Alexander Paulin 

Department of Mathematics
796 Evans Hall
University of California, Berkeley

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Calculus 1B (001 LEC) Spring 2016

Lectures: Mondays, Wednesdays and Fridays, 9am-10am, 155 Dwinelle Hall.
Office hours : MWF 2pm-4pm and TT 11am-12pm in 796 Evans Hall
Discussion sections: One hour every MWF. Here is a link with further details. You may only attend the discussion section for which you are enrolled.
Enrollment: For question about enrollment contact Thomas Brown.


Welcome to Calculus 1B! This a fantastic class with all sorts of interesting results and applications. There is a lot of material to cover so expect the pace to be quick. Everything related to the course will be on this website. We will not be using bCourses. There will be weekly homework (posted below) and a quiz in discussion section roughly every Friday. I have office hours everyday of the week so there should always be an opportunity to get my help if you need it. If you can't make any office hours, e-mail me and we'll find another time to meet. In addition to this I will be posting my own lecture notes on this website at the end of each week. You'll be able to link to them directly from the detailed syllabus below. I know 1B has a bit of a fearsome reputation. While some topics are undoubtedly challenging, if you work hard, attend the lectures and come to office hours you'll definitely do well. As I said this is a great class and I look forward to meeting you all on the 20th of January.

Discussion sections will begin on Wednesday the 20th of January.

Below you will find a link to homework 0. This is not really a homework assignment, instead it's a collection of warm-up exercises to remind you of the basics from 1A (or elsewhere). Theses are the core things you need to understand before begininning the course proper. I strongly advise you to do them. If there are any exercises you struggle with, make sure you spend some time revising those topics. The first proper homework (homework 1) with be posted on Friday the 22nd of January and will be due in section the following Friday.

Make sure to read the course policy and the detailed syllabus below.


The textbook for this course is: Stewart, Single Variable Calculus: Early Transcendentals for UC Berkeley, 8th edition (ISBN: 9781305765276, Cengage).

This is a custom edition containing chapters 1-9, 11 and 17 of Stewart's "Calculus: Early Transcendentals", 8th edition. The regular edition is also fine, it just contains extra chapters covered in math 53. The 7th edition is also acceptable, but you will need to watch for differences in the numbering of assigned homework problems. In chronological order, we'll cover the material in chapters 7, 8, 11, 9 and 17.


The Student Learning Center provides support for this class, including study groups, review sessions for exams, and drop-in tutoring. This is a fantastic resource, I definitely recommend you take advantage of it.

Grading and course policy

Homework 10%
Quizzes 10%
First Midterm 20%
Second Midterm 20%
Final Exam 40%

If the lowest (curved) midterm score is less than the (curved) final score, then it will be replaced by your final score. This grading policy allows you to miss one midterm without serious consequences. For example, if you scored 100% on everything except the second midterm, which you missed, then you would still get an overal score of 100%. You must, however, sit the final exam. It is your responsibility to make sure you have no schedule conflicts in exam week. Unless there are truly exceptional circumstances, there will be no make-up exams.

For more detailed information make sure to read the course policy.


Homework assignments are due each Friday in section. They will be posted here along with solutions. Your two lowest homework scores will be dropped. For more detailed information see the course policy.

Homework 0 (7th Edition) and Solutions (7th Edition).

Homework 1 and Solutions 1

Homework 2 and Solutions 2

Homework 3 and Solutions 3

Homework 4 and Solutions 4

Homework 5 and Solutions 5

Homework 6 and Solutions 6

Homework 7 and Solutions 7

Homework 8 and Solutions 8

Homework 9 and Solutions 9

Homework 10 and Solutions 10

Homework 11 and Solutions 11

Homework 12 (Not to be submitted) and Solutions 12


Quizzes will take place on Friday roughly every week in discussion section. They will last about 15 minutes, be of a similar difficulty to the homework and cover material from the preceding week. Your two lowest scores will be dropped from your grade. Here is the quiz schedule:

1 Week 2 (1/25 - 1/29)
2 Week 3 (2/1 - 2/5)
3 Week 4 (2/8 - 2/12)
4 Week 7 (2/29 - 3/4)
5 Week 8 (3/7 - 3/11)
6 Week 9 (3/14 - 3/18)
7 Week 12 (4/4 - 4/8)
8 Week 13 (4/11 - 4/15)
9 Week 14 (4/18 - 4/22)
10 Week 15 (4/25 - 4/29)

For more detailed information see the course policy


There will be two midterms, the first on Friday February 19 and the second on Monday March 28. The final exam will be on Monday May 9 (7pm - 10pm) .

For more detailed information see the course policy

Practice First Midterm: Good Solutions and Bad Solutions.

Here two past exams: first midterm 1 and first midterm 2. Here are solutions: solutions 1 and solutions 2.

Here are the solutions to the first midterm: solutions. Here are some statistics about the first midterm: Midterm 1 Stats (001).

Here are three practice exams. Because they are from Fall 2014 they do not cover power series and Taylor series. Your exam will cover this material. For practice on those I strongly advise you to do the appropriate homework problems and carefully study my solutions. Here is the formula sheet that will be on the first page of the exam.

Practice Second Midterm and solutions.

Here two past exams: second midterm 1 and second midterm 2. Here are solutions: solutions 1 and solutions 2.

Here are the solutions to the second midterm: solutions. Here are some statistics about the second midterm: Midterm 2 Stats (001).

Here are three practice final exams. Because they are from Fall 2014 they do not have any variation of parameters questions. Your exam will cover variation of parameters. For practice on this I strongly advise you to do the appropriate homework 12 problems and carefully study my solutions. Here is the formula sheet that will be on the first page of the exam.

Practice Final Exam and solutions.

Here two past exams: Final 1 and Final 2. Here are the solutions: solutions 1, solutions 2.

Syllabus and Schedule

In Math 1A or elsewhere, you studied functions of a single variable, limits, and continuity. You learned about derivatives, which describe how functions change, and which can be used to help find maxima and minima of functions. You also learned about integrals which describe the aggregate behavior of a function over an interval, such as the area under a curve or the average of a varying quantity. The derivative and the integral are tied together in the fundamental theorem of calculus, one version of which relates the integral of the derivative of a function over an interval to the values of the function at the endpoints of the interval.

In this course we will continue the study of calculus in three parts as follows:

  • The first part of the course is about techniques of integration (sections 7.1 to 7.8 of the book). As you should already know, differentiation is relatively straightforward: if you know the derivatives of elementary functions, and rules such as the product rule and the chain rule, then you can differentiate just about any function you will ever come across. Integration, on the other hand, is hard. Sometimes it is even impossible to integrate a given function explicitly in terms of known functions. We will introduce a collection of useful tricks with which you can integrate many functions. The hard part is to figure out which trick(s) to use in a given situation. For integrals which we cannot evaluate explicitly, we will learn how to find good approximations to the answer.
  • The second part of the course is about sequences and series (chapter 11 of the book). This can be regarded as the general theory of approximating things. This part of the course is subtle and involves new ways of thinking. It may be a lot harder than the first part, especially if you have seen some of the first part before.
  • The third part of the course is an introduction to ordinary differential equations (chapters 9 and 17 of the book). Here one tries to understand a function, given an equation involving the function and its derivatives. ("Ordinary" means that we consider functions of a single variable. Functions of several variables enter into "partial" differential equations, which you can learn about in a more advanced course.) The theory of differential equations is perhaps the most interesting part of calculus, is the subject of much present-day research, and has many real-world applications. Our study of differential equations will make use of most of the calculus we have done so far.

Here is the lecture schedule for the course:

WhenWhat Where
Week 1 (1/19 - 1/22) Introduction and Reminders 1 - 5
Integration by Parts 7.1
Integration of Rational Functions 7.4
Week 2 (1/25 - 1/29) Trigonometric Integrals (video) 7.2
Trigonometric Substitution (video) 7.3
Week 3 (2/1 - 2/5) Strategies of Integration (video ) 7.5
Approximate Integration (video 1,video 2) 7.7
Improper Integrals (video) 7.8
Week 4 (2/8 - 2/12) Arc Length (video)8.1
Area of a Surface of Revolution (video) 8.2
Probability (video) 8.5
Week 5 (2/16 - 2/19) Review (video)
First Midterm (on 2/19)
Week 6 (2/22 - 2/26) Sequences (video) 11.1
Series (video 1, video 2 ) 11.2
The Integral Test and Estimates of Sums (video) 11.3
Week 7 (2/29 - 3/4) Comparison Tests (video) 11.4
Alternating Series (video 1, video 2 ) 11.5
Absolute Convergence and the Ratio and Root Tests (video) 11.6
Week 8 (3/7 - 3/11) Strategy for Testing Series ( video ) 11.7
Power Series (video1, video 2) 11.8
Representations of Functions as Power Series (video) 11.9
Week 9 (3/14 - 3/18) Taylor and Maclaurin Series (video 1, video 2) 11.10
Week 10 (3/21 - 3/25) Spring Break
Week 11 (3/28 - 4/1)Second Midterm (on 3/28)
Modelling with Differential Equations (video)9.1
Direction Fields (video)9.2
Week 12 (4/4 - 4/8)Separable Equations (video) 9.3
Models of Population Growth (video) 9.4
Linear Equations (video) 9.5
Week 13 (4/11 - 4/15) Introduction to the Complex Numbers (video) Appendix H
Second-Order Homogeneous Linear Equations (video 1,2) 17.1
Week 14 (4/18 - 4/22) Second-Order Non-homogeneous Linear Equations (video 1, video 2, video 3) 17.2
Week 15 (4/25 - 4/29) Series Solutions (video 1, video 2) 17.4
Week 16 (5/2 - 5/6) Review (video 1, video 2)
Week 17 (5/9 - 5/13) Final Exam: MONDAY, MAY 9, 7-10Pm