640:151-152 - Calculus I for the Mathematical and Physical Sciences

Going from math 135 to math 152

Students who want to major in computer science, biochemistry, chemistry, physics or mathematics must take the calculus 151-152-251 sequence, as well as other math courses. Any student who wants to take 200 level or high math courses should also take that sequence.

However, there are always some students who take math 135 before they decide they will need the math 151-152-251 sequence. It is possible to go from 135 to 152, though with some effort; it is almost impossible to go from math 136 or math 138 to math 251, as the syllabi are very different. Therefore math135 students who think they might eventually need 251 are encouraged to take math152 immediately after math135.

What follows refers to math 135 as taught prior to Fall 2003, from the text Applied Calculus by Tan. A new text is in use for Fall 2003; information on the transition from the current math 135 to math 152 will be posted shortly.

There are three primary differences between math135 and math151. First of all, math151 pays some attention to theorems and proofs, in particular the proofs of the rules for differentiation and the fundamental theorem of calculus (which can be part of exam questions). Secondly, math151 goes more deeply into limits, both in terms of limit theorems and in terms of evaluation of limits. Thirdly, there is more emphasis on trigonometry and in particular on the inverse trig functions. (The inverse trig functions are of particular importance for integration; although it might look utterly unlikely, it turns out that the integral of 1 over 1+x^2 is arctan(x) + C, as is shown and used in math152. Inverse trig functions can be found in Appendix D of Stewart's text.)

The text for math151-152-251 is Single Variable Calculus Early Transcendentals, fourth edition, by James Stewart. It is imperative that a student who intends to take math152 get this book and study it before embarking on math152.

Limits are important because all of calculus is founded on the idea of a limit. Look at the definition of the derivative (limit of a quotient of the form 0/0) and the definite integral (limit of Riemann sums). In math152, you will see many more definitions involving limits: improper integration, limits of infinite sequences, sums of infinite series. Sections 2.2, 2.3, 2.6, and 4.4 of the Stewart text are particularly important for limits. The epsilon-delta definition of a limit is usually discussed in math151, but only very simple problems involving its use are given on exams, and it is not really needed for math152 or 251. On the other hand, there are two limit theorems that are used (implicitly if not explicitly) in math152 and math 251 but are not in Tan's book:

Theorem: If f is continuous at b and limx->a g(x) = b, then limx->a f(g(x)) = f(\limx->a g(x)) = f(b).

Squeeze Theorem: If f and g are defined on an interval containing a, with |f(x)| < g(x) on that interval and lim_{x\to a} g(x) = 0, then limx->a f(x) = 0.
Both of these ``make sense" and even seem obvious once you understand what they say. You don't need to know how to prove them.
There is also L'Hopital's Rule, a workhorse that provides a relatively easy way to evaluate limits of indeterminate forms of the form 0/0 or infinity/infinity. L'Hopital's Rule is often covered in math152 as well as or instead of math151, but students would be advised to try to learn it before taking math152. The main problem that students have with it is trying to apply it in cases where it does not apply! (You have to be SURE that the limit is of one of those two forms before using L'Hopital's Rule.)

In the following files there are some problems that you might want to try, to extend your ability to evaluate limits, to hone your skills at verifying formulas or otherwise proving things, and to practice using L'Hopital's Rule. You might also want to do extra problems on trig functions and inverse trig functions.