear Approximations and Differentials (Section 2.8)
- Exponential Growth in Biology or Exponential Decay of Radioactive Substance (Section 3.4)
- Velocity & Acceleration (Problems about this appear in Sections 2.4, 2.5.)
- Slope or Equation of the Tangent Line and/or Normal Line. (Problems about this appear in Sections 2.4, 2.5, 3.3.)

- Observe what is
**not**on this list: There won't be any problems on Exam X2 about**L'Hopital's Rule**. They will be on Exam X3.

**Mon Mar 11 – Fri Mar 15 is Spring Break: No class!**

**Mon Mar 18: **
Section 4.1: Maximum and Minimum Values
(Handout on the *Closed Interval Method*)

**Tue Mar 19: **Recitation **R09**

Students do this Class Drill about Identifying Extrema

Remember the definition of ** Critical Number** from the Monday March 18 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:

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

- \(f(x)=x^4-6x^2+5\)
- \(f(x)=x\sqrt{4-x^2}\)
- \((x)=xe^{(-x^2/8)}\)
- \(f(x)=\ln(x^2+x+1)\)

Remember the ** Closed Interval Method** from the Monday March 18 Lecture.

Used for finding the *absolute maximum value* and *absolute minimum value* for a continuous function on a closed interval.

**Step 1: **Confirm that the interval is closed and that the function is continuous.

**Step 2: **Find the critical numbers of the function

**Step 3: **Make a 2-column table.

- In the left column, put a list of important \(x\) values in increasing order:
- left endpoint
- critical numbers in the interval
- right endpoint.

- In the right column, put the corresponding \(y\) values.

**Step 4: **Identify the *greatest* and *least* \(y\) values in the list. These are the *absolute maximum value* and the *absolute minimum value*. Write a clear conclusion.

- Find the absolute max value and absolute min value of \(f(x)=x^4-6x^2+5\) on the interval [-2,3].
- Find the absolute max value and absolute min value of \(f(x)=x\sqrt{4-x^2}\) on the interval [-1,2].
- Find the absolute max value and absolute min value of \((x)=xe^{(-x^2/8)}\) on the interval [-1,4].
- Find the absolute max value and absolute min value of \(f(x)=\ln(x^2+x+1)\) on the interval [-1,1].

**Wed Mar 20: **
Section 4.2: The Mean Value Theorem Handout

**Fri Mar 22: **
Section 4.3: Derivatives and the Shapes of Graphs
(Quiz **Q6**)

**20 Minutes at the end of class****No books, notes, calculators, or phones**- Two Problems, 15 points each, printed on front & back of one sheet of paper
- One problem based on Suggested Exercises from
**Section 4.1**. - One problem based on Suggested Exercises from
**Section 4.2**.

- One problem based on Suggested Exercises from

**Mon Mar 25: **
Section 4.4: Curve Sketching
(Handout on Graphing Strategy)

**Tue Mar 26: **Recitation **R10**

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

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

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

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

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

- Verify that the function and the interval satisfy the three
*hypotheses*of the Mean Value Theorem. Explain clearly. - The next four questions are about finding the value of the constant \(c\).
- Find the formula for \(f'(x)\)
- Use your formula for \(f'(x)\) to build an expression for \(f'(c)\). The result should be an expression involving the letter \(c\), not the variable \(x\).
- Now compute the value of the number \(\frac{f(b)-f(a)}{b-a}\).
- Now, set the expression from Question
**(a)(ii)**equal to the number from**(a)(iii)**and solve the resulting equation for \(c\). - The next three questions are about illustrating your results of Questions
**(a)(iii)**and**(a)(iv)**. - Make a hand-drawn a graph of the function \(f(x)=\ln{(x)}\) on the interval \([1,4]\). You can use
**Desmos**to get the shape, if you want. - Label the
of the graph with their \((x,y)\) coordinates. Note that these coordinates are \((a,f(a))\) and \((b,f(b))\).*endpoints* - Draw the
that touches the graph at the endpoints \((a,f(a))\) and \((b,f(b))\). Label this line with the*secant line***slope \(m\)**that you found in Question**(a)(iii)**. - Find the point with \(x\) coordinate \(c\), where \(c\) is the number that you found in Question
**(a)(iv)**, label the point with the coordinates \((c,f(c))\), and draw the tangent line at the point. Does your tangent line appear to have the same slope as your secant line?

**Remark: **The upcoming Question **(b)(ii)** is not about the ratio \(\frac{f(b)-f(a)}{b-a}\). That ratio should not appear anywhere in your solution to Question **(b)(ii)** . Read the instructions for Question **(b)(ii)** carefully.

**Remark: **The number that you just found in part **(a)(iii)** is the number that should be the **slope of the secant line** and also the **slope of the tangent line**.

**Remark: **The resulting value of \(c\) that you just found in Question **(a)(iv)** is the \(x\) coordinate of the point of tangency for the tangent line. The number \(c\) is ** not** the slope of the tangent line. You already found the number that is the slope in part

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

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

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

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

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

- Find the intervals on which \(f\) is
*increasing*or*decreasing*. - Find the
*local maximum values*and*local minimum values*of \(f\). Note that these are \(y\) values, not \(x\) values. - Find the \((x,y)\) coordinates of all points on the graph of \(f\) where a
*local max*or*local min*occurs. Notice that this question differs slightly from the previous question. - Find the intervals on which \(f\) is
*concave up*or*concave down*. - Find the \((x,y)\) coordinates of all
*inflection points*of \(f\).

**[3] **Use the Handout on Graphing Strategy to sketch the curve \(y=xe^{-x}\)

**Wed Mar 27 **
Section 4.5: Optimization Problems

**Fri Mar 29: **
Section 4.5: Optimization Problems
(Last Day to Drop)
(Quiz **Q7**)

**20 Minutes at the end of class****No books, notes, calculators, or phones**- Two Problems, 15 points each, printed on front & back of one sheet of paper
- One problem based on Suggested Exercises from
**Section 4.3**. - One problem based on Suggested Exercises from
**Section 4.4**.

- One problem based on Suggested Exercises from

**Mon Apr 1: **
Section 4.6: Newton's Method
(Class Drill 1 on Newton's Method)

**Tue Apr 2: **Recitation **R11**

**[1] **(Suggested Exercise 4.5#17, similar to book Section 4.5 Example 3 and Wed Mar 27 Class Example.) 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.)

**[2] **(Suggested Exercise 4.5#22, similar to book Section 4.5 Example 5 and Fri Mar 29 Class Example.) 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.)

**[3] **(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.)

**[4] **Students work in pairs on this (Class Drill on Using Newton's Method)

**[5] **(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.)

**Hint for an outline of the solution: **

- Consider a line \(L\) that has axis intercepts at \((a,0)\) and \((0,b)\), where \(a,b\) are unknown
*positive*real numbers. Sketch the line \(L\), and compute the*area*\(A\) that the line cuts off in the first quadrant. Your expression for area \(A\) should involve the two unknowns \(a,b\). With just this expression for \(A\), it would be impossible to minimize \(A\). - Now, satisfy the additional requirement that the line \(L\) must pass through the point \((3,5)\).
- You now have two equations involving the constants \(a,b\). The first is an equation involving \(A,a,b\). The second is an equation just involving \(a,b\). Solve the second equation for \(b\) and use that result to eliminate \(b\) in the first equation. The result should be a single equation involving \(A,a\).
- Using
*calculus*, find the value of \(a\) that minimizes \(A\). (Remember that \(a\) must be*positive*.) - Find the corresponding value of \(b\).
- Find the slope \(m\) of line \(L\).
- Find the
*equation*of line \(L\).

**Wed Apr 3: **
Section 4.7: Antiderivatives

**Fri Apr 5: **
Section 4.7: Antiderivatives
(Quiz **Q8**)

**20 Minutes at the end of class****No books, notes, calculators, or phones**- Three Problems, 10 points each, printed on front & back of one sheet of paper
- One problem based on Suggested Exercises from
**Section 4.5**. - One problem based on Suggested Exercises from
**Section 4.6**. - One problem based on Suggested Exercises from
**Section 4.7**.

- One problem based on Suggested Exercises from

**Mon Apr 8: **
Section 5.1: Areas and Distances
(Lecture Notes)
(Class Drill on Riemann Sums)

**Tue Apr 9: **Recitation **R12**

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

You might not remember that *l’Hospital’s Rule* was covered in the last class meeting before Exam X2, so there were no problem about l’Hospital’s Rule on Exam X2. At the time, I said that there would be problems about l’Hospital’s Rule on Exam X3. Well, here we are.

**[8] (Suggested Exercise 3.7#1)** Consider the following limit, which is in \(\frac{0}{0}\) *indeterminate form*:
$$\lim_{x\rightarrow 1}\frac{x^2-1}{x^2-x}$$

- Find the limit using techniques from Chapter 1.
- Find the limit using
*l’Hospital’s Rule*.

**[9] (Suggested Exercise 3.7#3)** The following limit is in \(\frac{0}{0}\) *indeterminate form*. Find the limit using *l’Hospital’s Rule*
$$\lim_{x\rightarrow \left(\pi/2\right)^+}\frac{\cos x}{1-\sin x}$$

**[10] (Suggested Exercise 3.7#25)** The following limit is in \(\infty \cdot 0\) *indeterminate form*. Find the limit using *l’Hospital’s Rule*
$$\lim_{x\rightarrow \infty}x^3e^{-x^2}$$
**Hint: **Rewrite the product as a quotient, so that it will then be in \(\frac{0}{0}\) *indeterminate form*. Then use *l’Hospital’s Rule*.

**Wed Apr 10: **
Section 5.2: The Definite Integral
(Class Drill on Definite Integrals)

**Fri Apr 12: Exam **
**X3**
**Covering Sections 4.1 through 5.2**

**The Exam will last the full duration of the class period.****No books, notes, calculators, or phones**- The Exam for MATH 2301 Section 120 (Barsamian) will have 8 problems, printed on front & back of three sheets of paper.
- Problem about
**l'Hopital's Rule**(Section 3.7). - Problem about
**Derivatives and Shapes of Graphs and/or Curve Sketching**(Sections 4.3, 4.4). - Another problem about
**Derivatives and Shapes of Graphs and/or Curve Sketching**(Sections 4.3, 4.4). - Problem about
**Optimization**(Section 4.5). - Problem about
**Antiderivatives**(Section 4.5). - Problem about
**Antiderivatives involving**(Section 4.5).*motion in 1 dimension*. (*position*,*velocity*,*acceleration*) - Problem about
**Area and/or Distance**(Section 5.1) - Problem about
**Definite Integral**(Section 5.2)

- Problem about
- All problems are based on suggested exercises.
- Observe what is
**not**on the list for the Exam for MATH 2301 Section 120 (Barsamian)- Section 4.1
- Section 4.2
- Section 4.6

**Mon Apr 15: **
Section 5.3: Evaluating Definite Integrals

**Tue Apr 16: **Recitation **R13**

(the relationship between ** definite integrals** and

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

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

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

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

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

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

**[4]: **(5.3#59) An object moves along a line with velocity
$$v(t)=3t-5 \ \text{ for } \ 0\leq t \leq 3$$
where \(t\) is the time in seconds and \(v(t)\) is the velocity at time \(t\), in meters per second.

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

**[5]: **(5.3#60) An object moves along a line with velocity
$$v(t)=t^2-2t-3\ \text{ for } \ 0\leq t \leq 6$$
where \(t\) is the time in seconds and \(v(t)\) is the velocity at time \(t\), in meters per second.

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

**[6]: **(5.3#18) **Hint: **The **bad news** is that we have no antiderivative rules that work on a function as complicated as the function that is the integrand here. Your only hope is to ** rewrite the integrand** in a form for which our basic antiderivative rules

**[7]: **(5.3#9) **Hint: **You’ll have to rewrite the integrand as a sum of power functions before integrating.
$$\int_{1}^{4}\left(\frac{4+6u}{\sqrt{u}}\right)du$$

**[8]: **(5.3#23) **Hint: **You’ll have to rewrite the integrand as a sum of power functions and \(\frac{1}{x}\) functions before integrating.
$$\int_{1}^{e}\left(\frac{x^2+x+1}{x}\right)dx$$

**Wed Apr 17: **
Section 5.3: Evaluating Definite Integrals

**Fri Apr 19: **
Section 5.4: The Fundamental Theorem of Calculus
(Class Drill: Area Function)
(Quiz **Q9**)

**Mon Apr 22: **
Section 5.4: The Fundamental Theorem of Calculus

**Tue Apr 23: **Recitation **R14**

(the relationship between ** definite integrals** and

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

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

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

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

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

**[4]: **(5.4#6) The function \(g(x)\) is defined by the integral:
$$g(x)=\int_{3}^{x}e^{t^2-t} \ dt$$
Find \(g'(x)\).

**[5]: **(5.4#10) The function \(g(x)\) is defined by the integral:
$$g(x)=\int_{0}^{x}\sqrt{1+\sqrt{t}} \ dt$$
Find \(g'(x)\).

**[6]: **(5.4#10) The function \(h(x)\) is defined by the integral:
$$h(x)=\int_{0}^{\tan x}\sqrt{1+\sqrt{t}} \ dt$$
(Hint: You will need the ** Chain Rule**.)

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

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

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

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

**Wed Apr 24: **
Section 5.5: The Substitution Rule
(Handout on Substitution Method)

**Fri Apr 26: **
Section 5.5: The Substitution Rule

**Thu May 2: Combined Final Exam ****FX**
**from 4:40pm – 6:40pm**

- Section 100 (Mon, Wed, Fri 8:35am, Ntiamoah) will have its final in a Morton Hall room that will be posted here later.
- Section 110 (Mon, Wed, Fri 11:50am, Ntiamoah) will have its final in a Morton Hall room that will be posted here later.
- Section 120 (Mon, Wed, Fri 2:00pm, Barsamian) will have its final in a Morton Hall room that will be posted here later.

page maintained by Mark Barsamian, last updated Mon Apr 12, 2024