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Syllabus

HOMEWORK ASSIGNMENTS
Week 1 (1/3-1/7): Review Chapters 1, 2 and 3. Prove Proposition 4.8 (ii). Ex 4.20 (2).
Week 2 (1/10-1/14): Give an epsilon-N definition for the statement that a given sequence {xn} does NOT converge. Ex 4.29 (1,4,5). Prove Proposition 5.17 and 5.18.
Week 3 (1/17-1/21): Prove Theorem 6.6, Proposition 6.11, 6.17 and 6.18 (but replace "limsup" in Proposition 6.18 by "lim"), 6.26(3,5(iv))
Week 4 (1/24-1/28): 7.16 (3,5), Let f(x)=x^3 for x<0, x for x>0 and 0 for x=0. Is f continuous at x=0? (prove)
Week 5 (1/31-2/4): 8.20 (1,2,3,4)
Week 6 (2/7-2/11): 9.17 (1(vi),3,5,6)
Week 7 (2/14-2/18): 10.11 (3, 5)
Week 8 (2/21-2/25): 10.15 (4,5), 11.8 (2,4)
Week 9, Spring Break (2/28-3/4):
Week 10 (3/7-3/11): 11.11 (2,3), 12.12 (1,5)
Week 11 (3/14-3/18): 12.12 (2), 12.21 (complete 2, 3, 4, 5), 13 (complete Prop 13.7)
Week 12 (3/21-3/25): Is f:[0,1]->R defined by f(x)=1 if x is rational and 0 otherwise, Riemann integrable? (explain). Following example 13.17, show that f: [0,1]->R defined by f(x)=x is Riemann integrable. 13.26 (2,4)
Week 13 (3/28-4/1)due Fr 4/8: 13.26 (5,use 4), 13.34 (1iii,1vi,2)
Week 14 (4/4-4/8): 14.3 (1,4,6), 14.5 (1), 14.7 (1,5(i),6(iv))
Week 15 (4/11-4/15): NOT DUE 15.6 (1(vi),2,5), 15.10 (2,5)

Announcements:

Graduate students enrolled in this class, should turn in 1 solved HW problem every other week, on wednesdays following the past 2 weeks. (So the first HW is due on wed 1/20.)
Undergraduates should not turn in HW.
Exam 1 is on Monday 2-21, chapters 4-9.
We will have a review session on the material of Exam 1 on Friday 2-18 (I completely forgot about the review I promised on Monday 2-14. Sorry!!! Later today I will post a proof of the Ratio Test here.)
Here is the promised proof of the ratio test.
Exam 1 is closed book, formula sheets/calculators are not allowed.
There will be 4 or 5 problems similar to the homework problems (solving those and those that were not assigned will be the best way to prepare).
You should know everything discussed in class, but here is the minimal list of what you should master:
Chap 4: Convergence/divergence of sequences (epsilon-N definition, know how to apply). A sequence which is monotone and bounded, has a limit.
Chap 5: Subsequence (if a sequence has limit l, then so has every subsequence), sandwich theorem, Bolzano-Weierstrass Theorem, Cauchy sequence (definition), Cauchy sequences converge and vice versa.
Chap 6: Series. Important examples to be memorized: geometric series and harmonic series. The terms of a convergent series, converge to zero (necessary condition). Alternating series, comparison test, root and ratio test (understand, and know how to apply). Absolute and conditional convergence.
Chap 7: Functions, inverse functions, composite functions, bounded functions.
Chap 8: Limits (left, right, epsilon-delta definition; know how to verify for given functions, divergence to + or - infinity), continuity at a point (including characterization in terms of sequences), sandwich theorem.
Chap 9: Continuity on an interval, intermediate value theorem, image of compact interval under a continuous function is compact interval (and thus minima and maxima are achieved), Brouwer's theorem.
Here is the proof of the contraction mapping theorem.
Office hours on 4/28 and 4/30 are canceled.
There will be a review on the material of Exam 2 (chap 10, 11, 12 and 13) on Friday April 1. The purpose of this session is for you to ask me any questions you have on this material (both theory as excercises). Please prepare accordingly. We will continue the review on Monday April 4.
The mandatory course/teaching evaluation is scheduled on Monday April 4.
Exam 2 is on Friday 4-8, chapters 10-13. The format for this exam will be the same as for the first exam: closed book, no calculators and 4 or 5 problems.
Here is the proof of convergence of the Newton-Rhapson algorithm.
List of things to know for Exam 2.
Chap 10: Derivatives, NOT 10.4, properties of derivatives (differentiability -> continuity at a point),
sum, product, quotient, composition rules.
Chap 11: local minimum, maximum (necessary condition in Thm 11.2), Rolle's Thm, Mean Value Thm, Thm 11.7; Taylor's Thm.
Chap 12: Monotone functions, their limits (Thm 12.4 and Cor 12.5), differentiable monotone functions,
inverse functions, convex functions (definition and different characterizations), Thm 12.14, properties
of convex, differentiable functions (Thm 12.18 and 12.19).
Chap 13: Definition of the integral of continuous functions on compact intervals, elementary properties,
Fundamental Theorems of Calculus, Riemann integral (definition, know how to verify for simple examples),
applications of the fundamental theorems of calculus in Sec 13.19, improper integrals (due to infinite
domain of integration and due to singularities in points), integral test (Thm 13.32).
The final exam is in our classroom on in our classroom on Monday April 25 between 10:00 and 12:00. This is a comprehensive, closed book exam on everything covered in class on Chap 4-15. Calculators are not allowed.

Regarding Chap 14 and 15, here is what you should know for the final exam (for Chap 4-13, see above):
Chap 14: definition of logaritm, exponential and powers, and their properties (including those in the excercises: 14.3 (2), 14.5 (1,2), 14.7 (1,2)).
Chap 15: Power series, interval and radius of convergence (formulas for computing the latter), Taylor series (real analytic functions), continuity and differentiation of power series.