To sketch a slope field, you are given an equation and a graph with points plotted.
You have to plug the points plotted into the differential equation and depending on the outcome of the slope, that is what line you draw.
If the slope is positive, the line with be /
Negative will be \
A zero slope is a horizontal line __
An undefined slope is a vertical line |
Given that dy/dx= y-1 /x^2 sketch a slope field for the nine points. Plugging into the equation you get
(-1,0) = 0-1/(-1)^2 = -1 so \
(-1,1) = 0 __
(-1,2) = 1 /
(1,0) = -1 \
(1,1) = 0 __
(1,2) = 1 /
(2,0) = -1/4 \ except not as steep
(2,1) = 0 __
(2,2) = 1/4 / except not as steep
It's pretty simple except you have to try to make the lines of the slopes that are the same as similar as possible and the different ones have to be recognized is different.
We also learned how to find distance given velocity.
You have to integrate to get position the plug in each time value to the integral.
So an example would be
The integral of 2x with a distance of 0
2x integrated is = x^2
Then plug in each time value:
x(0) = f(0) - f(1) = -1
x(1) = f(1) - f(2) = -3
x(2) = f(2) - f(3) = -5
x(3) = f(3) -f(4) = -7
x(4) = f(4) - f(5) = -9
x(5)
Lastly, you add up the totals and get -25
However, since distance cannot be negative, I think it would just be 25
I'm not sure if this is a good example or not, but I tried.
I'm having trouble moving between graphs still and problems such as number 39 on the calculator portion of the AP test we took. It says The region S in the figure showen above is bounded by y=sec x and y=4. What is the volume of the solid formed when S is rotated about the x-axis.
Good luck tomorrow!
If the volume problem you are speaking of is the one I remember... you have to decide which equation is on top. Well, y=4 is on top of the secx. So you do pi times the integral of (4)^2 - sec^2(x) on the interval it said.
ReplyDeleteExactly what John said...people try to shift the whole graph down 4 when you just use washers to find the area...
ReplyDeletey=4
y=sec(x)
Once you graph it you find that y=4 is on top and y=sec(x) is on the bottom. Knowing that the formula for washers is π times the integral of the top equation squared - the bottom equation squared.
π integral ((4^2)-(sec(x)^2))
π integral (16-sec(x)^2)
π(16x-tan(x))
Before multiplying π in, you must find your interval so that you can plug in for x. To do this you just find your points of intersection between the graphs y=4 and y=sec(x). Once found you just find the definite integral then multiply by π and your done.