Chapter 3 Of course, Chapter 3 contains all the material necessary to pass the gateway exam. It is tempting to imagine that once the gateway has been mastered, so has Chapter 3. If you convince yourself of this, you'll be in for a rude surprise: We have already tested you on rote differentiation skills, and we don't plan to do it again. It is unlikely that you'll find questions which ask you simply to differentiate formulas. Expect story problems, interpretive problems, and various other complexities. Look over your group homework to find the kinds of complex word problems we expect you to solve.

3.1 I can differentiate polynomials, can you? I admit, this is mostly gateway material. Constant multiple, sum, and power rules allow us to differentiate many functions, and leave many still impossible. It's all pretty basic looking back. One warning: As you learned on the gateway, a constant is a constant is a constant. Even if it looks like ln(Pi)*sin(3e+4), it's just a constant. Treat it appropriately!

3.2 The only thing here which isn't essentially gateway training is the nice observation that the derivative of a^x is proportional to the value of a^x (even as x changes). We could now deduce this fact from the fact that d/dx (a^x) = ln(a) a^x, which even tells us the constant of proportionality. Despite my claim that this is "gateway training," it is still possible to write a story problem. See 3.2.36 for a typical one. Notice that this is just a "differentiate this function" problem inside a very thin plastic wrapper marked "Section 2.5". In other words, there are no new deep and difficult conceptual notions here.

3.3 Let me continue the warnings now. The product rule is not just about gateway skills, as you can see by looking over the problems in 3.3. Problems 3.3.40-3.3.44 are all new conceptual problems, testing not just your ability to differentiate and interpret, but also your ability to understand the inner workings of the product rule. Sometimes you must use the functional notation: h'(x) = f(x)g'(x) + f'(x)g(x), and sometimes you must use the equivalent Leibniz notation. Of course, it is still possible to write "thin plastic wrapper" questions. Every expression of the product rule has five terms. There's no rule against giving you "the wrong four", and asking you to get the last. These tricks are more common with chain rule questions, but they're possible here too.

3.4 Can you differentiate ln(sin(ln(x)))? How can you recognize a story problem which will require the chain rule? Oil slicks and rain drops are good examples here, but they are not the only ones. Often with chain rule problems, you must supply some formulas of your own, like Area = Pi * r², or the Pythagorean theorem. Don't panic--no weirdo formulas (surface area of a cone, anyone?) will be required. Tables and formulas are easy to quiz on too--compare 3.4.53 and 3.4.55 to 3.3.42 and 3.3.41. Can you do these? 3.4.65 illustrates the type of the new story problems available here.

3.5 It's hard to find anything deep here besides rote differentiation skills. Can you differentiate all trig functions? I hope so. Although the derivatives of inverse trig functions are not on the gateway, the functions can be quite helpful in setting up story problems in Chapter 4--thus, you should learn these derivatives as well! Story problems are possible only synthetically, as in 3.5.47-3.5.50. Notice that these problems just combine basic understanding of derivatives with the new skill. Still, it's very important to know how to do these things!

3.6 Why are there two sections on the chain rule? Because there are so many different kinds of applications and problems that we couldn't keep them constrained. They spilled all the way over 3.5 and filled up 3.6. Comfort with "Leibnizian" chain rule notation and so-called "related-rates" problems feature prominently here. There are numerous cookbook guides to solving "related rates" problems, but in my opinion these problems are the most natural outgrowth of the chain rule itself, not some special class. So just remember what the chain rule is, and be able to figure out how it applies to some weird problem. Also, remember that whenever you have any relationship between variables, you can differentiate it to produce some sort of relationship between their derivatives. So don't hesitate to differentiate those formulas! Examplary story problems on this material include 3.6.45-3.6.55.

3.7 What's the difference between implicit and explicit differentiation? When is the derivative of y² equal to 2y and when is it equal to something more complicated? Is (d/dx)x² equal to 2x(dx/dx)? Why is implicit differentiation often a more efficient alternative to regular differentiation? Why is it sometimes the only choice? How can you use implicit differentiation to locate important points on the graph of a relation, such as vertical-tangent and horizontal tangent-points? 3.7.{Example1}, and 3.7.24-3.7.28 are typical problems of this section.

3.9 What is the justification for using the tangent line equation as a stand-in for the original formula for the function? What can be gained by doing this? If I say "Calculate the equation of the tangent line at x=7", what's "a" and what's "x"? How can we use the formula of the tangent line to estimate the values of the function? There are many different types of problems here--it's hard to call just a few of them "typical." Certainly 3.9.{7,9,11, and 12} are good review problems.

4.1 This section is preparation for the rest of Chapter 4. It features vocabulary, and the so-called "first derivative test" and "second derivative test," the not-too-surprising observations that plenty of information about f' or f'' tells you enough to determine whether a critical point is a local max or min. Here we also review the graphical relationship between f, f', and f''. 4.1.36-4.1.43 are good examples reminding you how difficult this correspondence can be. 4.1.30-4.1.31 test your intuition for concavity, and therefore you intuition for graphing f while getting f'' just about right. The most important applications of 4.1, however, don't appear until 4.3, when we start finding maxes and mins in earnest.

4.2 What is a family of functions? What's the difference between a family of functions and a single function which has an unknown parameter in it? What kinds of procedures would you have to use to answer the question: "How does the parameter a affect the graph of functions of such-and-such type?" How would you make the best use of your calculator to do this? Sometimes we want a member of a given family to fit chosen properties, like having chosen critical points, roots, etc. How can you translate information given (e.g. "It must have a critical point at (1,2)") into equations that help you determine values of unknown parameters?

4.3 Find the max. Find the min. This section is about thoroughness. You might be tempted to find a critical point and just assume that it's the max or min you're looking for. Don't forget, though, to check endpoints! Don't forget to use either the first- or second-derivative test to determine the nature of your critical points! Don't forget to apply standards of "reasonableness" to the critical points found in a real-world problem (Radii can't be negative, Lengths too, etc). This section gives us our first opportunity to give massive story problems whose theme is the optimization of a particular variable. There are three or four main types of these, so practice hard! Another reason to practice is that setting a derivative equal to zero often gives you complex expressions that are difficult to solve. Keep your head! Think about factoring and don't divide both sides by anything that isn't guaranteed to be nonzero! (Triple negative! You know what I mean.) Fun and typical problems are everywhere: 4.3.7-4.3.20 are all excellent optimization-oriented story problems. The bird problems 28 & 29 are both excellent, and both very hard and conceptual.

4.4 Here sophisticated economic language can hide the mathematical subtleties. What is MR? Alright, it's the derivative of R, which is revenue, and we call it "Marginal revenue." But this leaves a lot unanswered: What is R a function of? MR is the derivative of R with respect to what? time? Is it fair to say that MR is the extra revenue obtained by selling one more item? How can you interpret MR and MC in terms of economic decisions? Does MR=MC always give the maximal profit?

There are two basic graphical tasks of interest here, illustrated by problems 5 and 9 of the section. Lots of vocabulary ("fixed cost") and economics make the section difficult.

4.5 Speaking of difficult sections! This is the "grown up" version of 4.3. You learned in 4.3 that a function can be optimized by finding its critical points, checking endpoints, and using first- or second-derivative tests, etc. Now you must apply all of those skills to problems in which you don't have a simple formula f(x)=x^2+x for the function involved. You must deduce this formula for yourself, and that requires geometric intuition. General tips are given on page 198, so I won't repeat. Common mistakes include:

Well, best of luck! See you all at the exam!