The due date for Lab 2 has been postponed until Wednesday, November 5th. For Monday's class, please bring your work thus far with you as I will leave some time to answer questions.

Thursday in Calc I, we spent some time using everything we've learned in Chapter 4 to develop a technique for graphing a function from beginning to end.

The main things we examined were:

• Domain -- the set of "good x"s for our function. This helps us set up a general locale for our graph.
• Intercepts -- where the graph hits the x- and y-axis.
• Symmetry -- whether the function is even (; graph is symmetric about the y-axis), odd (; graph is symmetric about the origin), or none of the above. If there are trig functions involved, we should check whether our function is periodic.
• Asymptotes -- vertical asymptotes generally occur where the denominator is 0 (but not the numerator); horizontal asymptotes are lines the graph follows arbitrarily closely on the far left or far right. Depending on the function, we may need to check for slant or oblique asymptotes.
• Intervals of increase and decrease -- take the first derivative and check where it's positive or negative.
• Local maxima or minima -- Use the first derivative test to examine critical points found in the previous step.
• Concavity / inflection points -- Take the second derivative and see where it's positive or negative. The inflection points will be the points in the domain where the second derivative changes sign.

This gives us enough to come up with a decent graph. If you don't feel like you have enough information to go on, plug and chug a few points.

We did two long examples on Thursday. Friday, we took a quiz. Monday, we'll move on to the Mean Value Theorem (the third of our "value" theorems).

I don't know how I did it, but I'm off by one example in each class ... I've done one more optimization example in the 7:50 class than I have in the 10:50 class. I think the 10:50 class asked more questions regarding the lab that's due on Monday. I'll have to fix things up this week.

Between last Thursday and Monday, we graphed a fairly non-standard example using the first and second derivatives to fill in some information on our graphs. We've also done two (or three) long optimization examples. One involved a farmer and his fencing, one involved the manufacturing of a metal can, and the last involved a dilemma involving rowing and running. (See in-class handout.)

I will try to catch up the 10:50 class on the rowing-running example on Wednesday and in both classes we will do some more graphing of functions we never thought we could graph by hand. See you then.