Calculus of One and Several Variables

Course Description and Grading Breakdown
This course will introduce the mathematical method through One Variable Calculus, with which the Caltech freshmen have reasonable familiarity. In Ma 1a, the underlying concepts will be stressed, as well as the need for checking the hypotheses precisely as to where the results apply. A main focus of the course will be on the writing of complete mathematical proofs, and also on how to do explicit calculations. Here are the list of topics that will be covered:
  • Mathematical Induction
  • Completeness
  • Functions
  • Integral for Step Functions
  • Integral for More General Functions
  • Applications of Integration
  • Limits
  • Continuity
  • Intermediate Value Theorem
  • Inversion
  • Extreme - Value Theorem
  • Integrability Theorem
  • Mean - Value Theorem for Integrals
  • Derivatives
  • Implicit Differentiation
  • Extreme Values
  • Mean - Value Theorem for Derivatives
  • Curve Sketching
  • Fundamental Theorem of Calculus
  • Integration by Substitution
  • Integration by Parts
  • Logarithm
  • Exponential Function
  • Integration by Partial Fractions
  • Taylor Series
TextbookCalculus, Volume 1 by Tom M. Apostol, ISBN: 0-471-00005-1

There will be weekly homework assignments, as well as a Midterm and a Final exam, each of the Take Home variety lasting three hours. The final grade will depend on a composite of these factors. To be precise, the homework will be worth 40 percent, while the final and midterm will each be worth 30 percent. No one will be excused from the final exam.

Course Meeting Time and Location
Lectures - Monday, Tuesday, Wednesday and Friday
10:00 - 10:55 am
119 Kerckhoff Laboratory of Biological Sciences (KRK)

Rectitation - Thursday
7:30 - 8:55 pm
119 Kerckhoff Laboratory of Biological Sciences (KRK)

Course Instructor Contact Information and Office Hours
210-3 Math Building (Building 15)
Email: vafaee at
Office Hour: Wednesdays 4:30 - 5:30 pm

TA Contact Information and Office Hours
210-O Math Building (Building 15)
Email: xxiao at
Office Hour: Sundays 7 - 8 pm (in 119 Downs)

Course Ombudsperson
Isabella Camplisson has volunteered to act as the Math 1a Section 1 Ombudsperson. If you have any feedback about the class, please contact Isabella who can pass the information along at the Ombuds meeting.

Course Policies
Homework policies:
Homework problems will be assigned weekly and posted online (usually by the evening of Monday).  The problems assigned in a given week are due the following Monday at 3:30 PM, turned in as one set.  Please turn in the problems to the math department's box for Math 001A (Section 1). The box is in Downs, near classroom 113, directly across from the elevator. 

No late submissions are allowed except for medical problems (a note is needed from the Health Center) or serious personal difficulties (a note is needed from the Dean's Office).

For privacy reasons, please staple a cover sheet to your assignments, meaning a sheet blank except for your name.  Your score will be displayed behind the cover sheet when the homework is returned to you.

You may work together on the problems, but you must write up your solutions individually.  In your solutions, you may use results from Apostol, or the results stated as problems already assigned.  Be explicit about citations when using such results; e.g., state the theorem number or the exercise number.

RecitationsPlease go to the recitations.  Each week, a portion of the recitation will be dedicated to introducing some computational (or even theoretical) aspect which the instructor will not have time to present in class.  In general, get to know your TA and bug her with your questions, and try to fill in all the holes in your understanding.

 Date PostedAssignment  SolutionDue Date 
 9/25Homework Set 1
 10/2Homework Set 2
 10/9Homework Set 3
 10/17Homework Set 4
 10/24Midterm Exam  10/30
 10/31 Homework Set 5 11/6
 11/7 Homework Set 6 11/13
 11/14Homework Set 7
 11/27 Homework Set 8
 12/2 Practice Final  

Midterm and Final Exam
Click here to download the final exam.

Collaboration Table
You may consult:  
Course textbook (including answers in the back)YESYES
Other booksYESNO
Solution manualsNONO
Your notes (taken in class)YESYES
Class notes of othersYESNO
Your hand copies of class notes of othersYESYES
Photocopies of class notes of othersYESNO
Electronic copies of class notes of othersYESNO
Course handoutsYESYES
Your returned homework / examsYESYES
Solutions to homework / exams (posted on webpage)YESYES
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESNO
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersNONO
Post about problems onlineNONO
For computational aids, you may use:


* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.

Faramarz Vafaee,
Dec 4, 2017, 6:38 PM
Faramarz Vafaee,
Sep 25, 2017, 6:15 PM
Faramarz Vafaee,
Oct 2, 2017, 9:21 PM
Faramarz Vafaee,
Oct 14, 2017, 7:57 PM
Faramarz Vafaee,
Oct 20, 2017, 10:15 PM
Faramarz Vafaee,
Nov 1, 2017, 12:38 AM
Faramarz Vafaee,
Nov 8, 2017, 8:15 AM
Faramarz Vafaee,
Nov 17, 2017, 1:02 PM
Faramarz Vafaee,
Nov 27, 2017, 6:12 PM
Faramarz Vafaee,
Oct 28, 2017, 5:26 PM
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Dec 2, 2017, 3:05 PM