On the next midterm or the final exam (or both) I may ask you to prove one or more of the following theorems:
6.2.1, 6.2.2, 6.2.3, 6.1.1, 6.1.2, 4.2.1, 4.2.2, 4.2.3, 4.2.4, 4.1.1, 4.1.2ab(not c), 3.5.1, 3.3.1, 3.3.Lemma3, 3.2.2, 2.2.1, 2.2.5, 2.4.1, 2.5.1, 2.5.3.
Final Exam, M Dec 12, 1:00-4:00pm. The exam will cover all homework problems (and their corresponding sections), plus proofs of the theorems listed above. You can give me a list of ten problems from HWs 1-8 and ten from HWs 21-29 to exclude from the final exam.
HW due M 12/5. Review all problems in HWs 24-29.
HW due F 12/2. Review all problems in HWs 18-23.
HW due W 11/30. Review all problems in HWs 9-17.
HW 29, due M 11/28. Review all problems in HWs 1-8. Also do these problems.
HW 28, due M 11/21. Read Sec 5.6 p.196-200 and the top half of p. 201. Do Sec 5.6: 1-4, 7-9.
HW 27, due F 11/18. Practice proving Theorems 6.2.1, 6.2.2, 6.2.3 without looking at class notes or the book. Do Sec 6.2: 15, 16; and p.270: 1a-e. Do 1a-e before doing 16. Give two proofs for part 1c: (i) using 1b; (ii) instead of using 1b, first show that each of the sequences {S2n} and {S2n+1} converges; then show they have the same limit.
HW 26, due W 11/16. Read Sec 6.2 p.234. Do Sec 6.2: 4, 13, 14. Also do these extra problems.
HW 25, due M 11/14. Read Sec 6.2 p.228-233. Do Sec 6.2: 8-12. Prove Theorem 6.2.1a-d without looking at your notes or the book's proof. Also do these extra problems.
HW 24, due F 11/11. Practice proving Theorems 6.1.1 and 6.1.2 without looking at the book or your notes. Do Sec 6.1: 8-10. Also do these extra problems. Here are some hints for #9.
HW 23, due W 11/9. Read Sec 6.1 p.226. Do Sec 6.1: 4-7. Also do the following problem. Prove or disprove: If lim sup ak is finite, then for all n, sup{ak: k > n} >= lim sup ak.
HW 22, due M 11/7. Read Sec 6.1 p.223-225. Do Sec 6.1: 1-3. Here are some hints. For problem 1e, use the following fact, without proof: For every epsilon > 0 , there exists an arbitrarily large integer n such that the interval [n - epsilon, n + epsilon] contains an integer multiple of pi/2 (this is equivalent to the theorem proved in project 5 of Ch 2). For problem 3, use Theorem 6.1.1.
HW 21, due F 11/4. Read Sec 4.6 p.151-153. Do Sec 4.6: 1-4.
Midterm 2, W 11/2. The exam will cover HWs 9-20 and their corresponding sections.
HW 20, due F 10/28. Read Sec 4.3 p.138. Do Sec 4.3: 12; plus these extra problems.
HW 19, due W 10/26. Read Sec 4.2 p.131-132; may skip proof Theorem 4.2.5. Do Sec 4.2: 7, 8, 11-13.
HW 18, due M 10/24. Read Sec 4.2 p.129-130. Do Sec 4.2: 1, 3, 4, 5, 6. #4: Canceled., you may assume that sin x is continuous and is increasing on (-pi/2, pi/2), and that (cos x)' = -sin x (but may not assume L'Hopital's rule). Also do the following problem: Suppose g is defined on an open interval containing c, and the limit of g(x) as x -> c^+ exists and is greater than w. Prove for all epsilon > 0, there is some x in (c, c + epsilon) such that g(x) > w.
HW 17, due F 10/21. Do Sec 4.1: 10, 11, 13, 14abce.
HW 16, due W 10/19. Read Sec 4.1. Do Sec 4.1: 1, 4, 6, 7, 9.
HW 15, due M 10/17. Do Sec 2.6: 12, 13; also do these extra problems.
HW 14, due F 10/14. Read Sec 3.5 p.107-1110. Do Sec 3.5: 5-8. Hint for #7: Let c be in (a,b); let s = sup f( [a,c) ); prove that if x_n -> c^-, then f(x_n) -> s. (Here's a proof.)
HW 13, due W 10/12: Read Sec 3.5 p.103-107. Do Sec 3.5: 1, 2, 3a, 4.
HW 12, due F 10/7: Read Sec 3.3 p.91-93. Do Sec 3.3: 6, 9, 14, 15. (Hint for 15: Use Theorem 3.3.4 and the Intermediate Value Theorem.)
HW 11, due W 10/5. Read Sec 3.3 p.87-91. Do Sec 3.3: 1-4.
HW 10, due M 10/3. Read Sec 3.2 p.83-85; you may skip the last definition in that section (we may cover it later). Do Sec 3.2: 9, 10, 12; also do these extra problems.
HW 9, due F 9/30. Read Sec 3.2 p.80-83. Do Sec 3.2: 1-5.
Midterm 1, W 9/28. The exam will cover HWs 1-8 and their corresponding sections. Start reviewing a week before; redo every homework problem (at least the ones you feel unsure about). Here's a copy of Midterm 1 with correspondence between HW problems and exam problems.
HW 8, due F 9/23. Do Sec 3.1: 8-13.
HW 7, due W 9/21. Read Sec 3.1. Do Sec 3.1: 2, 3, 5-7.
HW 6, due M 9/19. Read Sec 2.6. Do Sec 2.6: 1-11.
HW 5, due W 9/14. Read Sec 2.5. Do Sec 2.5: 1-9.
HW 4, due M 9/12. Do Sec 2.4: 8, 9, 11, 13, 14.
HW 3, due F 9/9. Read Sec 2.4; may skip the proof of Theorem 2.4.3. Do: Sec 1.1: 11. Sec 2.2: 7, 9, 10c. Sec 2.4: 1, 2, 5, 7. (Sec 2.2: may use 10ab for 10c. Note: 10a says: v.w <= ||v|| ||w||, which follows from the formula v.w = ||v|| ||w|| cos(theta); 10b is the triangle inequality: ||v+w|| <= ||v|| + ||w||. You've seen both in Linear Algebra.)
HW 2, due W 9/7. Read Sec 2.2. Do Sec 2.2: 3-6. Also do Sec 2.1: 3d, 4d, 6, 7.
HW 1, due F 9/2. Read Sec 2.1 (you should spend at least an hour reading the section slowly and making sure you understand everything). Do Sec 2.1: 2ac, 3ab, 4ab, 5, 8, 9.