Math 152:10-12, Fall 1999
General announcements
Office hours in CHM-207B (Douglass Campus) have ended for the semester.
The final exam was held on Thursday, December 16, 4--7 PM in
LOR-020. All students who were expected were present. Information
about grading progress is here. A calculator
was found after the exam. If you think you left one there, contact Paul Dreyer giving details.
Useful links
- Main course
page for 151 and 152. Follow this link and select "Old exams
(152)" from the menu on the left side for sample exams, including the
"Review final exam" for this semester, although the local copy
mentioned above may be easier to find.
Directory
Electronic addresses are collected here for your
convenience. You should seek help as soon as you notice a need. An
exam is always coming soon.
Course handouts
Adobe Acrobat format. You should be able to
view and print from your browser.
Copies of slides shown in lecture.
A two-per-page format is being used here, and typographical errors noticed
in lecture will be corrected, but no further editing will be done.
I also include a four-per-page format. This will be better if you
want to print the slides, but seems less useful for viewing.
- Lecture 1.Sections 5.5 and 6.1
with graphs of ex. 6.1.11 and ex. 6.1.23, and alternate form for printing.
- Lecture 2.Sections 6.2 and
6.3, andalternate form for printing.
- Lecture 3.Sections 6.5 and 7.1
with graphs of ex. 6.5.7 and ex. 6.5.9, and alternate form for printing.
- Lecture 4. Section 7.2 , and
alternate form for printing. (It was
planned to also cover Section 7.3, but this took longer than
expected. Only slides shown in lecture are included here.)
- Lecture 5. Sections 7.3 and 7.4, and
alternate form for printing. Lecture
planned for September 17, but washed out then and postponed until
September 20.
- Lecture 6. The rest of chapter
7, and alternate form for printing.
- Lecture 8. Report on exam
(which took the place of lecture 7) and introduction to chapter 10 and
alternate form for printing.
- Lecture 9. Sections 10.1 and
10.2, and alternate form for printing.
Only pages 1 thru 5 were shown in lecture, but all 8 pages are
included here for a preview of sections 3, 4, and 6 of chapter 10.
- Lecture 10. Sections 10.3, through
10.6, and alternate form for printing.
This completes work with numerical series prior to study of power
series.
- Lecture 11. List of problems
from earlier sections shown at blackboard, and topics from Sections
10.8 and 10.9, and alternate form for
printing.
- Lecture 12. List of problems
from Section 10.9 shown at blackboard, topics and problems from Section
10.10, and material from Section 10.11 that was prepared for the
lecture, but not shown; and alternate form for
printing.
- Lecture 13. End of Chapter 10,
and alternate form for printing.
- Lecture 15. Section 7.8,
Numerical integration, and alternate form
for printing.
- Lecture 16. Section 8.1,
Differential equations, and alternate form
for printing.
- Lecture 17. Section 8.1, arc
length, with a look back at volume integrals
and alternate form for printing.
- Lecture 18. Looking at an
integral using both numerical integration and series methods, and alternate form for printing.
- Lecture 20. Introduction to
parametric equations, and alternate form
for printing. The graphs are availiable individually: a cycloid; an
epicycloid; exercise 9.1.3; exercise 9.2.1;
exercise 9.2.7; an involute showing
some of the tangent lines that appear in its definition. These graph
were drawn using a program called gnuplot. This program is
available on eden, so you can experiment with it.
- Lecture 21. Second derivatives
and areas using parametric equations, and alternate form for printing.
- Lecture 22. Arc length for
parametric curves, and alternate form for
printing. The graphs are available individually: exercise 9.3.9;
exercise 9.3.17 an epithrochoid; exercise
9.3.18 the Cornu spiral from t=0 (the origin at the lower left) to
4 (inside the spiral), the alternating horizontal and veritcal
tangents are the points where t is the square root of an integer (this
is easy to prove -- try it).
- Lecture 23. Polar coordinates,
and alternate form for printing. The
graphs are available individually: a
cardioid; a limacon;
another limacon; one more limacon; a rose with 3 leaves; a
rose with 4 leaves; a rose with 20
leaves (this was not shown in lecture since the mesh used by the
graphing program was not fine enough to give a smooth curve); one of
the curves from exercise 86; and one of the
curves from exercise 87.
- Lecture 24. Areas and arc
length in polar coordinates, and alternate
form for printing
- No new slides were shown in the lecture on November 30.
An incomplete example from lecture is finished here with the aid of
Maple to do the calculations correctly and format the results. It is
available in a color version for viewing on
screen and a black and white version that
may print better. Two graphs were prepared that were not shown since
we were busy with other topics, but you may want to look at them: problem 15 and problem
23 from the review problems for chapter 9 on page 571.
Workshops
- #0 Review of 151, Sep. 02.
- #1 Substitutions, Areas,
Volumes. Sep. 09.
- #2 Areas, Volumes, Averages,
Integration by parts. Distributed in lecture on Sep. 14 for class of
Sep. 16. Although class cancelled because of weather, problem #1
should have been prepared for class of Sep. 23, and must be submitted
no later than class of Sept. 30.
- #3 Integrals, mostly
trigonometric. Sep. 23. Used as exam review.
- #4 Review chapter 7 and preview
chapter 10. Sep. 30. Problem 3 to be prepared for Oct. 14 based on
discussion of Oct. 07.
- #5 Chapter 10. Oct. 07. Problem
4 to be prepared for Oct. 14.
- #6 Sections 10.5 through
10.8. Oct. 14. Problem 3 to be prepared for Oct. 21, with additional
part (a') For each string of inequalities, show that they hold true
for n=4.
- #7 Practice exam. Distributed
in lecture on Oct. 19. Be prepared to discuss these problems on Oct. 21.
- #8 Series and numerical
methods. Oct. 28. Problem 2 to be prepared for Nov. 04.
- #9 Sections 8.1 and 8.2 with
use of numerical integration. Problem 1 or problem 4 is to be prepared
for Nov. 11.
- #10 More on Taylor Series and
an introduction to other graphing modes on the calculator.
Nov. 11. No problems need to be prepared from this set.
- #11 Parametric curves. Problem
2 is to be prepared.
- #12 Polar curves. Problem
2 is to be prepared.
Exam schedule:
- Tuesday, September 28 (chapters 6 and 7) Problem
distribution planned to be 35% applications of integrals, 65%
techniques of integration. Techniques were all introduced by section
7.4, but section 7.6 provides useful practice for an exam because it
gives integrals without telling you which technique to try first.
Trigonometric identities and reduction formulas will be given.
Although you will not be allowed your own formula sheet, you may write
formulas that you are concerned that you might forget in a convenient
place on the exam. At this point, you should be fluent in the use of
basic differentiation formulas, but may need help with organizing
integration by parts or setting up integrals to calculate volumes.
Trying to fit volume problems into the model of applying a
formula may not be the best way to go from a statement of the
problem to the first step in the solution. There are simple pictures
that guide the formulation of these problems, but not everyone will
see such a picture the same way. Since integration by parts is just a
variant on the product rule of differentiation, no statement as a
formula should be necessary.
- Friday, October 22 (chapter 10) Sections 10.1 through
10.10. The terms "sequence" and "series" have precise meanings that
must not be confused. Sequences arise in many different ways and
concentrate attention on role of limits. They apply to series through
the sequence of partial sums of the series. It is helpful
that we speak of the "limit of a sequence" and "sum of a series".
Some problems will aim at assuring that you have mastered the
vocabulary by being fairly easy in the correct interpretation while
revealing any misinterpretation. Unfortunately, other limits
associated with the terms of the series appear in the main
theorems of the subject, and good behavior of both sequences and
series are referred to as convergence. In early sections of
the chapter, convergence is established by giving the limit, but the
more important results are those that prove convergence by some sort
of comparison test. You can expect several problems that
depend on these tests. Moving from numerical series to power series,
questions about convergence turn into finding the interval of
convergence. Behavior at the endpoints of this interval is
always special, and leads to many of the numerical series that were
important examples earlier in the chapter. Doing algebra or calculus
with power series whose sums are known to produce new series leads to
a wide variety of problems. Taylor's theorem appears mainly to
introduce the series for the exponential and trigonometric functions,
but it can be used whenever repeated differentiation of a function is
possible. Since the theorem only asserts that repeated differentiation,
followed by evaluation at zero, gives the expected result, it will
mostly be in the background of all work, but not explicit in any
problem. By popular demand, the Maclaurin series for the sine and
cosine will be given with the exam, but no other formulas will be
supplied. The emphasis in this work is to build an intuition about
the convergence of sequences and series, so details that force you to
look at technical considerations of the theorems make the subject
appear more difficult. If a problem appears difficult, you have
probably gotten distracted by an irrelevant detail.
- Tuesday, November 09 (deferred topics and chapter 8)
Topics to be covered are: volumes (since they were not done well on
exam 1), power series (the weak point of exam 2), numerical
integration, separable differential equations, arc length. The
volumes that we concentrate on in this course are volumes of figures
formed by rotating a plane region about one of the coordinate axes.
The method of disks integrates the area of a cross-section using a
variable along the axis of rotation. Cross sections are
circles (or the region between two circles, whose area is the
difference of the areas of the bounding circles), and the area of a
circle is pi times the square of the radius. To use this, you need
the expression for the radius as a function of the variable of
integration. The method of shells integrates with respect to the
variable giving the distance from the axis of rotation. This
integrand is 2pi times the product of the distance from the axis and
the height of the region at that distance. To use this, the height
needs to be found as a function of the variable of integration. Both
of these formulas can be reconstructed by interpreting the integrand
as the area of the surface at a particular value of the variable of
integration. The source of the material on power series is spread over
several sections of the text. After using the ratio test to identify
the radius of convergence of a series, it was noted that algebra,
functional substitution, or calculus applied to series gave results
that were valid in the interval of convergence. This allowed the
geometric series to by used to produce series for ln(1+x) or
arctan(x). Multiplication of series requires that you think of the
series as long polynomials and consider all product of one term from
the first factor with one term from the second factor. Fortunately,
only finitely many terms are needed to get all contributions to the
terms of fixed degree in the product. Taylor's theorem gives a
formula for the coefficients of the series representing a function,
and an expression for the difference between the function an the
partial sums. The formula for the coefficients is the only formula
consistent with term-by-term differentiation and the interpretation of
the constant term as the value of the function at zero. (In order for
the general formula to be interpreted correctly, it is necessary to
define 0!=1. This is consistent with all properties of factorials,
although it can be disturbing if you think that n! must have a factor
of n.) An interesting check on the relation between Taylor's formula
and the multiplication of series is to compare the results obtained by
these two approaches to the coefficient of x in the product of two
Taylor series. Numerical integration uses an average of function
values to approximate the average of a function on an interval. To
get the integral, you need to multiply by the length of the interval.
Simpson's rule is worth knowing, since it usually gives better results
than the midpoint or trapezoidal rule with very little more effort.
In practice, between 50 and 100 points will be used, although all
features of the method can be illustrated with fewer
than 10 points. Separable differential equations are easily solved.
Typically, the notation of differentials is used to turn the equation
into a couple of integrals that express the general solution when you
add C to one of them. You can also find a particular solution that
satisfies a given initial condition. Arc length leads to a
special integral. In the examples used in this course, some trick
allows the integral to be evaluated in closed form. Usually, these
integrals define new functions.
- Friday, December 03 (chapter 9) "Key topics" 1 through 11
(except number 5) at the top of the chapter 9 review section give a
good idea of the topics covered in this segment of the course. One
topic not mentioned that does appear in exercises is the determination
of the second derivative of y with respect to x for parametric curves.
The key to such problems is that the chain rule tells you that the
derivative with respect to t of anything is the derivative of that
quantity with respect to x multiplied by dx/dt. With a parametric
description, derivatives with respect to t are easy to find since
everything is expressed in terms of t, so derivatives with respect to
x must be found by solving for them in this chain rule equation.
Remember to simplify expressions first. That makes it easier to
organize your work around solving the chain rule equation than it
would be to write a formula for the solution that you try to
substitute given expressions for x and y into. On this exam, you will
not be asked to show any graphs, but you should be able to get graphs
easily with your calculator and use the graph to guard against
accepting work flawed by serious errors in algebra or calculus. The
"line integrals" mentioned in lecture can simplify some of the area
calculations -- but not so much that you should try them without
practice. A plodding method that you understand will get you just as
good an answer, and you will be able to check it, so it may actually
save time. The actual calculation will not be very different. For
parametric curves, it is usually necessary to treat the top and bottom
of the region separately instead of integrating the height of the
region as a function of x as was done when "area between curves" was
first discussed. Our chief examples of parametric curves are: (1) x
and y are polynomials of low degree, as in review problems 17, 22 and
23; cycloids and trochoids, obtained by rolling a circle on a line or
another circle; involutes, obtained by following a pen at the end of a
string wrapped around a circle. These names may be help to recall
features of calculations done in problems in which these curves
appeared. Arc length problems also lead to integrals, although very
few of these can be calculated in closed form. Even those that can be
evaluated may require a trick, so you should be review the problems
assigned as homework. Polar coordinates appear as an alternate way to
describe points in the plane, and the equations for converting between
rectangular and polar coordinates should be practiced. These
convert coordinates of points, and can be used to transform some
equations of curves from one system to the other. Again, look at
examples and exercises to find the main examples. The fact that polar
coordinates are not unique causes some difficulties. This exam will
not require you to work deeply with this, but you should be aware of
the difficulties so you don't assume that you have made an error when
you get an answer in an unexpected form. The main examples of polar
curves are: roses; limacons (including the cardioid). These are always
given in polar form, since that forms gives the most useful geometric
information for drawing the curve. Converting to rectangular form
isn't difficult, but it is never done because the rectangular
form tells you less about the curve than the polar form. From one
point of view, a polar equation can be considered as a parametric form
in which the parameter is theta, the angle from the positive x-axis to
the point on the curve. All calculus problems about polar curves can
be done from this point of view, but formulas for area and arc
length are derived that use special considerations. These formulas
are easy to remember because there are pictures that identify areas of
sectors as (1/2)r^2 (d theta), and small distances as the square root of
dr^2 + r^2 (d theta)^2. While these pictures don't prove the
formulas, they help to remember them and guide their use.
- (Final) Thursday, December 16, 4 -- 7 PM, in LOR-020. You
may bring a one page (two sides) formula sheet for use during the exam.
Mail to:
bumby@math.rutgers.edu
Last updated: December 18, 1999