2007 November 13 / E-Mail Me
Carleton College Math 111 01-02, Fall 2007, Prof. Joshua R. Davis
Here I give you some tips about the final exam. You should also check out my little essay How To Study Calculus, if you haven't already. That page is about general study strategies. This page is about specific material that we have covered in our course. There is no coherent thesis or grand principle at work here; it's just some random thoughts that you might find useful. Disclaimer: You should not take this page as definitive; it does not constitute a promise or legal contract about the content of the exam.
Every calculus student should be proficient at computing derivatives. In most cases this is a routine and mechanical task. If you know the basic differentiation rules (sum rule, constant multiple rule, product rule, and chain rule) and the derivatives of the most popular functions (k, xn, ax, logb x, sin x, cos x — are there any I'm forgetting?) then you should be set. For example, I consider Exam 1 #1A and #1B to be routine differentiation problems. You should be able to do these and similar problems without really thinking. Practice.
Occasionally you run into a differentiation problem that can't be solved with the simple rules. Then you have to resort to the ultimate weapon — the definition of the derivative. The first few steps are mechanical, but the later steps may require some ingenuity or some skill with limits. One of my favorite examples is 2.7 #52. You should also try 2.7 #51.
On the final exam, you can expect several differentiation questions, to certify your basic competence.
Every calculus student should be proficient at computing antiderivatives. Antidifferentiation is not as straightforward as differentiation. First, many simple-looking functions don't have any simple antiderivative at all. Even for those that do, you might not have learned enough techniques in this course to find the antiderivatives. (For that, take Math 131.) The only major "technique" that we have learned is substitution, but it is not always easy to guess the best substitution; you have to make a few guesses.
On the final exam, you can expect several antidifferentiation questions, to certify your basic competence. Some of these may be posed as integration questions, where to compute an integral you first antidifferentiate and then apply the fundamental theorem.
What is calculus? Here is my attempt at a concise answer, for people like you who have studied the subject.
Calculus is primarily about derivatives and integrals. A derivative arises by taking slopes of secant lines that "approach" a tangent line. An integral arises by adding up areas of rectangles, as the number of rectangles "approaches infinity". Both ideas hinge crucially on "approaching", and that's what limits are for; they are the key concept underlying and unifying calculus. Despite the presence of limits, the derivative and integral are defined quite differently and don't apparently have much to do with each other. However, a remarkable result called the fundamental theorem of calculus reveals an intimate connection between them. This theorem is of tremendous scientific, historical, and humanistic significance, because it turns out that many practical problems can be phrased in terms of derivatives and integrals. Indeed, much of the technological advancement of the past 400 years would not have been possible without these ideas and the rich interplay between them.
I've asked you a few conceptual questions this term. Exam 2 #3 tries to get at the tension between (anti)derivatives and integrals described above. It was done almost verbatim in class. Exam 1 #3 was done almost verbatim in the homework. Exam 2 #1 is supposed to test your understanding of what an antiderivative is. Exam 1 #2 is really difficult. Exam 2 #5B is a typical example. It gets at the difference between a Riemann sum (a sum of the areas of rectangles) and an integral (the limit of some Riemann sums). What happens when you pass to the limit? (Answer: Your area estimates become exact.) How do you estimate an integral? (Answer: You retreat from the limit.) Are you able to express this in terms of sigma notation, with all of that delta-x stuff? That is, do you really understand what an integral is?
I suspect that you'll see another conceptual question or two on the exam. Certainly you should have the definitions of derivative and integral memorized, and understand them as well as you can.
Your education in mathematics is not useful unless you can recognize a mathematical problem when you see it in the wild. Interpreting practical, applied problems ("story" problems) into tractable math problems is a vital skill. So my exams usually include a couple of story problems, such as Exam 1 #7 and Exam 2 #4. To practice story problems, I recommend that you select a section of the book with a lot of them (such as the optimization and related rates sections) and march through the problems, setting them up but not solving them (unless you want to practice that too).
On the final exam, you can expect to see a few story problems.
The book has a lot of "graphical" problems, such as 4.9 #49-53. I haven't been asking these on our exams, but I do like them, because they can't be solved by someone who depends on formulas. They favor imagery over text. For once, the art majors are rewarded for their unusual abilities and the English majors are penalized for reading too much.
Exam 1 #4C is a related example. The easiest way to solve it is to understand that the bouncing in y(t) arises from the cos(1.5 t) factor, and that the 10 e-t just attenuates it (shrinks the amplitude as t gets bigger). This question tests your ability to move between formulas and pictures.
On math exams, it is common to include one or two problems that are especially difficult, with the expectation that very few students will solve them. Exam 1 #2 has been our most prominent example. There's really no way to prepare for these challenge problems, other than doing a lot of practice problems to exercise your brain, or maybe making it through the final stage of How To Study Calculus. I'm just mentioning them because they're one aspect of how math professors design exams.
There are a couple of topics (areas between curves, volumes) that we've covered since the second exam. There are a few earlier topics (such as the mean value theorem) that were omitted from exams due to time constraints. Such stuff is likely to appear on the final.
If you make the following common mistakes, then you'll be mad at yourself.