This week in Calculus we reviewed and learned new things. One thing I am not comvortable with is simplifying the second derivative, but I semi-understand how to do it..which really doesn't make sense.
The thing I am most comfortable with is the second derivative test. This test involves points of inflection and convavity, which is concave up or concave down. When taking the second derivative you must be aware that points of inflection only happen if there is a change in concavity.
To do the second derivative, first, you must take the derivative of the equation such as
6/(x^2 +3) = ( (x^2+3)(0) - [(6)(2x)] ) / (x^2 + 3)^2
Which gives you (-12x) / ( (x^2 +3) ^2) in it's simpliest terms. Next, you must take the derivative of this equation. Which would give you (36(x^2 -1)) / (x^2 +3)^3
This must then be solved to it's simpliest state set equal to zero which is (36(x+1)(x-1)) / (x^2+3)^3
Now, you have found the critical values: X = +/- 1. These are only POTENTIAL points of influction. Next, you must set up intervals (-infinity, -1) u (-1, 1) u (1, infinity)
Now, you must pick a number in each interval and plug it into the second derivative. This will tell you if the interval concaves up or down.
f''(-2) = positive = concave up
f''(0) = negative = concave down
f''(2) = positvie = concave up
Another way to write this is:
at (-infinity, -1) and (1, infinity) = concave up
at (-1, 1) = concave down
points of inflection = x=-1
x=1
These are the points of inflection because this is where the concavity changes. It is very important that you know points of inflection ONLY occur with changes of concavity.
Things I am still not comfortable with is finding average velocity, slope, and instantaneous speed. If anyone can help me understand it would make me pumped for life.
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ReplyDeleteOkay for instantaneous speed it's really really easy. Not too hard to remember.
ReplyDeleteSo what have we done all year so far? Take derivatives right?
So for instantaneous speed problems...simply take the derivative.
Well now you're wondering, okay, they asked me for instantaneous speed at x=2. What should I do with that?
Well, what else is there really to do? Just simply plug in 2 for all the x's in the derivative (or whatever variable they may use) and what you get is the instantaneous speed at the x value.
That's how I think about it. I just kind of figure all we have been doing is taking derivatives and plugging in, so why not here too? :-)
-John
For average speed [velocity] also means the slope, the derivative, and the prime.
ReplyDeleteSo basically, you have to remember that slope is:
y(2) - y(1)
-----------
t(2) - t(1)
so, if you look in your notes, on the first problem we did with average speed..it should be:
An anvil falls off the roof onto Ryan's head. What is its average speed during the first two seconds if y=(16t)^2 describes the fall?
so, you know that you have kinda like a limit with these because of the "first two seconds" as stated in the question..
so t(1) and t(2) will be (0,2).
now, taking t(1) plug into the equation they gave you. [which is y=(16t)^2] so that will give you y=0
now, taking t(2) plug into the equation they gave you. [which is y=(16t)^2] so that will give you y=64
remember how i said that slope is:
y(2) - y(1)
-----------
t(2) - t(1)
well take what you have and plug that into the formula
so you know that the bottom is 2-0 because at the begining we have the point (0,2) [(t(1),t(2))]
for the top..remember we pluged in 0 first..well 0 was t(1) so that means that whatever the answer was when we pluged in 0 to the equation they gave us will be the y(1) just like when we pluged in 2, that was t(2) so that means that whatever the answer was when we pluged in 2 to the equation they gave us will be the y(2).
so when you plug in it will be
64-0
-----
2-0
which is 64 divided by 2
so your answer will be:
32 __ per second
[the __per second .. well your answer will always have the unit of length over the unit of time..and the problem doesn't give us a unit of length so that's why i left a blank, however it does say "during the first two seconds" so we know it'll be per second]
HOPE THIS HELPS!