Ma 1a-sec1:  Calculus of One and Several Variables and Linear Algebra

ANNOUNCEMENTS

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Graded homework and quizzes will be returned to you in recitation section.  If for some reason you do not collect your classwork in recitation section, you have the option to pick up your graded work in the Section boxes located on the 2nd floor of Sloan.

However, due to the Family Educational Rights and Privacy Act (FERPA) regulations, we cannot place graded work in a public location, without your permission.

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Download and return this form with your first homework assignment.

 

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COURSE DESCRIPTION

This course has two main goals:

Both of these goals are mostly focused on the basic analysis of integrals and derivations, and some famous functions such as trigonometric functions and the exponential function.


PREREQUISITES

TBA


SCHEDULE

Classes: Monday, Tuesday, Wednesday, Friday 10:00 - 10:55 am, 151 Sloan
Recitations: Thursday 7:30 to 8:55 pm, 151 Sloan


INSTRUCTORS

Farzad Fathizadeh
OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS
Sloan 358
626-395-4355
farzadf at caltech.edu


TA's

Daniel Siebel
Sloan 356
dsiebel@caltech.edu


OFFICE HOURS

Daniel Siebel (TA): Sunday 8 to 9 pm, Sloan 356
Farzad Fathizadeh (Instructor): Thursday 10 am to 11 am, Sloan 358


POLICIES

Grades

40% homework, 30% midterm, 30% final

Homework Policy

Homework problems will be assigned weekly and posted online.  The problems assigned in a given week are due the following Monday at 4:00 PM, turned in as one set.  Please turn in the problems to the math department's box for Math 001A (Section 1).

No late submissions are allowed except for medical problems (a note is needed from the Health Center) or serious personal diculties (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.


TOPICS COVERED


TEXTBOOKS

Apostol, Tom M., Calculus, Volume 1, 1991, Wiley, ISBN:  0-471-00005-1.


LECTURE NOTES

Date Description
Mon Sept 26 Induction and examples
Tues Sept 27 More examples for induction
Wed Sept 28 Functions, integration, and examples
Fri Sept 30 An example of integration, continuity of functions and related examples
Mon Oct 3 More examples on continuity, limit of a function and examples, basis limit theorems
Tues Oct 4 Basic limit theorems, the squeezing principle and an example
Wed Oct 5 Continuity of the compositon of two continuous functios, statement of the intermediate-value theorem for continuous functions and main ideas of its proof
Fri Oct 7 Least upper bound and greatest lower bound, completeness, proof of the intermediate value theorem for continuous functions
Mon Oct 10 Continuatin of the proof of the intermediate value theorem, statemetn of the exreme value theorem for continuous functions
Tues Oct 11 Boundedness of continuous functions on closed intervals and its connection with the exreme value theorem
Wed Oct 12 Proof of boundedness of continuous functions on closed intervals
Fri Oct 14 Proof of the extreme value theorem for continuous functions on closed intervals, a second proof for the boundedness theorem for continuous functions using the method of successive bisection
Mon Oct 17 Proof of uniform continuity of continuous functions on closed intervals
Tue Oct 18 The small span theorem for continuous functions, the notion of upper and lower integrals of bounded functions on closed intervals
Wed Oct 19 Approximation of upper and lower integrals using partitions and the behavior of the approximations with respect to refinements of partitions
Fri Oct 21 Proof of integrability of continuous functions on closed intervals, mean-value theorem for integrals of continuous functions
Mon Oct 24 Derivative of a function and examples of direct calculations of the derivative
Tues Oct 25 More examples of derivatives and discussion of useful identities
Wed Oct 26 Direct calculation of derivative of trigonometric functions using trigonometric identities
Fri Oct 28 Proof of continuity of differentiable functions, and proof of the formulas for the derivative of the sum and the product of two functions
Mon Oct 31 Proof of the formula for the derivative of quotient of two functions, examples, and idea of proof of the chain rule
Tues Nov 1 Proof of the chain rule, and examples of derivatives using the chain rule
Wed Nov 2 More examples of derivatives using the chain rule, relative maximum and relative minimum of functions and general ideas about finding such extremums using the derivative of a function
Fri Nov 4 Proof of the vanishing of the derivative of a functions at interior exremums (provided that the derivative exists), proof of the Rolle's theorem and the mean-value theorem for derivatives
Mon Nov 7 Applications of the mean-value theorem for derivatives, the first derivative test for finding relative extrema
Tues Nov 8 The second derivative test for finding relative exrema, concavity of functions and inflection points of their graphs
Wed Nov 9 Convex functions and their applications, Graph sketching
Fri Nov 11 More on graph sketching, the idea of approximating a function with polynomials in the neighborhood of a point (Taylor series), justification for the l'hospital rule
Mon Nov 14 More on Taylor expansion and examples, its relation with the mean-value theorem for derivatives
Tues Nov 15 Relation between differential and integral calculus: the first and the second fundamental theorems of calculs
Wed Nov 16 Examples of definite integrals calculated using the fundamental theorem of calculus
Fri Nov 18 Calculation of anti-derivatives using the method of substitution based on the chain rule, and using the method of integration by parts based on the product rule for differentiation
Mon Nov 21 Exponential and logarithmic functions and their properties
Tues Nov 22 Derivatives and anti-derivatives of exponential and logarithmic functions
Wed Nov 23 The derivative of arctan(x) and its applications in calculating the anti-derivative of rational functions, a brief introduction to complex numbers
Fri Nov 25 Holiday
Mon Nov 28 Several examples of calculating antiderivatives
Tues Nov 29 Complex numbers
Wed Nov 30 Complex numbers
Fri Dec 2 More examples of calculating antiderivatives

HOMEWORK

Due Date Homework Solutions
October 3rd Set 1 Solutions
October 10th Set 2 Solutions
October 17th Set 3 Solutions
October 24th Set 4 Solutions
November 14th Set 5 Solutions
November 21st Set 6 Solutions
November 30th Set 7 Solutions
Not to be handed in Suggested problems on complex numbers

EXAMS

 


READING