Tuesday, December 18, 2007

Video Mini-Lesson: Cone in the Sphere Problem

As a result of the numerous views of a calculus problem I published in November, I decided to present the following video mini-lesson. As before, I had to break it up into parts to control the file size for uploading. I hope this has some value for those who were looking for a more detailed discussion of this question. Much of this is highly appropriate for precalculus students.

Note: Before playing the videos below, a correction and comments:
(1) In error, I referred to the cross-section of the cone as an isosceles right triangle. Make that isosceles only!
(2) The video and audio quality is far from perfect. Bear with me on this!
(3) I didn't discuss the case where the height of the cone is less than or equal to the radius. This will not produce maximum volume but should have been noted. I will have more to say about this later.
(4) There is so much more to discuss about this question, in particular, the result that the cone of maximum volume has height equal to (4/3)R or that the center of the sphere divides the altitude into a 3:1 ratio. These may be discussed in upcoming videos. In particular, as suggested in the videos below, there will be a treatment of the 2-dimensional analogue of this problem, namely, the isosceles triangle in the circle problem.
(5) These video 'mini' lessons are designed for the university or secondary calculus student (probably comes too late for the college final exam) or for anyone wanting a refresher. Beyond my personal style of presentation, there are pedagogical issues (instructional tips) that arise in the videos that might be of interest to someone teaching calculus for the first time.

If you're getting bored of watching the same chalkboard and my same drab outfit, well, it is a low-budget video! I hope you will let me know if this proves helpful and if you'd like me to continue these. As mentioned previously, I will also be employing other technologies for demonstration purposes.

Happy Holidays!


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19 comments:

Anonymous said...

i can't tell you enough how much i appreciate this! thanks a ton!

Dave Marain said...

You're welcome! Without a comment like yours, I never really know who gets to see these and whether they are be helpful.

In the future I'm planning on creating mathcasts for solving common calculus problems, perhaps on a different website. Sometimes, though, a video showing a person is easier to relate to than symbols on a whiteboard...

Anonymous said...

Thank you so much. This really helped me understand what in the world I was doing. Thanks again.

Anonymous said...

these vids were extremely helpful, as this problem has bugged me for a long time. I could actually understand it even without sound in the library! thanks a bunch!

Dave Marain said...

thanks, anon!
You're making me feel I wasn't wasting my time doing this. Actually, I miss teaching in the classroom so this was just a labor of love...
Interesting that you could follow it without audio...
I'm thinking of doing a few more calc vids. Perhaps more of those nasty related rate or max/min problems that drive everyone crazy. Any suggestions?
Also, don't forget to vote in our new poll. I'm very interested in hearing from educators and students (and parents too!).

Joseph from the Philippines said...

Thanks sir dave! I also had a problem like this in calculus and couldn't figure out how i can solve r in terms of h. Thanks for the vid! Hope you can create more and help students like me.Yes you can create more max/min vids or solve related rates or rectilinear motion problems since those problems really give most of us a hard time. Again thank you so much!

Anonymous said...

Hi,

I really appreciate being able to see this problem. I am working through the classic Calculus Made Easy written by S.P. Thompson first published in 1910 and I got stuck on this as I had accidently transposed r for R. It wasn't until I saw your value for h, I realised my mistake! I found the discussion about absolute limits for the boundary of the possible solutions very worthwhile and it will help direct my further learning.

A Physics teacher brushing up his calculus!

Dave Marain said...

Thanks, anon!
It is gratifying to know this amateur video has actually helped...

I'm thinking of doing a few more of these. Which would be most useful?

Max/min word problems
Related rates
Net vs. total distance
Areas, volumes
Other applications of the definite integral
Simple differential equations (exponential decay/growth)
Logarithmic functions
The Chain Rule
[Note that, with the exception of the Chain Rule, I am omitting straight technique/manipulation]

Anonymous said...

you sir, are a god

Dave Marain said...

Bless you! :)

Seriously, if this crude video actually helps a few calc students survive and make sense out of max/min problems, that's very rewarding to me. I'm thinking of doing a few more in the future:

1)Explaining implicit differentiation
2)Solving related rate problems

My problem is always finding the uninterrupted time to plan, take and prepare/edit the video. I'm sure my wife thinks I am out of my mind or that I enjoy starring in my own videos!

IBY said...

I am not a calculus student, but I love math problems, and I like this one. It may not help me in my class, but it gave me something to ponder about.

Dave Marain said...

Thank you, IBY. There were some 'open' questions I posed in that video which I haven't yet discussed. Perhaps another day...

Anonymous said...

Dave, although you don't know me, at this moment, you are my best friend in the world. I have been trying to solve this classic problem on my own, and yanking out my hair. It turns out I assumed that the cone would obey the same rule as the equilateral triangle and took the wrong path. Your video was the perfect teacher. I am trying to walk through Martin Gardner's revision of Thompson's book Calculus Made Easy. Wonderful book, but I get stuck on certain problems and I'm trying to get rid of bad analysis habits like the one I just mentioned. Your video was great great great!!!! Thanks!!!!!

Dave Marain said...

Thank you! I always need validation! It's nice to know this actually helps people learn some calculus or math in general.

Teacher in Australia said...

Good on ya, mate! There is a group of high school kids in North Queensland who found your tute very helpful. Well developed and well paced.

Dave Marain said...

Thank you for those kind words. Glad to help...

Johnboni Corpuz said...

Whoah... Sir, thanks you sooo much! That question is like 20% of my tomorrow's exam! ^_^

I really appreciate your tutorial... And I hope you can help a lot more students like me! ^_^

More power and God Bless! ^_^

Talina and Carla said...

Thank you so much for doing this. You helped my sister and I not only finish, but understand the problem. Thank you for devoting your time to this project!

Megan said...

Thankyou so much! Helped me very much to make sense of my math homework :) I'm putting your blog into favourites.