# MATH 408D: Differential and Integral Calculus II

### General Information

```      Instructor: Dave Rusin (rusin@math.utexas.edu)
Office hrs: TTh 2-3, W 12-3,  and by appointment, in RLM 9.140 .
(I am usually in my office during ordinary business hours but if you
want to be sure I'm available, let me know in advance.)

Text: Calculus (7th Edition, "Early Transcendentals" version) by
James Stewart. You may also use the all-electronic version of
that same book, or the "special UT edition" at the bookstore.

Class meets Tuesdays and Thursdays in PHR2.110, 3:30-4:45pm.
Eat a good lunch or you will be too tired to concentrate until 5pm!

Teaching asst: Ryan Merryman  (rmerryman@math.utexas.edu)
Office hrs: T Th F 2:00-3:00pm in RLM 13.152
These sections meet Mondays and Wednesdays with Ryan at, respectively,
Sect 56100: noon-1:00pm in RLM 5.120
Sect 56105: 3:00-4:00pm in RLM 5.118
Sect 56110: 4:00-5:00pm in RLM 5.118
Your final exam is Wednesday, May 7, 2:00-5:00pm, in WRW 102

```

Course webpage: http://www.ma.utexas.edu/~rusin/408D/

New

The Sanger Learning Center is sponsoring classes to support students' preparation for final exams. The reviews will be led by mathematics graduate students. Students who attend every day will receive a cumulative review of the exam's material. The M408D Final Exam Review class will run Mon-Thu (4/28 - 5/1), 6:00-7:30pm

Here is a discussion of how to PROVE that a function of two variables has a limit.

### Description

This course is a continuation of M408C and covers a variety of topics in the theory of functions of one or more variables: indeterminate limits, improper integrals, infinite sequences, power and Taylor series, parametric curves, and derivatives and integrals of vector and multivariable functions with applications. Its objective is to provide students with practical mathematical skills necessary for advanced studies in all areas of science and engineering.

Please note that "mathematical skills" here refers to more than algebraic manipulation (although you will be expected to do that kind of thing quickly and accurately). It is an explicit goal of this course to develop your mathematical intuition: many of the problems you will be asked to solve will require much more thought than symbol-moving. I also take it as an important step in your mathematical training that you learn to communicate mathematics well: what you write must hang together logically, and be presented with enough words to make the presentation comprehensible.

### Pre-requisites

The prerequisite is a grade of at least C- in Mathematics 408C or 408L, or suitable performance on an entrance exam (AP, IB, or CLEP). Please note that if you had a C- in Math 408C or 408L, you have the weakest background in the class and so you should be working hardest and getting the most help and feedback.

Your semester grade will be based on a number of components. This structure is designed to encourage you to stay actively involved in the course all the way through the semester. Any adjustments to the schedules or policies will be announced multiple times in lecture and via email and on the course website shown above.

Homeworks: these are done online using the Quest system, located at https://quest.cns.utexas.edu/. This will enable you to get constant feedback on how well you are understanding the material. The homework must be completed online by the date posted, typically about one week after it becomes available. You will accumulate points during the semester, and your "Homework score" will be your average score as reported by Quest (on a scale of 1-100). Note that Quest will subtract points for wrong answers, so think carefully before you answer or you could end up with a negative score!

Quizzes: There will be a quiz (almost) every week. As with the homeworks, this will give you a semester "Quiz score" of up to 100 points.

Exams: There will be 3 mid-term exams, to be held during the usual class period, and a final exam. Each midterm is worth 100 points and the final is worth 200 points. I expect the dates to be February 6, March 6 (the day before Spring Break), and April 10. The last day of class is May 1. Please mark on your calendars now the time and date of the final exam. (I don't know yet what room the final exam will be held in.) Textbooks, notes, and electronic devices (including phones and calculators) are not permitted during exams. The exams will be a mix of multiple-choice and free-response questions; the ratio will change as the semester progresses.

Your semester grade is based only on the number of points accumulated from the above mix of 700 possible points. I will use this conversion table:
 Point total Semester grade 640-700 A 620-639 A- 600-619 B+ 560-599 B 540-559 B- 520-539 C+ 480-519 C 460-479 C- 450-459 D+ 410-449 D 400-409 D- 0-399 F
If for some reason there is a deviation from this scale it will be applied uniformly to your lecture section, and it will be announced in class. WARNING While the letter grade distributions in my classes tend to look like those of other instructors, students often report that my tests tend to be difficult and long, and the numerical scores are not high. Here is what the mean scores were in a recent semester: (Standard deviations shown in parentheses.)

1. Homework: 88 (9)
2. Quizzes: 87 (13)
3. Test 1: 68 (13)
4. Test 2: 69 (14)
5. Test 3: 74 (16)
6. Final Exam: 156 (41)
7. Semester Total: 542 (83)
You will notice that this mean score roughly separates the A's and B's from the C's, D's, and F's above; after all A and B are supposed to mean "above average", right? (Mean is not the same as median; most of the students last semester got A's and B's; only 11% of the letter grades were D's and F's.)

No letter grades will be assigned to the midterms, quizzes, or homeworks, but you should keep track of where you stand: I will advise you of the class averages and you can use this data from other semesters as a rough guideline.

### Policies

Make-ups: it is in general not possible to make up missing quizzes or homework assignments after the due date. If you believe you will have to miss a graded event, please notify me in advance; I will try to arrange for you to complete the work early.

Students with disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.

Religious holidays: If you are unable to participate in a required class activity (such as an exam) because it conflicts with your religious traditions, please notify me IN ADVANCE and I will make accommodations for you. Typically I will ask you to complete the required work before the religious observance begins.

Academic Integrity. Please read the message about Academic Integrity from the Dean of Students Office. I very much prefer to treat you as professionals whose honesty is beyond question; but if my trust is violated I will follow the procedures available to me to see that dishonesty is exposed and punished.

Campus safety: Please familiarize yourself with the Emergency Preparedness instructions provided by the university's Campus Safety and Security office. In the event of severe weather or a security threat, we will immediately suspend class and follow the instructions given. You may wish to sign up with the campus alert programs.

Add dates: Students who enroll after the first week will not be able to make up the first homework and first quiz. If you followed proper channels and enroll after one or both of those assignments has passed, I will multiply your semester quiz/HW totals by the appropriate fraction to make up for this, but you must see me personally to arrange this when you enroll. There is no provision to adjust scores beyond the first week's assignments.

Drop dates: Jan 29 is the last day to drop the course for a possible refund; March 31 is the last day an undergraduate student may, with the dean's approval, withdraw from the University or drop a class except for urgent and substantiated, nonacademic reasons. For more information about deadlines for adding and dropping the course under different circumstances, please consult the Registrar's web page, http://registrar.utexas.edu/calendars/13-14/

Quest: This course makes use of the web-based Quest content delivery and homework server system maintained by the College of Natural Sciences. This homework service will require a \$25 charge per student per class for its use, with no student being charged more than \$50 a semester. This goes toward the maintenance and operation of the resource. Please go to http://quest.cns.utexas.edu to log in to the Quest system for this class. After the 12th day of class, when you log into Quest you will be asked to pay via credit card on a secure payment site. Quest provides mandatory instructional material for this course, just as is your textbook, etc. For payment questions, email quest.billing@cns.utexas.edu.

Computers: We don't make use of sophisticated software in this class, but if you find this interesting, you are welcome to use the department's computer facilities. Our 40-seat undergrad computer lab in RLM 7.122, is open to all students enrolled in Math courses. Students can sign up for an individual account themselves in the computer lab using their UT EID. We have most of the mainstream commercial math software: Mathematica, Maple, Matlab, etc., and an asortment of open source programs. If you come to my office you will see me use some of this software to help illustrate concepts. Please see me if you would like more information.

Bennett exam: The Bennett contest exam is a competition held at The University of Texas Mathematics Department at the end of every regular semester. (This semester that will be Saturday, May 3.) Participation is limited to students who are finishing the Calculus sequence that semester. That includes you! The questions are based on the topics covered in the Calculus courses, but require more than the usual amount of persistence and cleverness. There are cash prizes for the top scorers. Please plan to participate!

### Assistance with course work

In addition to visiting the instructor or the teaching assistant during office hours, you might want to make use the services of the Sanger Learning and Career Center. The Sanger Center offers several forms of tutoring to students throughout the semester (for students in select math courses, including M408D):
• Drop-In Tutoring -- A free, walk-in study environment supported by Sanger's mathematics tutors
• Appointment Tutoring -- Individualized one-hour meetings with one of the mathematics tutors
• Final exam review
A further description of the Sanger Learning and Career Center services can be found here: http://lifelearning.utexas.edu/ Students can now enroll in refresher classes and register for tutoring support by going directly to the Student Login portion of the Sanger website. Another service, offered free to students with majors in the College of Natural Sciences, is RHSG, the Residential Halls Study Groups. These are tutoring groups that meet weekday evenings at Jester and Kinsolving dorms. I encourage you to participate if you are looking for additional resources to succeed in this course.

### Schedule

This semester we will cover most of chapters 10-15 of the text, following this pattern (subject to minor variation):

• 4.4 Indeterminate Forms and L'Hospital's Rule (Review)
• 7.8 Improper Integrals (one day)
• 11 Infinite Sequences and Series (eight days)
• 11.1 Sequences
• 11.2 Series
• 11.3 The Integral Test and Estimates of Sums
• 11.4 The Comparison Tests
• 11.5 Alternating Series
• 11.6 Absolute Convergence and the Ratio and Root Tests
• 11.7 Strategy for Testing Series
• 11.8 Power Series
• 11.9 Representations of Functions as Power Series
• 11.10 Taylor and Maclaurin Series
• 11.11 Applications of Taylor Polynomials
• 10 Parametric Equations and Polar Coordinates (three days)
• 10.1 Curves Defined by Parametric Equations
• 10.2 Calculus with Parametric Curves
• 10.3 Polar Coordinates
• 10.4 Areas and Lengths in Polar Coordinates
• 12 Vectors and the Geometry of Space (four days)
• 12.1 Three-Dimensional Coordinate Systems
• 12.2 Vectors
• 12.3 The Dot Product
• 12.4 The Cross Product
• 12.5 Equations of Lines and Planes
• 12.6 Cylinders and Quadric Surfaces
• 13 Vector Functions (one day)
• 13.1 Vector Functions and Space Curves
• 13.2 Derivatives and Integrals of Vector Functions
• 14 Partial Derivatives (five days)
• 14.1 Functions of Several Variables
• 14.2 Limits and Continuity
• 14.3 Partial Derivatives
• 14.4 Tangent Planes and Linear Approximations
• 14.5 The Chain Rule
• 14.6 Directional Derivatives and the Gradient Vector
• 14.7 Maximum and Minimum Values
• 14.8 Lagrange Multipliers
• 15 Multiple Integrals (five days)
• 15.1 Double Integrals over Rectangles
• 15.2 Iterated Integrals
• 15.3 Double Integrals over General Regions
• 15.4 Double Integrals in Polar Coordinates
• 15.5 Applications of Double Integrals (optional)
• 15.10 Change of Variables in Multiple Integrals (if time permits)