ch3_videos

=Chapter 3 Help Videos and Tutorials=

// Not quite sure whether/how this is going to work out, but I thought I'd give it a try... Let me know if this is helpful! //

Implicit Differentiation
Based on a question I received from a student about problem #3 on the Section 3.5 Worksheet, I made these videos to help with the process of implicit differentiation. Please forgive my paint program's less-than-clear writing, as using a mouse to write is a real pain! Remember, this kind of problem (finding the derivative using implicit differentiation) requires three distinct steps: And, just like we found before, we can take derivatives of derivatives -- even if they were found using the implicit differentiation process. So, considering the above problem, we can find the //second// derivative with respect to x using a continuation of the implicit differentiation process.
 * **Part 1**: **@http://screencast.com/t/NmEwMWM2**
 * **Part 2**: **@http://screencast.com/t/MmM3MGY4Y**
 * 1) Apply the derivative to the entire equation (each term is treated separately).
 * 2) Group all of the terms with the differential of interest (e.g., "dy/dx") on one side of the equation, and all other terms on the other side.
 * 3) Factor out the differential and divide by the remaining factors to get the differential (e.g., "dy/dx") by itself.
 * Finding the second derivative using implicit differentiation: **@http://screencast.com/t/ODQ1MTFiMm**
 * I don't actually **//complete//** the process in the above video, mostly because I don't have the space on screen, but you can see how it gets set up (and where you have to go from there). The next step would be to substitute in the fraction everywhere that "//dy/dx//" appears in the second derivative and simplify (which is the boring, algebra part). This can be a really tedious, long process; that means there is increased likelihood that you'll make otherwise avoidable mistakes, so take your time and check your work!

Related Rates
Since we didn't get too much time with them before break, I wanted to try posting something that might help with the reasoning process behind these problems. Again, I'm still working out the bugs on a reasonable video-making system, but hopefully this helps. I start with Example 4 from the Chapter 3, Section 7 notes (page 46).
 * **Part 1**: **@http://screencast.com/t/NzNlYWI4Y2**
 * **Part 2**: **@http://screencast.com/t/YTNiMGUwMTg**
 * **Part 3**: **@http://screencast.com/t/NTBiM2RiZ**

In general, related rates problems require a few important steps:
 * 1) Start by determining what the known and unknown quantities are (e.g., distances, masses, volumes, money, etc.).
 * 2) Try to find an equation that relates your known quantities and can be differentiated with respect to (usually) time. That is, find an equation (e.g., the Pythagorean Theorem in the above example) that involves the quantities you know, and that you can take the derivative of.
 * 3) Use all the rules (chain, product, quotient, etc.) and implicitly differentiate the entire equation with respect to time (//d/dt//).
 * 4) Once you have the derivative, substitute all known quantities and rates into the derivative and solve for the remaining unknown quantity or rate.

Also, I took some time and worked through solutions for some of the examples in the 3.7 Notes. Here they are presented as tutorials in a PDF for you to look over and work with.
 * [[file:Related Rate Examples from Notes.pdf]]

// Take your time, work out the problems as you read, and feel free to **contact me** with any questions you have. //

Oh, and I also created some related rate practice problems that we'll be using in class -- they're linked to from the **Chapter 3** wiki page.