Analysis I MAT 320 MAT 640 Spring 2016
MAT320 Analysis I: 4 hours, 4 credits. Introduction to real analysis, the real number system, limits, continuity, differentiation, the mean value theorem, Taylor's theorems and applications. Riemann integration and improper integrals.
Prerequisite: Either Vector Calculus MAT226 or Departmental permission
Course Webpage: https://sites.google.com/site/professorsormani/teaching/AnalysisI-S16
Professor Sormani: google "Sormani Math" or go to http://comet.lehman.cuny.edu/sormani
Contact: sormanic@gmail.com (do not call the office and leave messages) Office Hours: 1:30-2:00, 5-6:00 M/W in Gillet 200A
Expectations: Students are expected to learn both the mathematics covered in class and the mathematics in the textbook and other assigned reading. Completing homework is part of the learning experience. Students should review topics from prior courses as needed using old notes and books.
Homework: Approximately four hours of homework will be assigned in each lesson as well as additional review assignments over weekends. Note that a single problem may take an hour. In the schedule below, homework is written below the lesson when it is assigned and *ed problems are more important.
Exams: There are three exams(20% each) and a final (40%).
Textbook: Mathematical Analysis: a Straightforward Approach by Binmore, 2nd Ed Cambridge University Press . Available free at https://archive.org/details/MathematicalAnalysis ISBN: 9780521288828
Materials on Reserve in the Library: There are books about proving techniques available on reserve in the library as well as a complete set of handwritten lecture notes taken by a Lehman College math major.
Tutoring: There is no tutoring for this course, but you may stop by professor's office hours regularly.
Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.
At the end of the course students should be able to:
1. find limits, sups and infs by applying theorems (as part of department objectives in math A, B & E)
2. prove that a sequence converges and a function is continuous at a point (as part of E, F & G)
3. write a proof by contradiction (as part of F & G)
4. state, apply and prove theorems related to Calculus including Riemann sums (as part of E)
5. write a proof by induction involving series (as part of F & G)
6. find Taylor series, prove convergence theorems and find radii of convergence (as part of B, E & F)
These objectives will be assessed on the final exam along with other important techniques.
Consult the course webpage for updated homework assignments.
Lesson 1 (2/1): Quantifiers HW: Complete the Quantifiers Worksheet, Read 7.1-7.5, Read 7.6-7.12
Lesson 2 (2/3): Proofs and Counter examples HW: Read 1.1-1.4, rewrite examples 1.5 and 1.6 as two column proofs, do 1.8/exercise *1,*2,*4, Rules of Proof website
Lesson 3 (2/8): Proof by Contradiction HW: Read 1.7 and rewrite as a two column proof, do 1.8/ *5, *6, Read 1.9-1.20, 1.12/1*,2,3*,5,6; 1.20/2*, 3*, 6*; 7.16/2*,3*,4*
Lesson 4 (2/10): Continuoum HW: Read 2.1-2.9; Do 2.10/1,2*,3,6*; Read 7.9-7.16; Do 7.16/3*,4*
Lesson 5 (2/17): Sup and Inf, Archimedian Property HW: Read 2.12-2.13; Do 2.13/1*; 7.16/6*; Read 3.1-3.6; Do 3.6/1*,2*
Lesson 6 (2/22): Convergence HW: Read 4.1-4.5, Do 4.6/1*,2*,3* (be sure to use epsilon in these)
Lesson 7 (2/24): Convergence and the Sandwich Lemma HW: Do 3.11/2*, 3*, Read 4.7-4.9 and go over class notes carefully, Prove that if an converges to A and bn converges to B then (2an + 3 bn) converges to (2A +3B). Read 4.10-4.12, Do 4.20/3*
Lesson 8 (2/29): Practice for Exam I
Lesson 9 (3/2): Exam I on Sequences: will have an upper bound proof, a sup/inf proof, an epsilon-N convergence proof for a specific sequence, an epsilon-N convergence proof about sums/differences/products of sequences
Lesson 10 (3/7): Proof by Induction HW: Read 3.7-3.9, Read handout, Do the four starred problems on the handout. Do 3.11/1i*,1ii*, 4*, (the handout is on my door) Extra Credit due 2/29: Do 1.20/4, 3.6/6,
Lesson 11 (3/9): Monotone Sequences HW: Read 4.14-4.16, Rewrite the proof in 4.17 and examples 4.18 as two column proofs, Read 4.19, Do 4.20/1*,2, 6* Prove that a sequence which is increasing and bounded above converges to its sup. Read the rest of chapter 4, do 4.29/2,4.
Lesson 12 (3/14): Subsequences, liminf, limsup, the Bolzano Weierstrass Theorem and Cauchy Sequences HW: Read 5.1-5.7, Read 5.16-5.19 on Cauchy sequences, Do 5.21/1*, 4*
Lesson 13 (3/16): Limits of Functions Read first section of handout from the original Larson Calculus textbook (available on my office door), Do 39-46 from handout. HW: Read 8.1-8.5, Do 8.15/2*,3*, Prove Prop 8.12 (i) using Defn in 8.3,
Lesson 14 (3/21): Continuity HW: read the rest of the handout from Larson, do 49,50, 72,73 and prove the squeeze theorem and the three special limits, Read 8.6-8.7, Do 8.15/ 6*, Read 8.8, Prove Prop 8.12 (ii) and (iii) using Theorem 8.8*, Read 8.13-8.14, Do 8.15/ 5*, Read 8.6, 9.1-9.3 Prove 9.4(i)(ii)(iii) see hint below the three statements, Read 8.16, Prove 9.5* and 9.6*, a sample exam has been distributed in class and is on the door of my office, email questions!
Lesson 15 (3/28): Pointwise and Uniform Limits of Functions, Continuity and the Arzela-Ascoli Theorem
(Apply for a paid Research Experience for Undergraduates if you wish)
Lesson 16 (3/30): Exam II will have a proof by Induction, an epsilon-N proof that a given sequence is Cauchy, an epsilon-delta continuity proof for a specific function, an epsilon delta proof continuity proof for a combination of functions and will also have short questions about given sequences using the definitions of bounded, increasing, decreasing, liminf, limsup, Thm 4.17, Thm 4.10, Thm 4.25 and Thm 5.2. Read 5.1-5.3, Read 5.8-5.10 and then read 5.11-5.14, do 5.15/4,5,6. For students told to redo parts of this exam, the redone parts will be averaged with the original grades on those parts and the total will be recomputed. The redone parts must be redone during office hours.
HW before 4/4: Read 9.10, 9.13-9.14, Prove 9.10 imitating the proof in 9.9, Read 9.12 Prove 9.12 for infimum*, Do 9.17/1, 2*,3,4,5,6.
Also HW in email: prove the following functions are continuous, check if they are uniformly continuous and equicontinuous:
1) f_n(x)=x/n on [0,1] 2) f_n(x)= nx on [0,1] 3) f_n(x) =xn/(n+1) on [0,1]
Lesson 17 (4/4): Uniform Continuity, Equicontinuity and the Arzela-Ascoli Theorem
HW: prove the following functions are continuous, check if they are uniformly continuous and equicontinuous:
1) f_n(x)=x/n on [0,1] 2) f_n(x)= nx on [0,1] 3) f_n(x) =xn/(n+1) on [0,1] 4) f_n(x) = x^n on [0,1]
Then show the first sequence converges uniformly to f(x)=0, the second sequence is unbounded and has no limit, the third sequence converges to f(x)=x uniformly and the last sequence converges pointwise to f(x)=0 for x in [0,1) and f(x)=1 for x=1 but not uniform convergence,
Finally prove the uniform limit of an equicontinuous sequence is continuous.
For students told to redo parts of Exam II, the redone parts will be averaged with the original grades on those parts and the total will be recomputed. The redone parts must be redone during office hours sometime in the next two weeks.
Lesson 18 (4/6): Differentiation HW: Read 10.1-10.3, 10.4-10.10, Do 10.11/2*, Read 10.12-10.14 Do 10.15/2*, 5*
read on your own: Mean Value Theorem and Extrema HW: Read 11.1-11.7, Do 11.8/2*,3*,4*, Read 11.9, Do 11.11/2*,3*
Lesson 19 (4/11): Area and Integration
Read the "Riemann Sum" article on wikipedia. First write the left sum, right sum, upper sum and lower sums for the following five integrals using evenly spaced intervals so that xi-xi-1=(b-a)/N (but don't try to evaluate the sum). Which is largest? Which is smallest? Second find the relationship between epsilon and delta for each of these uniformly continuous functions using the mean value theorem (Hint: delta=epsilon/M where M=max|f'|).
1) f(x)=4x integrated from 0 to 3
2) f(x)=5x+2 integrated from 3 to 8
3) f(x)=10-2x integrated from 1 to 4
4) f(x)= (x-2)2 + 1 integrated from 1 to 3
5) Prove the integral of a constant function f(x)=c from a to b is c(b-a), by taking the sums and explicitly evaluating them.
Finally estimate how large N must be to guarantee an error of epsilon' in each integral. Verify this works. Verify that the upper sum is larger than the lower sum and the left and right sums are in between, and that the upper sum minus the lower sum has an error less than epsilon' when N is chosen large enough.
Lesson 20 (4/13): Riemann Integral
Read 13.1-13.3, Write proofs for 13.4 and 13.6, Read 13.16-13.17, Show that a function which is 0 on irrational numbers and 1 on rational numbers is not Riemann Integrable,
Look over 13.9-13.15, Read 13.19-13.22 carefully, look over 13.23-13.25, Do 13.26/1, 2.
Lesson 21 (4/18): Limits, Improper Integrals and Logs
Read 8.18-8.19, Do 8.20/1*, 2*,
Read 13.27-13.33, Do 13.34/1*,
Read 14.1-14.2, Do 14.3/1*, Read 14.4, Do 14.5/1*, Read 14.6, Do 14.7/3*,5*
Lesson 22 (4/20): Review of Calculus I from an advanced perspective
HW: 1) Prove the chain rule using the definition of derivative (can read this proof in a calculus textbook)
2) Prove limit of sin(x)/x is 1 and limit of (cos(x)-1)/x is 0 using trigonometry and areas of triangles as in a calculus textbook but adding justifications from Analysis, then prove the derivative of sine is cosine and the derivative of cosine is sine.
3) Prove that sin^2(x) + cos^2(x) =1 by first taking f(x) = sin^2(x) + cos^2(x) - 1, then showing f(0)=0, then showing f'(x)=0 everywhere, then concluding f(x)= const=0 everywhere.
4) Prove that e^{x+y}=e^x e^y by first taking f(x)= e^{x+y}-e^x-e^y, then showing f(0)=0, then showing f'(x)=0 everywhere, then concluding f(x)= const=0 everywhere.
5) Prove u substitution by writing out the chain rule and integrating both sides of the equality.
6) Prove integration by parts, by first noting the product rule (uv)' = u'v + u v' and then integrating both sides of this equality.
Lesson 23 (5/2): Exam III on the Proofs of Calculus I: prove that a sequence of functions is equicontinuous as in the HW from Lesson 16, prove that a sequence of functions converges uniformly to a limit function as in HW from Lesson 17, find the Riemann Sum as in HW from Lesson 19, Prove that a function satisfies a certain equality using its derivative as in the HW problems 3 & 4 from Lesson 22.
Lesson 24 (5/4): Series HW: Review 4.4, 4.10, 4.17, then Read 6.1-6.3, write out proofs of 6.2, 6.3 Prove that the series 1/2 + 1/4 + 1/8 + 1/16+... =1 using proof by induction to verify the partial sums add up to 1-(1/2^k) and using an epsilon N limit proof to show those partial sums converge to 1. Read 6.4-6.5, Read 6.8, Write out a proof of 6.9,
Lesson 25 (5/9): Convergence of Series HW: Read 6.10-6.11, Write out a proof of 6.11 Review Cauchy sequences in 5.16, 5.17, 5.19, then Read 6.12-6.15 Rewrite the proofs of 6.16-6.19 (in 6.18 you may assume the limit is 0 rather than the limsup.
Lesson 26 (5/11): Taylor Series and Convergence of Functions
HW: Review Calculus textbook section on Taylor Series about x=0 and steps taught in class including the induction proof to:
1) find the Taylor series for e(3x) and check where it converges using the ratio test.
2) find the Taylor series for 1/(1-x) and check where it converges using the ratio test.
3) Find the Taylor series for Ln(x+1) and check where it converges.
4) Prove fn(x)=xn defined on [0,1] converges pointwise to a function which is 0 everywhere on [0,1) and is 1 at 1.
Does this function converge uniformly? Prove or disprove.
5/13: Extra Office Hours 1pm - 7pm
Lesson 27 (5/16): Review for the Final
The final will have two sections: short answers and proofs. In the proof section there will be two proofs:
write a complete proof that a function is continuous
write a complete proof that a sequence converges.
write the first line of a proof by contradiction,
start a proof by induction: base case and first few lines of the proof including the application of the induction hypothesis
find a Riemann Sum approximating a Riemann Integral up to a given error
find limits, sups and infs, limsups and liminfs without proof
find taylor series and verify convergence of the series
In order to complete the other section you will need to know the statements of all the important theorems and definitions we've learned this semester including sup, inf, bound, limit, bounded increasing sequences converge, sandwich lemma, subsequences of bounded sequences converge, Cauchy sequences, theorems about these, continuity, theorems about this, differentiation, mean value theorem, Rolle's theorem, Riemann integration of continuous functions, theorems about integration, improper integrals, series, convergence tests including comparison, ratio, root and alternating series tests, Taylor series, radius of convergence, uniform convergence. You will also need working knowledge of these concepts in the sense that you must be able to find the limit of various given sequences, the sup and inf of various sets and functions, the Taylor series for a given function and give its radius of convergence.
Lesson 28 (5/18): class cancelled replaced by 2-5pm workshop on 5/23
Final Exam (5/25): 1:30-3:30 pm The Final Exam will be given during Finals Week covering the entire course especially topics needed in future courses.