### Preparation for the final exam, Math 151:07-09, fall 2007

The final exam for sections 7, 8, and 9 of Math 151 will be given

### General background(extended discussion) Study & review material TV review session(12/14) Review session for students in this lecture (12/16)

General background
The final exam will be written principally by the course coordinator, not by your own lecturer (me). Please keep this in mind. Several students have written to me and talked to me about preparation for the final exam. So what follows is a mixture of what I have said and sent to them. The final exam is cumulative and will cover all of the course. But as I wrote earlier,

Although the exam will cover the entire course, it is likely that it will disproportionately overweight (there will be more coverage!) what we have done since the second exam: antiderivatives, definite integral, uses of the definite integral, methods of computing the definite integral, etc. There are several reasons for this disproportion: first, the material has not yet been tested, and second, for many reasons both theoretical and practical, this is the most important stuff in the course. If you wish to have a chance at doing well on the final exam, start now and do every suggested problem in the sections starting with 4.9.

The time you have for the course should rationally be devoted to preparing for the final exam. What follows are my recommendations for this preparation.

Do the textbook problems for the last chunk of the course. Two results are important for your further studies in math and in any applications: these are the definite integral and its computation with the Fundamental Theorem of Calculus (FTC), and the Mean Value Theorem (MVT). The MVT has already been covered in the course and in an exam (however, see below) so you've got to assume that essentially every aspect of FTC will be tested on the exam. So this means learning

1. Geometric meaning of the definite integral via areas (+ for over the axis, - for under the axis).
2. Computation using antiderivatives, especially using substitution.
3. Differentiating an integral with variable upper limit.
4. Approximation of the definite integral with Riemann sums.

ASSUME that EVERY question which was on one of my exams (and, more certainly, the course coordinator's review problems!) WILL actually appear on the final exam. BE SURE you can do clearly, correctly, and rapidly, all of the questions which have already appeared.

I don't believe that there is only a finite, short list of exam questions (it would be a rather ... inferior ... subject if that were true!) but I do believe that the instructors are human, and that the subject has certain rather prominent aspects. A consequence of both of these is that the instructors will ask about almost all of the prominent ideas in the course, and, since humans have habits (sigh!) probably they will tend to ask about them in the same or similar ways as has been done previously.

I also feel, personally, that an exam, especially in an introductory course such as 151, is not the place to show students novelties. So, seriously, I hope that in most of most of my exams, the problems or questions much like them should have been seen before. What the students write can be used as a valid measure of how well the material has been learned.

This may be too abstract for you as a strategy so let me give a specific example.

One important consequence of MVT is the following idea (two parts because there are two signs):
If f´>0 then f is increasing; if f´<0 then f is decreasing.
"We" (the instructors) would like to check if students understand and can use this. So on an exam, we might:

1. Give a formula for f and ask students to find intervals where f is increasing and where f is decreasing.
Skills tested: differentiation (taking the formula for f and getting a formula for f´), algebraic manipulation of f´ to find out its sign(s), and the MVT idea previously mentioned.
2. Give a formula for f´ and ask students to predict intervals where f is increasing and decreasing. Maybe graph a candidate f as a result.
Skills tested: algebraic manipulation of f´ to find out its sign (but usually less work than the previous method), the MVT idea, then implementation via graphing (to check if the student KNOWS what {in|de}creasing means graphically).
3. Give a graph of f´ and ask students to predict intervals where f is increasing and decreasing. Maybe again graph a candidate of f.
Skills tested: "Reading" a graph (not trivial at all). So this is asking if students can tell what {in|de}creasing means from the picture, and then use this information to discuss intervals. Then, again, go the other way, from the intervals of {in|de}crease to actively drawing a graph, and this asks for graphing ability.
I do think that Math 151 has a finite number of such ideas (although not a finite number of questions!). I was serious when I prepared the summaries of the subject matter before the first and second exams, and the summary for the last part of the course. I hope they these summaries show you what I think are the important ideas of Math 151.

Study & review material for the final exam
The course coordinator's review problems and answers covering material before the first exam A version of our first exam and some possible answers My outline of that portion of the course
The course coordinator's review problems and answers covering material before the second exam A version of our second exam and some possible answers My outline of that portion of the course
The course coordinator's review problems and answers for the final exam The formula sheet to be included with the final exam. My outline of the last portion of the course

Continuing comment about the formula sheet
Become familiar with what is on the sheet. Students who need to consult any formula sheets extensively tend to be students who are not adequately prepared. They generally don't do well.

Reminder
No texts, notes, or calculators may be used on this exam.

What you can do if you have questions or problems
Here are my suggestions. Work with other people and talk to them. Consult the textbook and look at the relevant worked out exercises. Consult the diary and see the problems worked out there. Consult the recitation instructor or the lecturer. The order of these suggestions is given in terms of how easily they can be done and how quickly (realistically!) useful help can be gotten.

TV review
There will be a televised final exam review for Math 151 on Friday, December 14 from 3 to 5 PM. Students can be part of the live audience in Livingston Learning Center, watch it on RU-TV, or watch it online. If they watch it live, they can call in with questions while the review is being conducted. RU-TV will replay the review several times before the exam and, it can be watched it on the internet any time. These problems will be the basis for the review, but other questions and problems may be considered.
The web page http://rutv.rutgers.edu/tutorials.shtml will have a link to the final exam review "live" and you should also be able to use that page to access the review later.

Review Session
I'll have a review session on Sunday evening, December 16, at 5 PM in SEC 206. This is not intended to be a substitute for your own work. You must prepare by doing homework problems, workshop problems, and the supplied review problems by yourself or with others. If I could do things by watching others, I would easily hit 50 major league home runs each year. Attendance at this session will not be adequate preparation for students who have done little work on their own.