12/12/10

Alright well this week all we did was work on our study guides in class...And yet, I'm still not done with them haha. That's pretty sad..anyway I guess I'll just give a few random examples like some we'll probably see on the exam. Soooo, here we go..

Ex. 1) Find the slope of the graph of the function at the given value.
f(x)=2x^2+(3/x^2) when x=3
*First you should notice that there's two things going on in this problem--a term and a fraction. So what you can do is either take the derivative of each one separately or find a common denominator and turn it into one fraction...I'm going to find a common denominator and then use the quotient rule
*So to find your common denominator you multiply 2x^2 by x^2 and now your new fraction is 2x^4+3/x^2
*Now using the quotient rule you get:
(x^2)(8x^3)-[(2x^4+3)(2x)]/(x^4)
=4x^5-6x/x^4
^From there you can take out an x in the numerator, so you'll be able to cancel an x on the denominator also
*So now you have 4x^4-6/x^3
*Now all you do is plug 3 into that equation^
*So you get 318/27....which reduces to 106/9

Ex. 2) Determine all values of x (if any) at which the graph of the function has a horizontal tangent.
f(x)=x^3+3x^2+4
*To find a horizontal tangent, all you have to do is take the derivative, set it equal to zero and then solve for x
*So for the derivative you get 3x^2+6x
*Set it equal to zero:
3x^2+6x=0
3x(x+2)=0
x=0 and x=-2

Ex. 3) Suppose the position function for a free-falling object on a certain planet is given by s(t)=-16t^2+v0t+s0. A silver coin is dropped from the top of a building that is 1380 feet tall. Determine the average velocity of the coin over the time interval [3,4].
*First off, since it's asking for the average velocity, that means you're going to have to find the slope. So you won't be taking the derivative at all in this problem
*So to find the slope first you need to find the y-values for 3 and 4. To do that you plug 3 into the original equation -16t^2 and then plug in 4 after
*So when you plug in 3 you should get -144
*And when you plug in 4 you should get -256
*Now using the points you just found--(3,-144) & (4,-256)-- you have to use the slope formula which is (y2-y1)/(x2-x1)
*So once you plug in your points you should end up with:
-256+144/4-3
=-112 ft/sec

**I don't understand a lot of the problems in our studyguide packets...(packets 2 and 3)