Math 1b Syllabus
Course outline.
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 8.2 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.
Lecture schedule and homework and reading assignments.
The following is a list of the topics for each lecture together with
the corresponding reading and homework assignments. Note that a topic
listed for a given lecture might be started in the previous lecture or
finished in the following lecture. Also, sometimes I will explain
things which are not explained in the book. In general, you are
responsible for whatever is covered in lecture. For the reading
assigmments, be aware that math books are meant to be read slowly, not
like novels. To make sense of what the book is saying, you may need
to do some calculations on scrap paper, and you may need to read the
same section multiple times. Also remember that homework from a
Tuesday lecture is due on Friday three days later, and homework from a
Thursday lecture is due the following Monday.
Part 1: Integration.
- (Tues 1/17) Brief review of Math 1a and introduction to Math 1b.
- Reading: review chapters 1 through 6 as needed, especially
sections 3.2, 3.5, 5.3, and 5.5.
- Homework: There is no official
assignment. However it might be helpful to try some of the review
problems at the end of chapter 5.
- (Thurs 1/19) Basic techniques of integration: substitution and
integration by parts.
- Reading: section 7.1. Also review
Appendix D if necessary.
- Homework: 7.1: 3, 5, 9, 10,
19, 27, 33, 35, 45, 64.
- (Tues 1/24) Trigonometric integrals and trigonometric
substitutions.
- Reading: sections 7.2 and 7.3.
-
Homework: 7.2: 1, 3, 6, 7, 21, 39, 46, 55. 7.3: 1,
5, 13, 19, 27, 30, 31.
- (Thurs 1/26) Partial fractions and rationalizing substitutions.
-
Reading: sections 7.4 and 7.5.
- Homework: 7.4: 3, 7, 9, 21, 23, 25, 35, 43, 45,
49, 55, 57.
- (Tues 1/31) Approximate integration.
- Reading: section 7.7.
- Homework: 7.5: 1, 5, 13, 23, 33. 7.7: 1, 5, 13, 15,
21, 22, 29, 31. (You will need a calculator for
some of these.)
- (Thurs 2/2) Improper integrals.
- Reading: section 7.8.
- Homework: 7.8: 1, 5, 7, 11, 13, 21, 22, 23, 49, 53, 55,
59, 61.
- (Tues 2/7) Arc length and area of surfaces of revolution.
- Reading: sections 8.1 and 8.2. (We will not cover the rest of
chapter 8 but you might find it interesting.)
- Homework:
8.1: 7, 9, 11, 13, 30, 35. 8.2: 5, 9, 13, 15, 25, 29.
Part 2: Sequences and series.
- (Thurs 2/9) Sequences and limits.
- Reading: For Tuesday, review for the first midterm. For Thursday,
read section 11.1.
- Homework: There is no official assignment, but the review
problems at the end of chapter 7 may be helpful. Some practice
midterms will be posted here.
- (Tues 2/14) MIDTERM #1. Will cover the lectures up to and
including 2/7.
- Homework: 11.1: 5, 9, 11, 14, 18, 22, 26, 49, 59, 65,
67.
- (Thurs 2/16) Series: definition, basic examples and properties
- Reading: section 11.2.
- Homework: 11.2: 3, 5, 9, 15, 23, 27, 34, 38, 41, 49, 52, 65.
- (Tues 2/21) Integral test and estimates for sums.
- Reading: section 11.3.
- Homework: 11.3: 3, 5, 7, 12, 13, 15, 25, 30, 33.
11.4: 1, 2, 10, 11.
- (Thurs 2/23) Comparison test and alternating series
- Reading: sections 11.4 and 11.5
- Homework: 11.4: 20, 28, 32, 35, 37. 11.5: 2, 4,
10, 13, 15, 17, 25, 27, 32.
- (Tues 2/28) Absolute convergence; ratio and root tests.
- Reading: sections 11.6 and 11.7.
- Homework: 11.6: 3, 4, 7, 8, 16, 23, 25, 27. 11.7:
5, 8, 13, 14, 19, 34.
- (Thurs 3/2) Power series
- Reading: section 11.8
- Homework: 11.8: 3, 8, 9, 13, 15, 20, 27, 30, 31, 35, 39,
40. If you have a graphing calculator, you might have fun with 32-34,
but that's optional.
- (Tues 3/7) More power series
- Reading: section 11.9
- Homework: 11.9: 3, 5, 9, 11, 15, 23, 27, 29, 37, 38(a,b).
- (Thurs 3/9) Taylor series
- Reading: section 11.10
- Homework: 11.10: 5, 8, 11, 14, 27, 28, 38, 40, 46, 53,
55, 58.
- (Tues 3/14): More Taylor series
- Reading: sections 11.11 and 11.12.
- Homework: 11.11: 2, 5, 11, 13, 17, 19. 11.12:
25, 26, 27, 35.
Part 3: Ordinary differential equations.
- (Thurs 3/16) Review for the midterm.
- Reading: review for the second midterm.
- Homework: There is no official assignment, but the review
problems at the end of chapter 11 are highly recommended. Some practice
midterms will be posted here.
- (Tues 3/21) MIDTERM #2. Will cover the lectures up to and including 3/14.
- Homework: none. The next homework will be due Monday 4/3.
- (Thurs 3/23) Introduction to ordinary differential questions.
- Reading: sections 9.1 and 9.2
- Homework: 9.1: 1, 3, 7, 10, 12, 14. 9.2: 1, 3,
4, 5, 6, 10.
- (Tues 3/28 and Thurs 3/30) Spring break!
- (Tues 4/4) How to solve separable differential equations.
- Reading: section 9.3
- Homework: 9.3: 1, 3, 6, 7, 9, 11, 14, 33, 34(b), 38, 39.
- (Thurs 4/6) Modelling growth and decay.
- Reading: sections 9.4 and 9.5
- Homework: 9.4: 3, 7, 9, 20. 9.5: 7, 12, 15.
- (Tues 4/11) Linear first-order equations.
- Reading: section 9.6.
- Homework: 9.6: 1, 3, 5, 7, 9, 12, 15, 16, 17, 23, 25, 33.
- (Thurs 4/13) Complex numbers and complex exponentials.
- Reading: Appendix G.
- Homework: G: 9, 12, 16, 18, 25, 33, 37, 39, 41, 43, 45.
- (Tues 4/18) Homogeneous second order ODE's.
- Reading: section 17.1.
- Homework: 17.1: 1, 3, 5, 7, 9, 13, 17, 19, 20, 22.
- (Thurs 4/20) Inhomogeneous second order linear ODE's.
- Reading: section 17.2 (undetermined coefficients).
- Homework: 17.2: 1, 3, 5, 7, 9, 13, 15, 17.
- (Tues 4/25) More inhomogeneous equations.
- Reading: section 17.2 (variation of parameters).
- Homework: 17.2: 19, 20, 21, 22, 23, 25.
- (Thurs 4/27) Applications of linear second order ODE's.
- Reading: section 17.3.
- Homework: 17.3: 1, 2, 3, 5, 7, 9, 10, 12.
- (Tues 5/2) Solving differential equations using power series.
- Reading: section 17.4.
- Homework: 17.4: 1-10.
- (Thurs 5/4 and Tues 5/9) Review.
- Reading: review all
covered sections.
- Homework: there is no official assignment, but
I recommend trying the practice finals, as well as the review problems
from the book, and unassigned exercises from any sections of the book
that seem difficult.
- (Friday 5/19, 12:30-3:30, room 155 Dwinelle) Final exam.
- The final will cover the whole course, with a somewhat heavier
emphasis on the last third.
- (Saturday 5/20) Summer vacation!
Up to Math 1b home page.