Analysis I MAT 320 Spring 2014
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
Meeting Times: Monday/Wednesday 7:50-9:30 pm Gillet Hall Room 333
Course Webpage: https://sites.google.com/site/professorsormani/teaching/analysisi-s14
Professor Sormani: google "Sormani Math" or go to http://comet.lehman.cuny.edu/sormani
Contact: sormanic@member.ams.org (do not call the office and leave messages)
Office Hours: Mon/Wed 5:50-6:30 pm and 7:00-7:50 pm (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 three 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 collected on the days of quizzes.
Quizzes: THE QUIZZES COUNT MORE TOWARDS THE GRADE THAN EITHER OF THE EXAMS! Students must submit completed *ed homework from the prior week in order to begin a quiz and that homework will be part of the score. Students who miss quizzes because they are absent, late or have not completed the homework will receive a 0 on that quiz. There will be no make up quizzes but there is one extra quiz. Note that a quiz may contain proofs and the proofs should be written in a two column format with statements and justifications. Very similar proofs will be on the homework that is required to take the quiz. If you are unsure of the proof of a problem, you may write a question mark next to the step you are unsure of.
Grade: 11 Quizes: 5% each, Midterm Exam: 15%, Final:30%.
Textbook: Mathematical Analysis: a Straightforward Approach by Binmore, 2nd Ed Cambridge University Press . Available free at https://archive.org/details/MathematicalAnalysis
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. ISBN: 9780521288828
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 and quiz information.
1/27: Quantifiers
HW: Complete the Quantifiers Worksheet, Read 7.1-7.5, Read 7.6-7.12
1/29: 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
2/3: Proof Workshop
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*
2/5: Continuoum and Quiz 1 on Basic Proofs (must show 1.8/1,2,4,5,6 are all completed to take this quiz)
HW: Read 2.1-2.9; Do 2.10/1,2*,3,6*; Read 7.9-7.16; Do 7.16/3*,4*
2/10: 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*
2/19: Convergence and Quiz 2 (on 1.20 and 2.10)
HW: Read 4.1-4.5, Do 4.6/1*,2*,3* (be sure to use epsilon in these)
2/20 (Thursday!): 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 2/24: Do 1.20/4, 3.6/6,
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*,
2/26: Monotone Sequences and Quiz 3 (on Convergence)
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.
3/3: Subsequences and Quiz 4 (on Induction)
HW: Read 5.1-5.7, start redoing all starred problems and quizzes to prepare for the midterm.
3/5: Bolzano Weierstrass Theorem,
HW: study for Midterm Exam (which will essentially be problems similar to those in Quizes 1-4)
3/10: Midterm Exam
HW: Quiz V will be about describing sequences using the definitions of bounded, increasing, decreasing, Thm 4.17, Thm 4.10, Thm 4.25 and Thm 5.2 as we did in class on 10/7. To prepare for this quiz, Read 5.1-5.3, go over lesson 10/7, Read 5.8-5.10 and examine what subsequence of a_n and b_n from 10/7 is monotone in the proofs there, then read 5.11-5.14, do 5.15/4,5,6.
3/12: Cauchy Sequences and Quiz 5 (as described above)
HW: ( Read 5.16-5.19 on Cauchy sequences, Do 5.21/1*, 4*
3/17: Limits of Functions
Read first section of handout from the original Larson Calculus textbook, 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,
3/19: Continuity and Quiz 6
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*
3/24: Continuity
HW: 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*, 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*,
3/26: Proving Workshop and Quiz 7 (group classwork due Wednesday)
Group classwork is: Prove 9.5 and 9.6, Prove 9.12 for infimum, Read 9.16, Do 9.17/2,3,4,5,6.
3/31: 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*
4/2: Mean Value Theorem and Quiz 8 (Continuity)
HW: Read 11.1-11.7, Do 11.8/2*,3*,4*, Read 11.9, Do 11.11/2*,3*
4/7: Review of Calculus I from an advanced perspective
HW: Study all work on continuity and differentiation,
For Quiz 9: 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.
4/9: Area and Integration (Hand in Quiz 9)
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.
Please do this much before 4/9. Finally (after 4/9) 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. All of this is Quiz 10 due 4/28 (group work is encouraged)
4/23: 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.
All this homework counts as Quiz 11 (due 5/5) (group work is encouraged)
4/25: Lehman College Grand Career Expo 2014 Fri., Apr. 25, Apex/Auxiliary Gym., 11-2pm
4/28: Limits, Improper Integrals and Logs (Hand in Quiz 10)
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*
All this homework counts as part of Quiz 11 (due Mon 5/5) (group work is encouraged)
4/30: 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, This homework is Quiz 12 (due Wed 5/7) (group work is encouraged)
5/5: Convergence of Series (hand in Quiz 11)
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. This homework may be submitted as Extra Credit due on 5/12.
5/7: Taylor Series and Convergence of Functions (hand in Quiz 12)
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.
These above will prepare you for Quiz 13 given in class on Monday.
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.
Catching up and improving grades on key quizes:
Students who scored under 80% on Quiz 4 on Induction or never took it, may come to my office hours on Monday 5/12 for a new one.
Students who scored under 4 Quiz 8 on Continuity or never took it, may come to my office hours on Wednesday 5/14 for a new one.
Problems like these will be on the final. As with all quizes taken late, a deduction of 1 out of 5 points (20%) will be taken from the quiz score.
My office hours on 5/12 and 5/14 are 3-4 pm and 6-7:50 pm in my office Gillet 200A.
In total this class has had 13 quizes. I will drop the two lowest quiz scores from your grade. Nevertheless Quiz 13 is very important with a related problem on the final!
Teaching evaluations: Don't forget to login to your Lehman email to fill out teaching evaluations. For information about accessing your Lehman email, contact the IT Center Help Desk in Carman 108, tel. 718-960-1111, or help.desk@lehman.cuny.edu. Do this before 5/12. It'll take 10 minutes.
5/12: Differentiating Taylor Series and Quiz 13 (in class on Taylor Series),
HW: Read Binmore 15.1-15.9, Do 15.6/ 1*, 2*, 6*
1) Show f_j(x)=x^j/j converges in C_o sense but not C_1 sense on [0,1]
2) Find Taylor series about x=1 for cos(pi x) and find interval of convergence.
3) Find Taylor series about x=2 for 1/(1-x) and find interval of convergence.
5/14: Review for Final
The final will have two sections: short answers and proofs. In the proof section there will be two proofs:
a proof that a function is continuous at a point
a proof that a sequence converges.
In the short answer section, other proving techniques will be tested including
write the first line of a proof by contradiction,
outline a proof by induction
suggest theorems that might be relevant to prove a statement
fill in the justifications for a proof about integrals
find limits, sups and infs without proof
find taylor series and verify convergence of the series
In order to complete this 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.
Extra Office Hours on Monday 5/19 3-4pm and 6-10pm.
Final Exam: The Final Exam is Wednesday May 21 8:00-10:00 pm Gillet 333.
Lehman College, Department of Mathematics and Computer Science,
Professors are not permitted to accept any gifts from students.