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:

• finding critical points, but forgetting to check endpoints, or otherwise verify that they are the global max or global min sought.
• solving for one variable, circling, e.g., "r=9", and stopping without calculating the values of the other, more relevant variables, or making a conclusion sentence.
• Getting a nasty derivative in the form of a fraction, A/B=0, and not knowing what to do. (set A=0).
• Dividing both sides by, e.g., (x-2), thus effectively destroying one solution of the equation (namely x=2).
• Not being able to set up the problem

Often students get two equations and want to solve for one variable in terms of the other. But which to solve for? For example, in problem 7, should we solve for h in terms of w, or vice versa? Trust me: It never matters in the end. It may be easier one way than another, but both ways always work.

4.6 Related rates are reall not "new" material--just, in most cases, applications of the Chain Rule. If x and y are related, then their rates of change are related. Often these problems involve taking derivatives with respect to time, even when time is not a variable in the problem. In those cases, we must remember that the variables (e.g., x, y) are "implied" to be functions of time, so we must use the Chain Rule when differentiating with respect to time (or implicit differentiation, if you like to think of it that way). Diagrams are often very useful for these problems. Pay attention to whether a variable is increasing or decreasing with time, so the sign of your derivative is also correct. Work several of these, so you will recognize them on the final. They are not impossible!

5.1 Compare Chapters 2 and 5. In each, an important new concept is introduced, and in each case, it's easiest to understand in terms of position and velocity ("How do we measure velocity?" vs. "How do we measure distance traveled?") Understanding just 2.1 or just 5.1 is different than understanding it in the context of the larger chapter. If you understand all of Chapter 5, then a sensible attitude toward 5.1 is "Well, obviously this is true because velocity is the derivative of position." How can the formula for the difference between the right- and left-Riemann sums be used to plan for a given accuracy?

5.2 Areas, areas, and more areas! How many ways do we need to be able to calculate areas? Certainly from a table, using Riemann sums. From a graph, using Riemann sums. From a graph, using simple geometry. What are those limits all about? Is the limit of the left- and right- Riemann sums the same? What special thing happens when f(x) is below the x-axis? What tricks can we therefore use for odd functions? What interesting tricks exist for even functions?

5.3 Well, you saw it coming--the Fundamental Theorem of Calculus! Here's a hint: Thinking of writing the Fundamental Theorem on your notecard? If you do that I'll worry, because this theorem, expressible in a dozen different ways, is without a doubt the biggest insight in the book. You should know it by a thousand problems, in a hundred different scenarios. We're talking back-of-hand, here, folks. Better to write your own name on your notecard, "lest you forget!"

Everything else sounds so petty after the Fundamental Theorem, but here goes: Use units to understand the meaning of an integral, and understand that average value formula. What is the graphical significance of the average value? Numerical? What is its equation, and how can you use it?

5.4 I really believe this section's numerous "boxed formulas" summarize themselves. Only one caveat: Understand this stuff not only symbolically (so you can do algebra-like computations with integral signs) but also graphically.

6.1 This section can be confusing. Didn't we already learn how to graph f, given a graph of f'? Does this ever-so-advanced Chapter 6 material add anything to Chapter 2, or does it just reiterate? It adds one important quantitative skill to a mix of qualitative skills already learned: Interpreting those areas! Since the area under (any piece of) the rate graph is a total change in something, the size of the area gives quantitative information about exactly how high certain points on the graph of the antiderivative are.

Does a function have only one antiderivative, or many? How do they differ, algebraically? graphically?

6.2 Whereas 6.1 is about using known areas (and the fundamental theorem) to compute values of the antiderivative, 6.2 uses known antiderivatives (and the fundamental theorem) to compute values of areas. You have to know those rules, know how to use them, and know the relationship with everyone's favorite theorem.

Now, what was it I said about relying on a single resource? Shouldn't you be re-reading your classroom notes?

-jason