![Page 1: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/1.jpg)
Welcome to Math 1bCalculus, Series, and
Differential EquationsThomas W. Judson (Course Head)
Spring 2006
![Page 2: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/2.jpg)
About Math 1b
![Page 3: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/3.jpg)
About Math 1b
Speaking the language of modern mathematics requires fluency in infinite series, integration, and differential equations.
![Page 4: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/4.jpg)
About Math 1b
Speaking the language of modern mathematics requires fluency in infinite series, integration, and differential equations.We will learn to model practical situations using integrals and differential equations
![Page 5: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/5.jpg)
About Math 1b
Speaking the language of modern mathematics requires fluency in infinite series, integration, and differential equations.We will learn to model practical situations using integrals and differential equationsWe will learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations
![Page 6: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/6.jpg)
About Math 1b
Speaking the language of modern mathematics requires fluency in infinite series, integration, and differential equations.We will learn to model practical situations using integrals and differential equationsWe will learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equationsWe will develop both conceptual understanding and the ability to apply it.
![Page 7: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/7.jpg)
Learning Objectives
![Page 8: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/8.jpg)
Learning Objectives
Integration and Applications of the Definite Integral—Techniques of integration, numerical integration, areas, volumes, arc length, applications to the natural sciences and economics.
![Page 9: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/9.jpg)
Learning Objectives
Integration and Applications of the Definite Integral—Techniques of integration, numerical integration, areas, volumes, arc length, applications to the natural sciences and economics.
Infinite Sequences and Series—Sequences, series, testing for convergence, power series, and Taylor series.
![Page 10: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/10.jpg)
Learning Objectives
Integration and Applications of the Definite Integral—Techniques of integration, numerical integration, areas, volumes, arc length, applications to the natural sciences and economics.
Infinite Sequences and Series—Sequences, series, testing for convergence, power series, and Taylor series.
Differential Equations—Modeling, differential equations from a numerical, an analytical, and a geometrical approach, systems of differential equations
![Page 11: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/11.jpg)
The Textbook
James Stewart. Single Variable Calculus: Concepts & Context, third edition. Brooks/Cole, Belmont CA,
2005.ISBN 0-534-41022-7.
![Page 12: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/12.jpg)
Calculators
![Page 13: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/13.jpg)
Calculators
Any serviceable graphing calculator
![Page 14: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/14.jpg)
Calculators
Any serviceable graphing calculator
If you plan to purchase a graphing calculator, we recommend at TI-84, TI-86, or TI-89
![Page 15: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/15.jpg)
Calculators
Any serviceable graphing calculator
If you plan to purchase a graphing calculator, we recommend at TI-84, TI-86, or TI-89
Calculators are allowed on homework but not on exams
![Page 16: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/16.jpg)
Techniques of Integration
•
!u dv = uv !
!v du
•
!sin2 ! d!
•
!1
x"
4x2 ! 9dx
•
!5
(2x + 1)(x ! 2)dx
Predator-Prey Systems
dx
dt= "x ! #xy
dy
dt= !$y + %xy
Tacoma Narrowsy!! + "y! + #y + $y+ = F (t), (1)
where y+ = max(y, 0).
Resonancex!! + &2
0x = A cos &0t.
1
![Page 17: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/17.jpg)
Applications of Integration
How much work is done when pumping out all of the full contained in cylindrical tank?
![Page 18: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/18.jpg)
Sequences and Series
•
!u dv = uv !
!v du
•
!sin2 ! d!
•
!1
x"
4x2 ! 9dx
•
!5
(2x + 1)(x ! 2)dx
Predator-Prey Systems
dx
dt= "x ! #xy
dy
dt= !$y + %xy
Computing &
&2
6= 1 +
1
22+
1
32+
1
42+ · · ·
Representing Functions with Power Series
sin x = x !x3
3!+
x5
5!!
x7
7!+ · · ·
Resonancex!! + '2
0x = A cos '0t.
1
![Page 19: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/19.jpg)
Differential Equations
Brief Article
The Author
January 29, 2006
Spring-Massmx!! + bx! + kx = 0
Predator-Prey Systems
dx
dt= !x ! "xy
dy
dt= !#y + $xy
Tacoma Narrowsy!! + !y! + "y + #y+ = F (t), (1)
where y+ = max(y, 0).
Resonancex!! + %2
0x = A cos %0t.
1
![Page 20: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/20.jpg)
Prerequisites for Math 1b
![Page 21: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/21.jpg)
Prerequisites for Math 1b
A satisfactory score on the Math Placement Exam
![Page 22: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/22.jpg)
Prerequisites for Math 1b
A satisfactory score on the Math Placement Exam
Math 1a
![Page 23: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/23.jpg)
Prerequisites for Math 1b
A satisfactory score on the Math Placement Exam
Math 1a
Math Xa and Xb
![Page 24: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/24.jpg)
To Get a Second Opinion
![Page 25: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/25.jpg)
To Get a Second Opinion
If you still need advice about which math course to take, see a math advisor
![Page 26: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/26.jpg)
To Get a Second Opinion
If you still need advice about which math course to take, see a math advisor
http://abel.math.harvard.edu/sectioning/index.html
![Page 27: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/27.jpg)
To Get a Second Opinion
If you still need advice about which math course to take, see a math advisor
http://abel.math.harvard.edu/sectioning/index.html
The Online Placement Exam
![Page 28: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/28.jpg)
Sectioning
![Page 29: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/29.jpg)
Sectioning
Math 1b is taught in small sections
![Page 30: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/30.jpg)
Sectioning
Math 1b is taught in small sections
You must section by 12:00 PM on Thursday, February 2
![Page 31: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/31.jpg)
Sectioning
Math 1b is taught in small sections
You must section by 12:00 PM on Thursday, February 2
Sectioning directions can be found at http://abel.math.harvard.edu/sectioning/index.html
![Page 32: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/32.jpg)
GradingComponent Percentage
Homework 20%
Techniques of Integration Exam
10%
Midterm I 20%
Midterm II 20%
Final Exam 30%
![Page 33: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/33.jpg)
Exam Dates
![Page 34: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/34.jpg)
Exam Dates
Techniques of Integration Exam—Wednesday, February 22 at 6–7 PM in Science B
![Page 35: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/35.jpg)
Exam Dates
Techniques of Integration Exam—Wednesday, February 22 at 6–7 PM in Science B
Techniques of Integration Exam Retest—Tuesday, February 28 at 8–9 PM in Science B)
![Page 36: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/36.jpg)
Exam Dates
Techniques of Integration Exam—Wednesday, February 22 at 6–7 PM in Science B
Techniques of Integration Exam Retest—Tuesday, February 28 at 8–9 PM in Science B)
Midterm I—Tuesday, March 14 at 7–9 PM in Science B
![Page 37: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/37.jpg)
Exam Dates
Techniques of Integration Exam—Wednesday, February 22 at 6–7 PM in Science B
Techniques of Integration Exam Retest—Tuesday, February 28 at 8–9 PM in Science B)
Midterm I—Tuesday, March 14 at 7–9 PM in Science B
Midterm II—Tuesday, April 18 at 7–9 PM in Science B
![Page 38: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/38.jpg)
Exam Dates
Techniques of Integration Exam—Wednesday, February 22 at 6–7 PM in Science B
Techniques of Integration Exam Retest—Tuesday, February 28 at 8–9 PM in Science B)
Midterm I—Tuesday, March 14 at 7–9 PM in Science B
Midterm II—Tuesday, April 18 at 7–9 PM in Science B
Final Exam—Tuesday, May 23
![Page 39: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/39.jpg)
Homework
![Page 40: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/40.jpg)
Homework
Everyone does the same homework
![Page 41: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/41.jpg)
Homework
Everyone does the same homework
There are two homework schedules—one for MWF and one for TuTh classes
![Page 42: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/42.jpg)
Homework
Everyone does the same homework
There are two homework schedules—one for MWF and one for TuTh classes
Homework is due at the beginning of class
![Page 43: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/43.jpg)
Homework
Everyone does the same homework
There are two homework schedules—one for MWF and one for TuTh classes
Homework is due at the beginning of class
We drop your three lowest scores
![Page 44: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/44.jpg)
Pre-Reading Questions andPre-Class Surveys
![Page 45: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/45.jpg)
Pre-Reading Questions andPre-Class Surveys
Pre-reading problems are assigned for each class period
![Page 46: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/46.jpg)
Pre-Reading Questions andPre-Class Surveys
Pre-reading problems are assigned for each class period
Pre-reading problems are marked with an asterisk
![Page 47: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/47.jpg)
Pre-Reading Questions andPre-Class Surveys
Pre-reading problems are assigned for each class period
Pre-reading problems are marked with an asterisk
Pre-Class Surveys are assigned for each class period (Q&A Tool 2.0)
![Page 48: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/48.jpg)
Pre-Reading Questions andPre-Class Surveys
Pre-reading problems are assigned for each class period
Pre-reading problems are marked with an asterisk
Pre-Class Surveys are assigned for each class period (Q&A Tool 2.0)
Pre-Class Surveys are 5% of your homework grade
![Page 49: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/49.jpg)
Take Advantage of the System
![Page 50: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/50.jpg)
Take Advantage of the System
Each Math 1b section has an undergraduate course assistant
![Page 51: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/51.jpg)
Take Advantage of the System
Each Math 1b section has an undergraduate course assistant
CAs hold 90 minute problem sections each week
![Page 52: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/52.jpg)
Take Advantage of the System
Each Math 1b section has an undergraduate course assistant
CAs hold 90 minute problem sections each week
TF Office Hours
![Page 53: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/53.jpg)
Take Advantage of the System
Each Math 1b section has an undergraduate course assistant
CAs hold 90 minute problem sections each week
TF Office Hours
New location for the MQC
![Page 54: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/54.jpg)
1. Which of the following are equal to
!
5
1
ln x
xdx?
Please circle all of the correct answers. You do not need to justify your solution.
(a)5
"
i=1
ln xi
xi
!x, where !x = 4/n and xi = 1 + i!xi.
(b) limn!"
5"
i=1
lnxi
xi
!x, where !x = 4/n and xi = 1 + i!xi.
(c) limn!"
n"
i=1
lnxi
xi
!x, where !x = 4/n and xi = 1 + i!xi.
(d) (ln 5)2/2
(e)1
2
#
ln 5
5
$2
!1
2(ln 1)2
(f)1
52!
1
12
(g) ln(ln(5)) ! ln(ln 1))
(h)ln 1
1+
ln 2
2+
ln 3
3+
ln 4
4
2
![Page 55: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/55.jpg)
2. Put the following in ascending order (with “=” or “<” signs between each expression.You do not need to justify your solution. [Hint: Think about which expressions arepositive, which are negative, and which are zero. A picture may be helpful.]
(a)
!
6
2
ln t dt
(b) ln 2 + ln 3 + ln 4 + ln 5
(c) ln 3 + ln 4 + ln 5 + ln 6
(d) zero
(e) ln(2/6)
(f) limh!0
ln(2 + h) ! ln 2
h
3
![Page 56: Welcome to Math 1b Calculus, Series, and …abel.math.harvard.edu/archive/1b_spring_06/Course...Calculus, Series, and Differential Equations Thomas W. Judson (Course Head) Spring 2006](https://reader033.vdocuments.site/reader033/viewer/2022060213/5f054ec07e708231d4125145/html5/thumbnails/56.jpg)
Important Addresses
Thomas W. Judson (Course Head) [email protected]
Course Web Site http://www.courses.fas.harvard.edu/~math1b/
Sectioning Directions http://abel.math.harvard.edu/sectioning/index.html