Calculus Video: Optimization (Max-Min) - NEW Improved Version!

Well, I felt badly about that error I made in the original version from a few days ago (which will now be deleted). I also decided to change some portions, electing to solve for the critical values using algebra rather than by the graphing calculator. Finally, I took a tighter view of the whiteboard so that the writing will appear larger. There will be some glare on the board which I hope will not be too distracting. I hope you will find this more helpful and again I apologize for any confusion caused. If you stored the original video, I would ask you to delete that.

HAPPY HOLIDAYS!

The problem in the video below demonstrates important concepts as well as the standard procedure for solving optimization problems. There is also a brief discussion of a heuristic I have found very useful when teaching these kinds of applications. As always I depend on you to share your thoughts. I keep saying this knowing there might not be too many comments!

A "VERY BIG" Pre-Turkey Day Math Challenge for Middle or HS

Just a 'little' last-minute challenge before Turkey Day -- similar to many you've seen before on this blog and elsewhere...

Determine the exact digits of 100²⁰⁰⁸ - 100¹⁰⁰⁴.


Comments:
Students in middle school or higher will often (or should) employ the "make it simpler and look for a pattern" strategy. Some students will be able to apply algebraic reasoning (factoring, laws of exponents, etc.) to evaluate. It's worth letting students, working in pairs, 'play' with this for awhile, followed by a discussion of various methods. Then challenge them to write their own BIG exponent problem!

HAPPY THANKSGIVING!

Update 11-21-08

Some thoughts...

1. The new calc videos are getting many views but not comments. I'm depending on others to comment on the quality -- is it readable, is it sufficiently audible on your browser? Is the writing too small or too sloppy to be viewed? By the way, I tried using a whiteboard and colored markers, but, unfortunately, the lighting and contrast prevented this. I'm still working on adjusting the camcorder settings. Apart from the technical aspects, is the content helpful? Let me know...

2. Part of me is getting a sense that these videos belong on YouTube or simply a separate website that can be linked to from my blog. Or am I fooling myself here -- perhaps I need to abandon this blog and simply develop a website for instructional uses. I know that I haven't been a very good member of the blogosphere of late. I should be far more attentive to the contributions of the excellent boggers out there. I'm not really sharing and for that I feel guilty.

3. Let me try to get past this catharsis and remind everyone of our MathAnagram for Oct-Nov-Dec. Thus far we have only received one response. For those who forgot, here it is:

CHARMED ERA

Here is the link to the details.

4. I am putting the finishing touches on another optimization (Max-Min) calculus problem involving a ladder. It demonstrates many of the complications students confront in Calc I or AP Calc and builds on the ideas developed in the "cone in the sphere" problem from last year.

5. Those who are involved with teaching or writing Algebra I or Algebra II curriculum should definitely take another look at the Achieve site for the End of Course Exam for Algebra II. It is frequently updated and now contains more information for the upcoming Algebra I test. It is clear to me from studying the benchmarks, illustrative examples and the released items that there is a considerable raising of the bar going on here. Traditional Algebra 2 topics are being moved to Algebra I (more operations on radicals, more sophisticated absolute value expressions, more graphing of both linear and non-linear functions, more probability and data analysis, etc.). Furthermore, traditional precalculus topics are being moved to the regular (not honors) Algebra 2 curriculum (exponential and log functions, piecewise functions, etc.). How are teachers going to incorporate this more advanced material? As the middle school curriculum moves forward, students should be better prepared to handle a faster pace and more advanced work at the high school level. Further, there will simply be less time in September for the review of everything students forgot (or claim they never learned!). We simply cannot afford weeks of review of prior material at the beginning of the year. Yes, folks, I'm saying this knowing full well that many educators are dealing with youngsters whose basic arithmetic skills seem sorely lacking and whose recall of basic algebra material is weak. Sorry, kids, we need to move on. If you're confused in class, ask questions, take good notes and if it still doesn't make sense, be prepared to spend lots of time after school. No one provides more extra help than math teachers do! I guess after this rant, there may still be a reason for me to continue this blog, at least for now!

Update on Videos...

Sorry for some of the confusion here. Apparently, Blogger did accept the complete video of the Calculus Video Pt. 2: Related Rates. All 16+ minutes! Therefore, I deleted the 3 segments that had appeared below the first one (which repeated in segments what appeared in the first one). Since many of you read my posts in Google Reader or something similar, I would imagine that you do not see edits that I sometimes make to posts after they are published. It would be helpful to me if you could comment on this and let me know how edits do appear in a reader. I'm guessing you only see the original.

I would also appreciate it if you could let me know about the quality of the audio/video as it appears in your browser. I use Firefox and it seems ok although, when compressed in iMovie and by Blogger, the video of course loses some quality. These videos are low-tech, amateurish with an inexpensive digital camcorder and with less than professional lighting! I think the inexpensive chalkboard, my children's chalk and a dishrag for a towel are the true finishing touches! Despite all of this, I'm hoping that some might derive some benefit from these.

For teachers, the Pt. 1 video was intended to demonstrate multiple representations (aka, The Rule of Four): The problem and solution of finding the equations of the tangent lines were expressed verbally, symbolically (algebraic), graphically (the parabola), and numerically using a tabular approach to x, y and dy/dx. This approach was very important to me and I strongly advocate its use in the classroom. And, yes, it was all done without calculator technology, although the graphing calculator (with ViewScreen) would enhance the lesson.

Calculus Video Pt2 - 11-14-08: Related Rates

As promised, here is a video of a related rate application. This one is fairly straightforward but it does demonstrate the procedure to follow when doing the more sophisticated types.

These videos build on the foundation videos I posted which developed the Leibnitz form of the derivative and the Chain Rule as well as the basics of implicit differentiation. I strongly encourage the student to view these first. If you're more familiar with the topic and want a quick review, then start anywhere!



As always, I depend on my readers to make suggestions to improve the quality of these videos, both mathematically and technologically. Please indicate any errors I may have made so that I can correct them or at least indicate where they are in the video.

Finally, there are DVD sets of Calculus lessons out there as well as other videos you can find on YouTube or by searching. I am interested in knowing if you feel there is a reason for my producing more of these. I hope you enjoy it and, if it helps just one student feel a bit more comfortable with these topics, then it has been worth the effort!

Calc Videos : Chain Rule, Implicit Differentiation Pt. 1

I 've been promising this for awhile now so I thought I would at least upload the first 3 video segments. The more complicated implicit differentiation example will follow...

Note: There are usually some minor glitches in these productions, no exception here!
(1) In the first video below I inadvertently referred to one of the variables as 'dependent' rather than independent.
(2) On a few occasions the writing near the bottom of the board gets cut off but it should still be possible to follow along.

The three video segments below are linked. I had to split the original video to make it manageable.

Update 11-11-08

In deference to our veterans I am not publishing a new post other than a simple statement of gratitude to the heroes in our Armed Forces. Thank you for protecting us...

Just a brief addendum...
I'm putting the finishing touches on a series of calculus videos of the Chain Rule, Implicit Differentiation and Related Rate applications. This may come late for our AP or Calc I students but perhaps it will help on a midterm or final exam...

From "I Had a Dream" to "Yes We Can!" Congratulations to Pres-Elect Barack Obama

The truth is in the numbers...
He may not have gotten there with us but I know Dr. King is smiling. Today is a day of joy for all of us. A joy that that begins to eradicate the negativity, the fear and the cynicism we've had to abide for the past two years.

The road ahead is unknown but I can begin to see the light at the end of the proverbial tunnel...

Congratulations to the United States of America and all of its people!

Why not predict the election with a simple probability problem...

Up to now, I've chosen not to bring the election into this blog, however with the outcome to be decided, hopefully, in a few hours, perhaps we can make our predictions based on the following:

The election will be decided by a fair 8-sided die whose faces are numbered from 1 through 8.
The die will be rolled 5 times.

Candidate A wins if the number 7 appears at least once in the 5 rolls.
Candidate B wins otherwise.

You can decide the identity of each candidate! It would be interesting if the popular vote turns out to be this close...

'Odds'n'Evens' Week of 11-4-08 Catching Up...

Haven't had one of these overviews for awhile now...

1. Have you had a chance to work on our MathAnagram for Oct-Nov-Dec?

CHARMED ERA

Look here for details.
2. I am still interested in your thoughts on methods of teaching addition and subtraction of mixed numerals. The original post is here and reader comments were fascinating. Perhaps a poll or survey to collect even more input?
3. As some of you know I ran a series of brief calculus videos about a year ago. They are viewed quite often and I'm considering a few more. I did a search of online calculus tutorials/videos and came away with the feeling that there is a need for more. Some of the best tutorials come from the Visual Calculus series developed at University of Tennessee at Knoxville (utk), but these are not videos. Here is a sample of one of their best lessons (on implicit differentiation). I do not mean to slight the other excellent tutorials I found but I don't have time here to review them in depth. At various sites I viewed flash tutorials, one or two videos, lots of text but if you felt my video (even with the one error I made about an "isosceles" right triangle!), was helpful, let me know by emailing me or commenting on this post. I've been thinking about the Chain Rule, Implicit Differentiation and Related Rate applied problems. These are all connected so I might do a series of videos to demonstrate the underlying ideas and applications. Again, I would appreciate your input to encourage this effort since it is labor-intensive.
4. Any more thoughts about the recent post I published enumerating the topics tested on the recently released items from the Achieve/ADP Algebra 2 End of Course Exam? People are reading it but not commenting, a fairly common phenomenon!
5. No more thoughts about the Right Triangle in the Square Investigation? TC suggested a possible generalization but I'm wondering if anyone tried this in the classroom...
6. Perhaps not worthy of a full post but here's a mini-logic problem that reinforces both algebraic reasoning and the issue of demonstrating that an if-then statement is false:

If y² > x², then y > x.

PROVE THIS IS FALSE.
Of course there are many variations on this and one could also use this to demonstrate the converse error, but sometimes it's nice to include proof in our algebra classes too.