11/21/10

Soooo this past week we started off by learning how to find horizontal asymptotes, which are in fact the same as vertical asymptotes except the fact that they are horizontal...duh. But anyway, you find those by taking the limit as x goes to infinity of whatever function you are given. And, you use your limit rules as x goes to infinity: if the degree of the top = degree of bottom, then the answer is the coefficients; if the degree of the top > degree of the bottom, then the answer is either -infinity or infinity; and if the degree of the top < degree of the bottom, then the answer is zero. Then once you get what the limit is, your horiztonal asymptote would be written as y="whatever the limit is"..Then we worked with curve sketching which involves an INSANELY LONG process before you even draw the graph...Then on Thursday we started learning optimization, which is kind of like related rate but a little different. Here are some examples of what we did this week:

Ex. 1) y=x(sqrt of 4-x) .."x times the square root of 4-x"
Directions: Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes.
*Well obviously I won't be sketching a graph on here, but to find all these things they're asking for you're going to have to do it in steps..
Step 1.) Find the domain and range of the function.
Domain: Back to advanced math..you take the inside of the square root and set it equal to zero, solve for x, and then set up a number line and plug in numbers from each side.
So when you solve for x you get x=4...plug in numbers on both sides of 4 on the number line...numbers to the left work, numbers to the right do not.
So your domain should be (-infinity,4]
Range: I'm not so sure about this since there's a square root..If I remember from advanced math correctly I think the range would be the shift..which there is none..
Sooo maybe the range is (-infinity,infinity)? or possibly (-infinity,0)u(0,infinity)?
Step 2.) Find vertical and horizontal asymptotes (if any) and find x and y intercepts.
*Well there are no vertical asymptotes because the function is not a fraction and I'm pretty sure there aren't any horizontal asymptoes either..so that step is done
*X-intercepts: Set the function equal to zero and solve for x
x=4
x-intercept is (4,0)
*Y-intercepts: Plug in 0 for x and solve for y
y-intercept is (0,0)
Step 3.) First Derivative Test!!!!!!!!!!
*We should all know how to take a derivative by now so here's what you get:
y'=(-3x+8)/2sqrt of 4-x
*Now take the numerator of the fraction and set it equal to zero and solve for x
-3x+8=0
x=8/3
*Now check for differentiability...and the function is in fact differentiable everywhere
*Now set up your intervals like this:
(-infinity,8/3)u(8/3,infinity)
*Now plug numbers in between those intervals into the 1st derivative
*for the first interval you can plug in 0 and you should get a positive number; for the second interval you can plug in 3 and you should get a negative number
-So that means you have a function that is increasing then decreasing...which means there is a max at x=8/3
^To find the y-value for that, all you do is plug 8/3 into the original equation and you should get something like 3.079
Step 4.) Second Derivative Test!!!!!!!!!!
*And again, I think we all know how to do that so here's what you get:
y''=(3x-16)/4(4-x)^3/2
*Now you set the top of the fraction equal to zero and solve for x
3x-16=0
x=16/3
*Now set up your intervals
(-infinity, 16/3)u(16/3, infinity)
*Now plug in values
*You can plug 0 into the 1st interval and you should get a negative number; and you can plug 6 into the second interval and you should get a negative number again...which means it's concave down and concave down
***And that's all the steps! Then you have to plot all the points and sketch the graph.

**Now for what I didn't understand this week...OPTIMIZATION! I honestly don't get how to find the equations first off..it seems like you pretty much pull them from thin air. I don't know, I guess I just need some work with it..