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.
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COURSE DESCRIPTION
This course has two main goals:
- Learning how to do explicit calculations.
- Discovering some fundamental mathematical properties, and in particular seeing some mathematical reasoning.
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.
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
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
- 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
- Complex Numbers
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 | |
October 10th | Set 2 | |
October 17th | Set 3 | |
October 24th | Set 4 | |
November 14th | Set 5 | |
November 21st | Set 6 | |
November 30th | Set 7 | |
Not to be handed in | Suggested problems on complex numbers |