Harvard University,FAS
Fall 2004

Mathematics Math1b
Fall 2004

Introduction to Functions and Calculus II
and Differential equations

Course Head: Robin Gottlieb

Office: SciCtr 430
Email: gottlieb@math.harvard.edu
Mainpage Syllabus Sections Calendar Homework Exams Supplements Links

Homework

HW # Assignment Due Solutions
Primer Problem Set 0 M. September 27 (MWF) or T September 28 (TTh) Primer Solution  
Problem Set 1 Section 5.2 #18,32,35
Section 6.1 # 2,25,36
# 1,2 on Integration Handout B
Start #19 on the handout. It is due in a week.
W Sep 29 (MWF) or
Th Sep 30 (TTh)
Solution 1  
Problem Set 2 #3, 4, 5 on Integration Handout B
Section 6.2 #2, 10, 12, 23
OptionalReading: Supp. 27.2 (pp. 843-850)
F Oct 1 (MWF) or
T Oct 5 (TTh)
Solution 2  
Problem Set 3 # 6, 7, 8, 9, 10 on Integration Handout B
Section 6.2 #25, 28
Optional Reading: Supp. 28.1 (pp. 853-6)
M Oct 4 (MWF) or
T Oct 5 (TTh)
Solution 3  
Problem Set 4 # 11, 12, 13, 14, 19 on Integration Handout B
W Oct 6 (MWF) or
Th Oct 7 (TTh)
Solution 4  
Problem Set 5 #15, 18 on Integration Handout B where on #18 it should read x=0 rather than t=0 (or change all x's to t's if you prefer)
Section 6.3 #1, 4
Section 6.4 # 2, 6, 11
For extra credit do #16 on the integration Handout
Optional Reading: Supp. 28.2 pp. 865-7
F Oct 8 (MWF) or
T Oct 12 (TTh)
Solution 5  
Problem Set 6 # 17 on Integration Handout B
Section 6.5 #4, 7, 8, 10, 11, 14
Optional Reading: Supp. 28.2 (pp. 867-71)
W Oct 13 (MWF) or
T Oct 12 (TTh)
Solution 6  
Problem Set 7 Section 5.5 # 22, 30, 31, 45, 55, 62
#20 on Integration Handout B
Optional Reading: Supp. 25.2 and 25.3 (pp. 787-804)
F Oct 15 (MWF) or
Th Oct 14 (TTh)
Solution 7 
Problem Set 8 Section 5.6 #4, 6, 9, 14, 18, 22, 28
#21 on Integration Handout B
Extra credit: Section 5.6 #34
Optional Reading: Supp. 29.1 pp 877-885
M Oct 18 (MWF) or
T Oct 19 (TTh)
Solution 8  
Problem Set 9 Section 5.7 #16, 18, 24, 27 (you can do this more easily via substitution: the choice is yours), and #29.
# 22, 23 on Integration Handout B
Optional Reading:Supp. 29.3 p. 898-902 and, if you like, 29.2 p. 886.
W Oct 20 (MWF) or
T Oct 19 (TTh)
Solution 9  
Problem Set 10 Section 5.10 #6, 7, 55
5.9 # 2
# 24, 26 on Integration Handout B
Also on the integration handout B you can do #25 for extra credit. Extra credit: 5.10 #50 and 5.9 # 35.
Optional Reading:Supp. 29.4 p. 903-914 and, if you like, 26 p. 805 - 816.
F Oct 22 (MWF) or
Th Oct 21 (TTh)
Solution 10  
Problem Set 11 Section 5.10 #2, 20, 24, 29, 49, 51
M Oct 25 (MWF) or
T Oct 26 (TTh)
Solution 11  
Problem Set 12 Section 6.7 # 4ab, 6 (refer to Example 2)
Extra credit: 6.7 #10
W Oct 27 (MWF) or
T Oct 26 (TTh)
Solution 12  
Problem Set 13 #1, 3 on Series Handout A
F Oct 29 (MWF) or
Th Oct 28 (TTh)
Solution 13 
Problem Set 14 # 2, 4, 7, 8 on Series Handout A
8.1 #39; 8.2 #1, 2, 16, 18, 20
optional reading: Supp 18.1, 18.2, 18.3
M Nov 1 (MWF) or
T Nov 2 (TTh)
Solution 14 
Problem Set 15 8.2 #34, 48, 51
8.3 # 4, 10, 19, 20
W Nov 3 (MWF) or
T Nov 2 (TTh)
Solution 15  
Problem Set 16 8.3 #1, 2, 5, 9, 15, 16, 18, 22
# 9 on Series Handout A
optional reading G: 30.5 pp. 964- 972
F Nov. 5 (MWF) or
Th Nov 4 (TTH)
Solution 16  
Problem Set 17 In the supplement on the web do S 30.1 # 2, 9, 11 (hint for (b) - look at the graph of tan x), 12, 13 (use your work from 2
# 10 on Series Handout A
M Nov 8 (MWF) or
T Nov 9 (TTh)
Solution 17  
Problem Set 18 8.7 in Stewart #7, 13, 18, 20
# 11, 12 on Series Handout A
In 30.1 do # 20. Reading: 8.7 pp. 613-mid 614. (We'll return to 8.7.) In the supplement on the web middle of page 947 through mid p.948.
W Nov 10 (MWF) or
T Nov 9 (TTh)
Solution 18  
Problem Set 19 8.4 #12, 13, 19 (explain your reasoning carefully)
# 13, 14, 15 on Series Handout A
F Nov 12 (MWF)
or
T Nov 16 (TTh)
Solution 19 
Problem Set 20 8.4 #20, 22, 31, 32, 33(notice that this is the Taylor series for e^x)
# 17, 18, 19 on Series Handout A
optional reading: G:30.5 convergence tests pp. 964-977
M Nov 15 (MWF)
or
T Nov 16 (TTh)
Solution 20 
Problem Set 21 8.5 #4, 12, 16, 18, 20
8.6 #11
#20, 21 on Series Handout A
W Nov 17 (MWF)
or
Th Nov 18 (TTh)
Solution 21  
Problem Set 22 8.7 #11, 23, 29, 38
# 22 on Series Handout A
F Nov 19 (MWF)
or
Th Nov 18 (TTh)
Solution 22  
Problem Set 23 This is NOT to be turned in. Some of these problems will be done in a coursewide review. Answers will be posted.
8.5 #23a; 8.6 #6, 16; 8.7 # 31; 8.9 #20, 22, 23
#25 on Series Handout A
Concept Check on p. 640 (Stewart) #3,4,10, 11a-d and the True/False Quiz # 1-7 and 11
before Monday, Nov 22 (MWF/ TTH) Solution 23  
Problem Set 24 Take the exam.
M Nov 22 (MWF/ TTh)
Problem Set 25 Enjoy Thanksgiving Break!
 
Problem Set 26 #1, 2,3,4, 5, 7 on Differential Equations Handout
Stewart: 7.1 #7, 11
Stewart 7.2 #3-6
(use the web applet under the supplements page: dfield to help with the handout and to check your answers to 7.2.)
W Dec 1 (MWF)
or
Th Dec 2 (TTh)
Solution 26 
Problem Set 27 #6, 8, 9 on Differential Equations Handout A Stewart 7.3 #4, 16, 29, 33, and 36.
F Dec. 3 (MWF)
or
T Dec 7 (TTh)
Solution 27  
Problem Set 28 Supplement 31.3 (p. 1014-1016) #1,2,3,5,6, 7
Stewart 7.3 #34 and 7.4 #19
Relevant reading is section 31.3 (pp. 1002-1017 in the supplement. At this point we've covered 31.1-31.4 in the differential equations chapter in the supplement.
M Dec. 6 (MWF)
or
T Dec 7 (TTh)
Solution 28  
Problem Set 29 Read the First Order Linear Diff Eqns handout under "supplements" and do problems 1,2 (with the condition that $x>0$), and 3 at the end of the handout.
Do #10 (part (b) should refer to 10(a)(i) not 4ai), 11, 12, on Differential Equations Handout A version II
W Dec. 8 (MWF)
or
Th Dec 9 (TTh) Note: TTH sections:
Your TF may assign a couple of problems from PS #30 as well. You'll find out in class.
Solution 29 
Problem Set 30 Stewart 7.6 #1,2 plus Chapter 7 Rev. p. 559 #20 (in (d) the trajectories are closed) and 21
Do #13, 15 on Differential Equations Handout
Be prepared to put your answer to #15 on the board the date the assignment is due. Full credit for 15 will be given as long as you turn in a thoughtful answer.
. Read the supplement 31.5 (pp. 1024 - 1040) and Stewart 7.6
F Dec. 10 (MWF)
or
T Dec 14 (TTh)
Solution 30
Problem Set 31 In the supplement on p. 1042-1045 do #9,(in the note sent Friday I removed 13bc), 14
Do #14, 16 on Differential Equations Handout

Stewart: 7.1 #5. (And look back at #4 that you did for the first ppre-class reading assignment. We'll refer to these examples in class on the week of the 13th.)
M Dec. 13 (MWF)
or
T Dec 14 (TTh)
Solution 31  
Problem Set 32 From the supplement on p. 1044 do #13 - where in (a) dy/dt = x, not -4x (as written there are no answers to (a)).
Plus, do the following two problems:
1. Consider the system
dx/dt = -.1x -.1x^2 + xy
dy/dt = y - 0.1 y^2 - xy .
Let x(t) be the number of thousands of animals of species X and y(t) be the number of thousands of animals of species Y.
(a) Would this model a competive system? predator/prey? symbiotic?
(b) Do a phase plane analysis. Then look at the pplane program to see what the trajectories look like.
x\ (c)Suppose x(0) = 0 and y (0) = 1. What do you expect in the long run? Suppose instead that x(0) and y(0) are both positive. What do you expect in the long run?

2. Consider the system
dx/dt = -y and dy/dt = x. (similar by not identical to Ex. 31.26 on p. 1031 of the supplement.
(a) Do a phase plane analysis.
(b) Show that x(t) = 4 cos t and y(t) = 4 sin t satisfy the system of differential equations. Verify that x^2 + y^2 = 16. At t=0 (x(t), y(t)) gives the point (4, 0). Note that as t increases x decreases and y increases and the point (x(t), y(t)) travels around the circle of radius 4 in a counterclockwise direction. (You'll do more of this in Math 21a) Read the supplement 31.6 (pp. 1045 - 1047)
W Dec. 15 (MWF)
or
Th Dec 16 (TTh)
Solution 32  
Problem Set 33 In the supplement do #12 and 13 on p. 1050
Do #19,20, 21, 22 on Differential Equations Handout
Read the supplement 31.6 (pp. 1048 - 1049)
F Dec. 17 (MWF)
or
T Dec. 21 (TTh)
Solution 33 
Problem Set 34 Do #23, 24, 25, 26 on Differential Equations Handout
Do the handout on the supplements page entitled :last homework problem.
M Dec. 20 (MWF)
T Dec. 21 (TTH)t
Solution 35  


Math 1b, Fall, 2004, math1b@fas.harvard.edu, Department of Mathematics, Faculty of Art and Sciences, Harvard University