9/11/10

Alright, so this week the first thing we learned was differentiability (the ability to take a derivative). As in, if a function is not differentiable, then you shouldn't bother trying to take the derivative of it :) To review, a function is not differentiable when there is a "corner" in the graph (which is in absolute value and piecewise functions), when there is a vertical tangent line (when the exponent is an odd number over an odd number), when there is a cusp (graph when exponent is even # over an odd #), and when the function/graph is not continuous (if there's a jump, removable, or vertical asymptote). So, I think that's pretty easy to understand. Also this week we learned the shortcut to finding a derivative!! YAYYYYYYYYYY. And I must say, it is sooo much easier than the long way, however I get confused with some fractions (which I'll say in the "I don't get this stuff" part of this blog)
Anyway, here are a few examples of what I understood the most this week:

Ex. 1) (4x+1)^2 (0,1)
^Find the slope of the graph of the function at the given point.
*The first things you should notice are the key words-"slope of the graph" because that's telling you to take the derivative. Now that you know the shortcut, you can use that here.
*But first, you have to factor the equation. So you "square the first, square the last, 2 times the first times the last"
*So you get: 16x^2 + 8x +1
*Then all you do is take the derivative of each term
*So the derivative of 16x^2 would be 16(2)x = 32x
*d/dx of 8x would be 8
*d/dx of 1 (CONSTANT RULEEEE!) = 0
*So your equation for the slope is 32x + 8
*To get the slope, all you have to do is plug in the x coordinate of the point they gave you.
*So you get 32(0)+8 = 8

Ex. 2) Discuss the differentiability of f(x)=4x+1/x
*Since you may not know what the graph of this looks like, you can type the equation into your y= in your calculator to find out.
*You should see that there are vertical asymptotes (going to infinity and negative infinity) at x=0.
*Therefore, the function is not differentiable because it isn't continuous.
**Hint Hint!!--You really didn't have to type it in your calculator to know there are vertical asymptotes. Since you can't factor the function any more than it already is, and there's still something left in the denominator, setting that equal to zero will give you your vertical asymptotes..

**As for what I didn't understand this week. It's not much actually, I'll just have ALOT of questions to ask on Monday. For instance, I'm having trouble find the derivative of problems like this: (4x^3+3x^2)/(x) ..basically when it's an equation over another equation. I keep working the problems like that over and over, and I can't get the answer they have on calc chat...sooo if anyone can help, that would be appreciated.