Harvard University,FAS
Spring 2004

Mathematics Math1b
Spring 2004

Introduction to Functions and Calculus II
and Differential equations

Course Head: Robin Gottlieb

Office: SciCtr 429
Email: gottlieb@math.harvard.edu
Mainpage Syllabus Sections Calendar Homework Exams Supplements Links

Homework

HW # Assignment Due Solutions
Primer Problem Set 0 Due Thu Feb 6 (TTh) or Fri Feb 7 (MWF) Primer Solution  
Problem Set 1 Section 5.5# 30, 31, 45, 55, 62
Integration Handout #1, 2 (with #1 not due until Problem Set 4 is turned in.)
(TTH classes: Hand in Problem Set 0 with this.)
Reading: Text 5.5 and read about trig integrals on pp. 403-404
W Feb. 5 (MWF) or
Th Feb 6 (TTh)
Solution 1  
Problem Set 2 Section 5.6 # 4, 6, 9, 14, 18, 22, 28
(MWF classes: Hand in Problem Set 0 with this) Extra credit: 5.6 #34
Reading:5.6
F Feb. 7 (MWF) or
T Feb 11 (TTh)
Solution 2  
Problem Set 3 Section 5.7 # 16, 18, 24, 27 (It is easier using substitution - the choice is yours), 29
Integration Handout # 3, 4
Reading:5.7
M Feb 10 (MWF) or
T Feb 11 (TTh)
Solution 3  
Problem Set 4 Integration Handout #5, 6, 7
Plus hand in #1 from the Integration Handout
Reading: 5.7
W Feb 12 (MWF) or
Th Feb 13 (TTh)
Solution 4  
Problem Set 5 Integration Handout # 8, 9
5.2 # 32, 35
6.1 #2, 25
Reading: 5.2 and 6.1
F Feb 14 (MWF) or
T Feb 18 (TTh)
Solution 5  
Problem Set 6 Integration Handout # 11, 12
6.2 #21, 23, 42
Reading:6.2
Optional reading: supplement 27.2 pp. 843-850
Directions for accessing the Supplement can be found under "supplement" in the menu above..
W Feb 19 (MWF) or
T Feb 18 (TTh)
Solution 6  
Problem Set 7 Integration Handout #13, 14, 15, 16, 17
Extra Credit #18 on the Integration Handout
Reading: Supplement pp. 827-842.
F. Feb. 21 (MWF) or
Th. Feb 20 (TTh)
Solution 7  
Problem Set 8 Section 6.2 # 28, 37
Integration Handout #19, 20, 21.
Reading: Supplement 28.1 pp. 853-6
Stewart: 6.2 pp. 453-5.
M. Feb 24 (MWF) or
T Feb. 25 (TTh)
Solution 8  
Problem Set 9 Section 6.5 # 4, 7, 8, 10, 12, 14
Integration Handout #22
Reading:Section 6.5
Optional: supp. 28.2 pp. 867-871.
W. Feb 26 (MWF) or
T. Feb 25 (TTh)
Solution 9  
Problem Set 10 Section 6.5 #13
Integration Handout # 26
Section 5.10 # 2, 6, 12, 14, 20, 51
(Note: one of these requires no computation. Look and think before you plunge in.)
Reading: 5.10
F. Feb 28 (MWF) or
Th Feb. 27 (TTh)
Solution 10  
Problem Set 11 Section 5.10 # 24, 29
Section 6.7 #2, 4(ab), 6
Extra credit: 6.7 #13
Reading: 6.7
Solutions will be posted by March 1 to assist
you in your studying for the exam.
Solution 11  
Problem Set 12 Section 6.4 # 2, 11 Section 6.3 # 4
Integration Handout # 23, 24
Extra credit: Integration handout # 25.
Reading: 6.3 (omit Ex. 4) and 6.4
F. Mar 7 (MWF) or
Th. Mar 6 (TTh)
(Series problems will also be due on these days, so start early.)
Solution 12  
Problem Set 13 Series Handout #1, 3, 5, 6
F Mar 7 (MWF) or
Th Mar 6 (TTh)
Solution 13  
Problem Set 14 Section 8.1 #39; 8.2 #1, 2, 12, 16, 18, 20
Series Handout #2, 4, 7, 8
Reading: Supplement 18.1, 18.2, 18.3. If you need work on summation notation then read 18.4.
Stewart 8.1 and 8.2.
M Mar 10 (MWF) or
T Mar 11(TTh)
Solution 14  
Problem Set 15 Section 8.2 #34, 48, 51 and
8.3 # 1, 2, 5, 15
Series Handout #9 (including part (c TYPO! in part (c) the series should start at n=2 not n=1. ln 1 is zero!) - the extra credit is extra credit).
Extra credit: 8.2 #52 Reading:8/3 (through p. 588 Ex.5
W Mar.12 (MWF) or
Th Mar 13 (TTh)
Solution 15  
Problem Set 16 Section 8.3 #3, 4, 9, 10, 16, 18, 19, 20, 22
Reading: 8.3
F Mar 14 (MWF) or
T Mar 18 (TTh)
Solution 16  
Problem Set 17 In the supplement on the web: Section 30.1 #1, 2, 9, 11, 12, 13(use your work from #2), 16
Series Handout # 10
Reading:Section 30.1 from the supplement on the web.
M Mar 17 (MWF) or
T Mar 18 (TTh)
Solution 17  
Problem Set 18 Section 8.7 #7(ignore references to R_n(x) here and throughtout the assignment) 13, 18, 20, 22
Series Handout # 11, 12
Reading: 8.7 pp 613 -mid 614
(In the supplement on the web middle of p. 947-mid 948)
W Mar 19 (MWF) or
Th Mar 20 (TTh)
Solution 18  
Problem Set 19 Section 8.4 #12, 13, 19 (explain your reasoning carefully)
Series Handout #13, 14 (typo! Fix the series for sin x), 15
Reading:8.4 pp. 592-597 (alternating series and absolute convergence
F Mar 21 (MWF) or
T Apr. 1 (TTh)
Solution 19  
Problem Set 20 Section 8.4 #2, 20, 22, 31, 33 (notice that this is the Taylor series for e^x)
Series Handout #17, 18
Reading:8.4 pp. 597-8 (ratio test)
M Mar 31 (MWF) or
T Apr. 1 (TTh)
Solution 20  
Problem Set 21 Section 8.5 # 4, 12, 14, 16, 20
Series Handout #20, 21
Reading:8.5
W Apr 2 (MWF) or
Th Apr 3r (TTh)
Solution 21  
Problem Set 22 Section 8.6 # 9, 11
SEction 8.7 # 11 (no R_n(x)), 29, 36
Series Handout #22, 24
Reading:8.6 and 8.7
F Apr 4 (MWF) or
T Apr. 8 (TTh)
Solution 22  
Problem Set 23 Section 8.7 # 34 section 8.8 # 1, 9 section 8.9 #22.
Series Handout #23a-c (no Taylor remainder) and 25.
Reading:8.8 and 8.9
M Apr 7 (MWF) or
T Apr. 8 (TTh)
Solution 23  
Problem Set 24 Problems for Review: (answers to be posted before the exam)
Concept Check on p. 640 (stewart) #3, 4, 10, 11a-d
True/False quiz # 1-7 and 11 on p. 640
8.6 # 6, 16 and 8.5 # 23a and 8.7 #38 and 8.9#23
W Apr 9 (MWF) or
Th Apr. 10 (TTh)
Solution 24  
Problem Set 25 Study for the exam
 
Problem Set 26 Differential equations Handout #1, 2, 3, 4, 5
(use the web applet under the supplements page: dfield.)
7.2 # 3, 4, 5, 6 (you can check this using the web applet!)
M Apr 14 (MWF) or
T Apr. 15 (TTh)
Solution 26  
Problem Set 27 Differential equations Handout #6, 7, 8, 9
7.1 # 7, 11; and 7.3 #4, 10
Reading: 7.1, 7.2 (without Euler's method) and 7.3 or the supplement 31.1, 31.2 and 31.4
W Apr 16 (MWF) or
Th Apr. 17 (TTh)
Solution 27  
Problem Set 28 From the supplement on the web, 31.3 (pp. 1014 -16) #1,2,3,6 and
From Stewart 7.4 #19bcd
Reading: Supplement 31.3 (p.1002-1012)
F Apr 18 (MWF) or
T Apr. 22 (TTh)
Solution 28  
Problem Set 29 Differential equations Handout #10, 11, 12
First Order Linear handout #1, 2, 3
7.3 # 34, 36
Reading: First Order Linear Handout
M Apr 21 (MWF) or
T Apr. 22 (TTh)
Solution 29  
Problem Set 30 Differential equations Handout #13, 15 Use the pplane applet to check your work.
Be prepared to put your answer to 15 on the board on the date the assignment is due. Full credit will be given for 15 as long as you turn in a thoughtful answer.
7.6 # 1, 2 plus Chapter 7 Review #20 and 21 where in (20(d) the trajectories are closed curves.)

TTH classes: do Stewart 7.1 #4 and 5 for Thursday as well.
Reading: Stewart 7.6 and the supplement 31.5 ( pp 1024-1040).
W Apr 23 (MWF) or
Th Apr. 24 (TTh)
Solution 30
A corrected version of the phase diagram in #13 can be found here.

A detailed and exemplary solution to #15 can be found here

Problem Set 31 In the supplement on p. 1042-5 do #9. 13bc, 14
Differential equations Handout #14, 16
MWF classes: In Stewart 7.1 do #4, 5
F Apr 25 (MWF) or
T Apr. 29 (TTh)
Solution 31
The solution to 13b should be (vii), not (xi).
Problem Set 32 Differential equations Handout #19, 20, 21, 22, 23
Reading: Supplement 31.6 pp. 1045-middle of page 1047.
M Apr 28 (MWF) or
T Apr. 29 (TTh)
Solution 32  
Problem Set 33 Differential equations Handout #19b, 20b, 24, 26 (23 was listed again by error - you just did it.)
Supplement p. 1050 #12, 13, 17
Reading: Supplement 31.6 pp. 1047 1049.
W Apr 30 (MWF) or
Th May 1 (TTh)
Solution 33
In 24(d), you are asked to solve y'' - 9 = 0. The given solution is incorrect. A correct method would simply integrate twice to obtain the answer. i.e. y' = 9t + c1 followed by y = (9/2)t^2 + tc1 + c2.
Problem Set 34 Differential equations Handout #25
Supplement p. 1050 # 16
I. Use series to solve y'' = -4y where at t=0 y = 2 (NOT 0) and y'=0.
II. Analyze 2 differential eqns from two perspectives: first, using what you know about solving second order differential equations and second, writing each as a system of first order differential equations. Details will be found on the supplements page under "last problem of the term".
Reading: Supplement 30.4 (pp. 960-961) or Stewart 8.10.
F May 2 (MWF) or
T May 6(TTh) (in your CA's mailbox)
Solution 34  


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