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!

Back-to-School Geometry - Rectangles, Squares and Deeper Challenges

The connections between geometry and other rich areas of mathematics are boundless. Here is a fairly straightforward set of problems that can be explored as far as your eye and mind can see. On the surface, we have three rectangles each of which has a half-diagonal of length 6. Students can be asked to find the area of each without using any trigonometry. On a deeper level, one can ask students to explain or prove why the square has the greatest area for a given
diagonal length. This is straightforward using the well-known trig formula for area of a triangle, K = (1/2)absin C, however the challenge here is to use non-trig methods (although the student can use special right triangles) to compute the areas and demonstrate the maximum. The maximum piece is more sophisticated and the idea of bringing this in before precalculus and calculus has many benefits.

The instructor might begin by asking students to draw any rectangle with a diagonal of 12. How many such rectangles could there be? Which one would appear to have the greatest area? Ok, now let's explore a few special cases.

This problem allows the creative student to devise a visual way of explaining the maximum. It also allows the instructor to bring in the Arithmetic Mean-Geometric Mean Inequality for enrichment. So many methods and approaches are possible...

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!

Drum Roll Please: The Debut of TC's Total Challenge

As you may have read in an earlier comment, I've invited one of MathNotations' most dedicated and talented contributors to go beyond commenting and share some of his creative ideas and insights by being an occasional guest blogger - he has graciously accepted.

For his inaugural offering, tc is challenging you and/or your students to solve a classic calculus problem using non-calculus methods. I have made a few minor edits, but the activity is essentially what tc sent to me.
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I give you tc's Total Challenge I:

One of my math professors in college used to say there were three ways
of tackling any problem: the right way, the wrong way and the Navy way
(correct, but extremely roundabout).

In this exercise, we will look at three ways (not necessarily the ones
named above) of doing the following problem:

Determine the rectangle of maximum area that can be inscribed
in a circle of given radius r.

Let the inscribed rectangle have sides a and b. The diagonal of the rectangle passes through the center of the circle (this can be shown, but you can assume it is true).

(1) Express r in terms of a and b.

(2) Express the area in terms of a and r.

(3) Instead of maximizing the area, we can maximize the square of the area.
(a) Express the square of the area as a quadratic in a² (you may want to substitute c for a²).
(b) By completing the square, determine the value of a for which the area is a maximum.
(c) Determine the value of b and the maximum area.
(d) What conclusion can you draw about the rectangle of maximum area?
(This is the first way, which I call the Algebra way)

(4) Divide the rectangle into 2 congruent triangles, using a diagonal. Draw a half
diagonal that intersects this diagonal.
(a) Write an inequality for the area of one of these triangles in terms of r alone. The inequality should be of the form Area ≤ _______.
(b) If you can achieve equality, then you have maximized the area of the rectangle! Find out when this occurs, and if it does, find the lengths of a and b. (The Geometry way).

(5) Method 3 - the Calculus way of course.
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Additional comments from DM:
(i) thanks, tc!
(ii) tc's geometric approach in (4) also suggests a connection to the famous AM-GM Inequality. Visit this link and see if you can make the connection. This is not obvious.
Hint: Apply the AM-GM to a² and b².