Mr. Slater's Blog

November ASMA

Today we did the November ASMA Contest, and students turned in their six AP related rates free response problems.  We then had time to briefly look at one or two of the optimization problems students worked on for homework from last night.  Tonight's assignment continues working with optimization:  Pages 267-269: # 27, 31, 35, 36, 37, 39, 40, 41, 44, 54, and 59.  Remember to draw and appropriately label a picture whenever possible, and look for the "est" word - that will tell you that a maximum or minimum is needed, and what equation you need to set up.  Once you've finished the setup, the actual derivative you need to find is usually pretty straight-forward, and the algebra and arithmetic necessary to solve it is fairly simple.

Normal Curve

A normal distribution of data means that most of the numbers in a set of data are close to the "average," while relatively few data points tend to one extreme or the other.  Since the standard deviation is a statistic that tells you how tightly all the various data points are clustered around the mean in a set of data, we can compare data from one sample to another by using the mean and standard deviation to convert data to z-scores (a common scale).  For a nice, non-technical, resource about the normal curve (written by a journalist, not a mathematician), look at: http://www.robertniles.com/stats/stdev.shtml

The assignment for tonight is:  Page 266:  #1-19 odds and 18, and Page  267: #1-17 odds and 18-20.

Optimization

The most common use of derivatives is optimization - i.e., determining maxima or minima.  Since the first derivative allows us to find critical points which can be tested to see if they produce a maximum or a minimum, that's where our optimal points will occur - unless they happen at the endpoints of our interval, since most real world problems have limited domains.  Read through the half-dozen examples in your textbook in Section 4.6, and then do Pages 265-266: #1, 4, 5, 6, 8-13, 19, 21, and 23.

Linear Approximations

Back in "BC days" (before calculators), some calculations were extremely tedious and involved to do, taking lots of paper and time to approximate - i.e., finding sqrt(65) or the sin 65.  Using the local linearity trait of functions we discussed much earlier in the semester allows us to use the tangent line to a curve as a means of approximately values of the function, usually accurate to two or three (or more) decimal places.  Section 4.1 looks at applications using these linear approximations, as students worked on Pages 218-219: #9-27 odd, 31, 32, 35, 37, and 41-49 odd. 

I also checked yesterday's assignment - the related rates worksheet that students had the class period to work on and then finished the odd problems for homework.  Few students had done any of the problems on this most important topic, so I handed out a set of six free response related rates problems from AP Exams from 1988-1995 - note how common a topic related rates seems to be?  These problems are to be written up in detail, to be turned in on Thursday for a grade.  (Since many students aren't doing homework for practice, I guess we'll need to collect homework for a grade...)

Related Rates, yet another day

After having worked on Section 3.11 for the past three days, I checked homework from this section to see how things were going on this most important topic.  Many students had attempted few (if any) of the problems, despite having had all of last Thursday and Friday in class to work on these assignments - hence the lack of questions being asked in class!  Students received a worksheet of additional, "interesting". related rates problems, and were once again given most of the class to work on and discuss with partners - I will be checking this in class tomorrow.