On Wednesday in Calc I, we started looking at what the second derivative says about f. As it turns out, the second derivative tells us about the way the graph of f bends as it passes through an interval. A function f is concave up if it is curved like the outside of a right-side-up bowl -- its graph sits atop all its tangents. On the other hand, a function f is concave down if it is curved like the outside of an upside-down bowl -- its graph sits beneath its tangents. What we found is that f is concave up on an interval if and only if for every x in that interval and that f is concave down on an interval if and only if for every x in the interval. The points where the second derivative changes sign are called inflection points.

We then noticed that at a local maximum where the graph passes through the maximum point smoothly, that the graph is concave down at that point. Similarly, if our graph passes smoothly through a local minimum point, then the graph is concave up at that point. This led us to the fact that we could possibly compute the second derivative at a critical point and classify the critical point as a local maximum or minimum based on the result. To wit:

Second Derivative Test: Let be continuous at a critical point c:

• If and , then we have a local minimum at c.
• If and , then we have a local maximum at c.

If doesn't exist, or if doesn't exist, or if , then we must revert to the First Derivative Test.

We looked at a "kitchen sink" example, where we could use the Second Derivative Test at one critical point, but not another, and we came up with a full graph. We'll do a lot more of this in Section 4.5.

On Thursday, we'll start looking at optimization problems.

Your second lab assignment has been posted to the Computer Labs page. It is due on Monday, November 3rd.

Please pay strict attention to write-up rules, as well as the penalty for handing in a stylistically incorrect lab.

We may or may not have time for you to work on this during class time, so download now, start soon. Do not wait until the last minute. If you do so, and the network crashes, the deadline passes anyway. Plan and act accordingly.

This week, the architects and engineers in class have field trips on Wednesday. In addition, we have a short week with many people leaving on Thursday.

Let me make the following clear:

• There will be class held on Wednesday. I plan to proceed through the rest of 4.2 and 4.3, but you should check this space to see just how far we went.
• If you go on the field trip, that doesn't automatically make you immune to the material covered that day. You are responsible for it and will be eventually quizzed and tested on it.
• There will be class held on Thursday. I don't care if I'm "the only one of your professors who didn't cancel class Thursday." We will proceed through new material for which you will be held responsible.
• If you are under threat of being dropped for missing too many classes, you definitely should be here Thursday.

Having said all that, I hope the field trips go well and that you get a lot of knowledge and enjoyment out of them.

Monday in Calc I, we started examining what derivatives could actually tell us about a function. The long-term outlook will be to (1) draw better graphs than we used to in our Precalculus classes and (2) use the techniques to solve applied problems.

We started by looking at what could tell us about the original function f. We quickly came to the idea that if for all x in an interval, then the function f is increasing on that interval, and if for all x on an interval, than f is decreasing on that interval. (A function that is solely increasing or decreasing on an interval is said to be monotonic on that interval.)

We looked at an example or two of how to determines intervals of increase or decrease. (Such intervals are always open, by the way.) We noticed along the way that the critical points acted as boundaries for the intervals of increase and decrease, and also that if changed sign while passing through the critical point, that critical point could be classified as either a local maximum or a local minimum. This was the essence of the First Derivative Test. (Note that if doesn't change sign while passing through the critical point, then the critical point can be rejected as a possible local extremum.)

On Friday in 121, we continued our discussion of ways to find absolute and local extreme points. Our biggest problem was that we had lots and lots of points to sift through. But we found a trend on Monday that local maxima and minima turned up where the graph of had a "turnaround" of some kind, either a smooth one or a sharp turn. The smooth ones, where are called stationary points, and the sharp turns, or "cusps," occur at singular points, where doesn't exist. Together these two types of points make up the critical points for f. The critical points are the candidates for locations of local maxima and minima. In fact, if a function has a local maximum or minimum at c, then c must have been a critical point. (Note that all critical points must be in the domain of f.)

We then looked at a couple of examples. We saw that in the case where can be written as a fraction, the critical point question becomes one of when the numerator is 0, and when the denominator is 0.

We then applied this technique to the Extreme Value Theorem and came up with a procedure to find the absolute maxima and minima of our function f on a closed interval [a,b]:

1. Find the values of f at the critical points inside the interval [a,b].
2. Find the values of f at a and b.
3. The largest of the above values is the absolute maximum value on [a,b]; the smallest is the absolute minimum value.

We'll discover next week how to classify critical points as maxima, minima, or none of the above.