MATH 275: Calculus - Topics in Calculus I

Fall 2012

Meetings: TR 9:30AM-10:45PM, Ingraham 222 Van Vleck B119
Instructor: Benedek Valkó
valko at math dot wisc dot edu
409 Van Vleck
Office hours: Monday 3:30-4:30, Tuesday 1-2 or by appointment.

TA: Jo Nelson (nelson at math dot wisc dot edu)   
Discussion Sections: MW 8:50AM-9:40AM (301), MW 11:00AM-11:50AM (302) 

I will use the class email list to send out corrections, announcements, please check your email regularly.

Course description:
This course is the first semester of the Calculus Honors sequence developed by the Mathematics Department at the UW. The goal of the sequence is to provide highly motivated and well-prepared students with an opportunity to go beyond the traditional approach to the subject to develop a deeper understanding of this fundamental area of mathematics and to appreciate its power and beauty. The material covers essentially the same topics as the standard first semester calculus course, but the material is discussed in greater depth, and with much more emphasis on mathematical ideas. The course will be challenging, and the student might find it surprisingly different at the beginning. But it is also meant to be a lot of fun and to provide the students with the kind of clear and precise thinking that is characteristic of mathematics and that will be useful for them in almost any subject they pursue.

Textbook: T. Apostol, Calculus, Volume 1. 2nd edition.

Prerequisites: Personal invitation or invitation from the instructor.

Course Content: We plan to cover Chapters 1-6 with part of Chapter 9. Additional topics might be included. The main topics are

Evaluation: Course grade will be based on work in the discussion section (20%), two evening midterm exams (20%) and a final exam (40%). The final grade will be computed according to the following scale
A: [100,89), AB: [89,87), B: [87,76), BC: [76,74), C: [74,62), D: [62,50), F: [50,0]
There will be no curving in the class, but the instructor reserves the right to modify the final grade lines.

Discussion section grades consist of homework and quizzes. Attendance in one of the weekly discussion sections is required.
You will not be allowed to use calculators during the quizzes and exams.

1st midterm: Wednesday,  October 10, 7:15PM-8:45PM, Social Sciences 6102 review sheet
2nd midterm: Wednesday, November 14, 7:15PM-8:45PM, Social Sciences 6102 review sheet
Final exam: December 18, Tuesday, 12:25PM - 2:25PM, Ingraham 120 review sheet

There will be no make-up exams in the course. If you know that you cannot make it for one of the evening exams then you should contact me ahead of time.

Homework and quizzes: Homework will be assigned every week and it should be handed in at the Wednesday discussion section at the start of the lecture on Thursday (starting with the 2nd week). Your TA will decide how much of the assignment will be graded. Your TA may also decide to give quizzes during the discussion section.
The assignments will contain bonus problems. These will be more challenging than the regular homework and will have flexible deadlines. You will need to present the solution of the bonus problem in person (send me an email to set up an appointment). You may schedule a presentation up to the last week of the semester.

Instructions for the homework assignments:

How to succeed in this course: It helps immensely if you keep up with the material during the semester by reading the textbook and following the lectures. I will post section numbers of the textbook for upcoming lectures, it is beneficial to read ahead.
Although reading the textbook is important, the best way to master the material is through problem solving. This of course includes the homework assignments, but you shouldn't stop there, there are more than enough practice problems in the text book. (If you run out of problems to solve, I'm happy to supply more!)
For some students studying in groups helps a lot, let me know if you need help with setting up a study group.
If you feel lost, do not hesitate to ask for help. I'm available in my office hours (see above) or by appointment. Do not wait until the end of the semester if you think you are in trouble!


The homework assigned in week N is due in the Wednesday discussion section Thursday lecture in week N+1.
Dates Covered topics Homework Suggested reading for next week
Week 1
(9/4, 9/6)
Field and order axioms of real numbers, integers and rational numbers (I 3.2-3.7) 
complex numbers: basic operations, conjugate, modulus, argument, geometric representation, polar coordinates (9.1-9.6)
Parts 3 and 4 from the Introduction
Week 2
(9/11, 9/13)
Least upper bound, induction (I. 3.8-3.11)
HW2 Part 4 from the introduction, 1.1-1.2
Week 3
(9/18, 9/20)
Sums and products, inequalities (triangle inequality, Cauchy-Schwarz inequality) (I. 14.6-14.8)
functions, area (1.2-1.6)
HW3 1.8-1.14
Week 4
(9/25, 9/28)
Intervals, partition, step functions, integral for step functions, properties of the integral (1.8-1.14)
HW4 1.16-1.20
Week 5
(10/2, 10/4)
Integrals of more general functions, upper and lower integrals, monotonic functions are integrable, the integral of x^p (1.16-1.25)
Extra bonus problems 2.1-2.16
Week 6
(10/9, 10/11)
Basic properties of integrals, applications of integration (area between graphs, volume, average value) (2.1-2.19)
First midterm exam solutions
HW5 3.1-3.4
Week 7
(10/16, 10/18)
Limits, continuity, basic limit theorems (3.1-3.5)
HW6 3.5-3.9
Week 8
(10/23, 10/25)
Composite functions and continuity, Bolzano's theorem, intermediate value theorem, inverse functions
(3.7-3.13) write-up on infinite limits
HW7 3.15-3.19
Week 9
(10/30, 11/1)
Extreme values of continuous functions, uniform continuity, integrability of continuous functions, mean value theorem
HW8 4.1-4.7
Week 10
(11/6, 11/8)
Derivative of a function, algebra of derivatives, the derivative as a slope (4.1-4.7)
no hw this week
Week 11
(11/13, 11/15)
Chain rule, implicit differentiation, derivative of the inverse, relative min/max (4.8-4.13)
Second midterm exam solutions
HW9 4.14-4.16
Week 12
Rolle's theorem, Mean Value theorem for derivatives, finding relative extrema
no HW this week
4.17-4.20, 5.1
Week 13
(11/27, 11/29)
Convexity, Curve sketching, extremum problems
Fundamental theorem of calculus
4.17-4.20, 5.1-5.6
HW10 5.7-5.9, 6.1-6.7
Week 14.
(12/4, 12/6)
Integration by substitution, integration by parts, the logarithm function
5.7-5.9, 6.1-6.8
HW11 6.12-6.16
Week 15.
(12/11, 12/13)
The exponential function, hyperbolic functions, inverse trigonometric functions, integration by partial fractions
no HW
review for the final exam