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.

Course handouts

Adobe Acrobat format. You should be able to view and print from your browser.

Writing up workshop problems. A page of hints distributed in lecture of Sept. 10.
First part of syllabus. September 03 to September 24 (techniques of integration and applications).
Second part of syllabus. October 01 to October 19 (sequences and series).
Third part of syllabus. October 26 to November 05 (numerical methods and other applications).
Fourth part of syllabus. November 12 to November 30 (Parametric equations and polar coordinates).

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