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I found this but don't know if it is on the same topic, nor do I know how to derive it.
Nice explanations, mikau.
I've used these ideas myself to try to do problems on ILovePhysics.com, when I didn't know the formula.
Okay, now I'll give away how I came up with this.
I started off with the assumption that x^-.99 is similar to x^-1,
and then integrated.
I noted that the results didn't line up as I expected, so I figured it was the "+C" you get when you do an integral.
Anyway, I used the ln(55) for a test, and found I was off by around a 100.
Then I decided to average two curves.
the integral of x^-1.01 and x^-.99, so I did that.
Then my equation was pretty good, but I changed the 100's to 1000's to make it more accurate.
Which is like starting off with x^-1.001 and x^-.999, probably.
So these curves above and below ln(x) are not exactly equidistant, so that's why the answer is not perfect.
Here is a graph of x^3 and cos(x^3).
The slope is zero at the inflection point of x^3 at the origin, but the slope of the inflection points of the cos(x^3) are slanted slopes along the x-axis where the function crosses the x-axis.
So inflection points can have slopes other than zero, but as you said, the second derivative is zero
because that says the slope is not changing at that point.
The slope is in limbo for a second.
6 green, 12 blue, 12 yellow makes 30 out of 36.
So sell three lots of 10 to sell 30 tons.
$3000
Drawing a graph was helpful. y=20-2x, y= 16-x , y=3x, for x between 0 and 2.
Hi VR Hawks, didn't Louis Armstrong die the day you were born?
Wow! That sun-locator and lighted area of the globe is truly awesome and useful.
I love that ganesh. I think it's cool the mathmatician thought about the side of the sheep that was being viewed. That surprised me.
A PVC pipe is produced every 10 seconds per machine, once setup correctly. During setup, the operator should take measurements. And during operation, measurements should be taken to ensure high quality. Inner diameter could be
check non-evasively at the ends, and throughout by pumping water through the pipe and see if flow rate is as expected. The operator could insert of trumpet mouthpiece in the end and make a sound just for the fun of it. Drill holes could be made, and
an entire crew could play music on the 10 foot pipe, and then measurements could be taken through the holes, by inserting
skinny pointed rods with ruler markings on them.
Hi Justin. This may not help, but who knows?
Define the bottom or top of the parabola at (d,D)
I'll rename your (x1,y1) to be (a,A), just because I feel like it.
And (x2,y2) is (b,B). And (x3,y3) is (c,C).
Also your "a" multiplier, I will call "N", as it makes the parabola narrower.
Now if you get the point (d,D) and find N, then the equation for the parabola will be
(y-D) = N (x-d)^2
Okay, let's begin:
There are distance relationships in a parabola to do with squares,
hence we can write these equations:
N(a-d)^2 = |A-D|
N(b-d)^2 = |B-D|
N(c-d)^2 = |C-D|
I came up with this approximation for the ln(x).
1000/2(x^(1/1000) - 1/(x^(1/1000)))
or shown in LaTex as:
On the graph, click on it for bigger, the green dots are the above function,
while the red dots is the ln(x).
I just made this.
On the bottom there are dotted lines because I was going by increments in
a double loop checking for true statement for every (x,y) at intervals of 0.01
for positive x and y below 4.
Maybe the problem of my being logged in making a worksheet is that I am using the Opera Browser.
It is also incompatible with my ability to write emails at rock.com
Next time I make a worksheet, I'll use Internet Explorer instead.
What does "cross over" mean?
Also, I made a worksheet with the word flashlight in the title, and I was logged in, but it's gone. Can't find it. Oh well I have it stored at home if you can't find it.
Also, there still exists two Tricky Subtraction worksheets, and one has a simple question and a long answer, which I'd like to keep, but the other one has a really long question section, and a shorter answer. This one could be deleted.
...
Oh I think I know what you mean about crossing over root and suffix, sort of like if the word was ripeness, you wouldn't want to use the "en", use "ness" instead. Yeah, that could get tricky to program, add in priorities if possible.
The grade two Braille is working really nicely. I can read it since I studied it many years ago and am quite familiar. What are the 70 abbreviations you have programmed. I'll test them all in sentences. I saw "this", and "that", and "the", "er", "en", "in", etc.
I noticed that the graphic image of Braille, when more than a few lines, the rest gets cut off, just so you know. I love the variations of displaying the Braille with/without empty dots. Awesome work!!!
(off the subject, but the word receive is spelt recieve when you send out a web card.
I looked at the copy and it looks great except I left out an "of" or something in one sentence, but that's minor. Yes, I think I realized I might not have been logged in later. How do you add in newlines and indents in the editor? Use <HTML> or just press <enter> and <tab> when editting?
The reason I used <pre> is because the <Enter> keys I did press were not shown when I was first making the Tricky Subtraction worksheet. Also check out Google today! The title "Google" is in Braille, what a coincidence!
I just made a worksheet twice called Tricky Subtraction.
It isn't searchable yet, probably because it isn't approved yet.
Feel free to change it or have me rewrite it if it is still around and not deleted.
Excellent work, MathIsFun!! I don't know who would use it. But it certainly is a nice thing to do because then families of a blind person, can play around with your translator. How people use either text dots or a bitmap image is endless. But someone would surely find it useful for something if you made it. Also, you are currently translating into grade 1 braille, which is single letters. In grade 2 braille, which is what the Braille Reader's Digest is in, they have other characters for common letter combinations like "er", "ing", "en", "in", "the", etc. I didn't try a period on your thing yet, but I noticed the exclamation point and question marks were both the same and as a lowered "e". Actually, they are lowered "f" and lowered "h". Period is lowered "d".
I'm just guessing, but I think for question "c." you can't take an average because when you want to multiply by 4 or more, what number do you pick 4.3, 4.2, 4.1, or 4, etc. I would call the formula a weighted average.
( 0 x 7083 ) + (1 x 18621) + (2 x 22414) + (3 x 5533) + (4 or more times 2797)
____________________________________________________________________
(7083 + 18621 + 22414 + 5533 + 2797)
That horizontal line is the divide by sign. It's a fraction.
In all the light green graphs of previous posts, the point (e,0) is wrong, it is (e, -1) like in krassi's hand drawing before he raised it by 1.
Also here is some more information in a new plot showing three regions to be rotated, the hardest one being the skinny area I have labelled region C.
slope = y' = e^x - 1/x - x^x(1+lnx)
By looking at a graph, it looks like x=.67 is about horizontal slope, so
try .67 for x to see if is nearly zero slope., and I get slope = 0.003267, which is awesome!!
Now try 1/e to see if it is slope of -1.
try x = .367879441, and I got slope is -1.2736, thus the tangent with slope -1 is to the right a little bit.
What is this number for x??
Will try to find this x with a BASIC program. Got x = 0.4112922
From this we can conclude that "area 4" is non-zero.
So we have 4 regions to rotate around y=-x+1 to find the volume.
Okay, I found one way in this pdf:
www.bitjungle.com/~isoent/isoent-ref.pdf
\textgreater
I figured out a way to do it by guessing.
It works on this LaTex practicer (http://www.artofproblemsolving.com/LaTeX/AoPS_L_TeXer.php), but not here.
. . . .. . .. . ..
. . . . . . .. The lowered "f" ( .. ) is an exclamation point.
. . .. . .Oops, my mistake. Nice explaining!