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I always use the integrator when I'm helping someone out, because with integration I tend to make stupid little mistakes, so I always make sure I'm right... so I don't mislead the person asking the question.
Did I do something wrong? I was kind of just winging it, and my answer matched The Integrator's.
People will always tell you you're wrong. Sometimes they're right, sometimes there not.
I suggest you take down the names and numbers of everyone who told you you'd fail, that way when you graduate with a math major you can call all of them and say something like, "Told you so."
You may in fact hate it, you may not. Just do it, and if you're not happy, switch majors.
Hmmm, I brought this up in July, and it was pretty much ignored. And now people seem interested.
I knew about triangle numbers being on Pascal's Triangle, but I didn't know tetrahedral numbers were.
And I'd never heard of pentatopic or figurate numbers. Interesting.
Distributive property
The more traditional way of writing it is the way you did, I just didn't realize that at the time.
It seems right to me, although like I said before, I have no formal training in statistics. It certainly "feels" right though.
Why not just combine all the different threads and let him "play in his playground" by himself. If it offends you, don't read it If it doesn't offend you, well that's fine too.
I agree, that's a huge assumption. He did however say, "Note, there is a 50/50 chance of the father being BB or Bb."
He actually said "BB or BB" but I'm assuming he meant "BB or Bb"
Have you ever taken a formal probability class? I haven't, what little I know is ether intuitive, or I've read about it somewhere.
You have two possible punett squares:
Square A:
| b | b
----+------+-----
B | Bb | Bb
----+------+-----
b | bb | bb
Square B:
| b | b
----+------+-----
B | Bb | Bb
----+------+-----
B | Bb | BbSo the odds are
(6/8) = 75% Bb (Brown)
and
(2/8) = 25% bb (Blue)
Assuming, that is, that there is a 50%-50% chance that the Dad is brown either dominate or recessive.
Personally I would just make all the Punett squares needed the write down all the possible answers. It's tedious and probably the longest possible method, but if you can spare the time (ie the test is not timed), it's a good way to know for sure you're right.
I read this book called, "An Imaginary Tale: The Story of √-1" by Paul J. Nahin. It's pretty good, but there's one part in the book that I got kind of hung up on.
In this post, "x" will always be a real number.
He shows that by using the differential arc length formula:
And by using the distance formula:
That for any real curve y, you can find the ratio of the distance of the arc length to the distance of the secant, obviously that ratio will always be more than 1 because the shortest distance between to points is a strait line.
But then he uses the complex function, y(x) = x² + ix and shows that the ratio is less than one. Personally I think that real cool.
The issue I'm having though, is that I think his equations are wrong. If I were to graph y(x) = x² + ix I would graph it so that x was the domain, y was the real part of the range, and z was the imaginary part of the range (making the y,z plane the complex plane). But then the arc length would be a different formula altogether, and the ratio would we greater than 1 (like it is for functions whose domain and range are entirely real).
Is my reasoning way off? What would be the arc length formula?
Woah, cool. Care to explain what the answer is? I have the feeling it'll be a bit over my head...
Most of that made sense to me (I had to read through it a few times) but what does that line over the limit mean? That's the first time I've seen that.
(Just a comprehension check. Omega_3(S) = {x, y, x², y², x³, y³, xy, x²y, xy²})
Do they have a firm understanding of the Cartesian Plane? And a firm understanding of Algebraic manipulation? If so, I guess either the graph or the equation is a good place to start....
It's hard to come up with something at a 4th grade level. I'd say try to introduce them to the ideas using graphs. Because functions are abstract concepts, it might be harder for younger kids to get their head around.
Unless of course they're exceptionally smart kids. I myself was in an accelerated math class and I didn't deal with these concepts until 7th grade.
sinx cosy + 3 cosx siny = 2 sinx siny - 4 cosx cosy
dividing through by cosx:
tanx cosy + 3siny = 2 tanx siny - 4cosy
dividing through by cosy:
tanx + 3 tany = 2 tanx tany - 4
tanx +4 = 2 tanx tany - 3 tany
tanx + 4 = (2 tanx - 3) tany
(tanx + 4)/(2 tan x - 3) = tany
EDIT: Ahhh, I was too slow. I am glad to see I was right though.
side a = √2
side c = 2
Use the law of cosines to get b in terms of the cosine of the angle opposite side c:
Use the quadratic formula:
From that we can deduce that if gamma is 90° then b is √2.
Now we can use a modified version of Heron's formula:
seeing how Area (A) is 1, we can say:
Subbing our original values for a and b:
so, b is ±√2 or ±√10
Now, we've seen already that b is √2 when gamma is 90° But what of the other three values? I don't know how to say it technically, but it has something to do with the fact that they're negative angles (note √10 > pi).
This is an excellent question, and is something that I think needs a lot of discussion. Personally I think of a function like a person, and a graph is sort of like a personality trait of the function, it tells you certain things about it, but not everything. Another personality trait might be whether it's odd or even, what its derivative or integral is, what its domain and range are, etc...
A function is an abstract concept, we can't see or feel it so we use these "personality traits" to better understand what we're working with.
Out of curiosity, are you asking this for a pure mathematics cause (math for the sake of math) or an applied mathematics cause (math to be used somewhere else)?
All you need to do is apply integration by parts.
After a little work with google, and that uselessly vague hint, I figured it out:
Don't stress. That's the advice, really. I personally have never studied for a test (except Spanish class) and I've always done well.
That's not because I'm exceptionally smart or anything, it's because I have a relaxed attitude about it. You have to rely on what you know and trust yourself . I know it sounds dorky, but it's true.
I took an art class once, (I'm a horrid artist) and I had to do a quick sketch of what my final project should look like, and when all was said and done, my rough sketch (that I did in a few minutes) was MUCH better than the project I spent hours on. I asked my art teacher about it, and he said it was because I was just casually doing the sketch, not concerned about what it'll come out like. When was working on the final project I tried really hard, and really stressed about not making a mistake, and it really came out bad.
So the trick is, just kind of chill. Study the material until you are comfortable, but when you start to freek out, just stop, read a good book, have something to eat, whatever.
x*tan-¹[x] - ln[1 + x²]/2
I don't know how to show this, integrals like this always send me strait to a table or The Integrator.
Now at this point it's starting to look worse, but bare with me...
Now it looks really bad, but by shifting some constants around we get:
Now if you pay close attention, you'll see that there's an (∫e^(2x) cos(3x) dx) term on both sides so we can :
∫ tanh(2x+1) dx = (1/2)ln(cosh(2x+1))
tanh(u) = -i*tan(iu) ( i = √(-1) )
u = 2x+1
du = 2 dx
thus:
(1/2) du = dx
∫ -i*tan(iu) (1/2)du = (-i/2)∫tan(iu) du = (1/2)(ln(|cos(iu)|) + C
cosh(u) = cos(iu)
(1/2) ln(cosh u) + C = (1/2)ln(cosh(2x+1))
I disagree londof+, I thing JaneFairfax's proof kind of kills any chance of another answer.
So then there can be only one Mandelbrot Set, but there are an infinity of Julia sets, is that right?
Thanks for the code by the way, that really clears things up.
Does anyone know the difference?
I almost put this in the Help Me section, then I thought about the code section, but I thought that this might be appropriate here. I understand how they are created, this wiki was particularly helpful (specifically the code part) but I can't figure out the defferance between the Mandelbrot Sets and the Julia Sets. And because there's no code part on the Julia Stes page I can't even take a guess. The process looks the same in both cases for me.
Unfortunately the formal definitions are simply too technical for me.