IMPLICIT DERIVATIVES:
1. Take the derivative like normal.
2. For each y-term, you put y' or dy/dx behind it.
3. Solve for dy/dy or y'.
36x^2 + 2y = 9
36x + 2(dy/dx) = 0
2(dy/dx) = -36x
-36x/2
dy/dx = -18x
steps to SECOND IMPLICIT DERIVATIVES
1. Take the derivative of the first derivative
2. Put d^2y/dx^2 for dy/dx
3. Simplify
4. Plug in values
so an example using the same equation is:
we take the first implicit derivative first
36x^2 + 2y = 9
36x + 2(dy/dx) = 0
2(dy/dx) = -36x
-36x/2
dy/dx = -18x
You have to identify what you are looking for and what you are given. Not only does this make it easier on you, it's kind of necessary, especially when you want those points on free response questions (or so I'm told). You also have to realize that when you are doing related rates, you have to put dy/dt or dx/dt or whatever whenever you are taking the derivative of some variable in relation to time (hence the t). So given that:
Given xy = 4
you want to know what dy/dt equals given x = 8 and dx/dt=10.
Take the derivative (product rule):
dx/dt y + dy/dt x = 0
Plug in everything:
dy/dt = -10y/8
= -5y/4
I need help on the problems with I, II, and III...you know what I mean? How do I attack the problem???
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Well, depending on what they are asking, I'm pretty sure you test each one (I, II, and III)
ReplyDeleteMost of the time, when they ask you which of the following is true, most of the time they ask you to find if it is continuous. You can check this to see if there are any asymptotes, jumps, cusps, corners, etc. If they ask if it is differentiable, the want you to take the derivative of each function and see if when solved for x, if they give you the same answer. If they ask for local max or mins at a certain number, you use the first derivative test and plug in like normal. Hope this helps!
ReplyDeletethe ones with I, II, and III usually asks if the two piecewise functions have limits, are continuous, and are differentiable at a certain point. For limits, you plug in the number into both equations and see if they wind up with the same solutions. Continuous is pretty much the same thing. And differentiable, for one see if there is an equal to sign somewhere in the piecewise, if there isn't, then there's a removable. Then take the derivative and and plug in the point. If they're the same solutions then it is differentiable at that point.
ReplyDeleteYOu see what I,II,and III are all asking and you test them to see if their true or false
ReplyDeleteFor these, just plug in the answer choices and see which ones are true.
ReplyDelete