21we must hav an IN-BUILT colouring system for Prophet Sir, all whose posts are automatically coloured and even Stored in a separate area Reading : "PROLIFIC PROPHET"
Either you will have this in a trice or you will slog it out for several minutes:
Evaluate \int_0^1 \frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-c)(x-a)}{(b-c)(b-a)} + \frac{(x-a)(x-b)}{(c-b)(c-a)} \ dx
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26 Answers
39sry sir. i meant same as prophet sir meant, thought people would get it. he explained it better way.
21
1sir my bookmarks are overflowing due to all your posts...[9].
afer reading all this on screen im completely [7]
this applies to almost all ur posts...
I have to get ur posts printed and read them on paper for better grip
frankly its hell lot difficult for me to concentrate on screen...(habitually)
341Sometimes when you see a problem, it helps to get inside the mind of the author of the problem. Of course, you guys putting x=a etc. you get the result. Do you think the author had to do this.
So lets see how this is immediately obvious:
Lagrange Interpolation Formula:
I will give the formula for the quadratic case and its easily extended to any degree polynomial.
The theorem says that if we know if P(x1) = a, P(x2) =b and P(x3) = c, then a quadratic P(x) satisfying the given conditions is uniquely determined as
P(x) = \frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)} P(x_1) + \frac{(x-x_3)(x-x_1)}{(x_2-x_3)(x_2-x_1)} P(x_2)+ \frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)} P(x_3)
So, for an nth degree polynomial we will need the values taken by the polynomial at n+1 points
Now you can see that all the author has done is to put
P(x_1) = P(x_2) = P(x_3) = 1
62I cantthink of any..
i guess we have to wait for theProphet..
or may be b555 seems to have discovered it?!
1@ prophet sir
"
Can anyone tell me why is the polynomial written in this peculiar way, instead of taking the coefficients to the numerator?
"
well becoz it would make it a hell more complicated......hehe sorry for bad joke .... but sir... u've in ur post only mentioned that
"(oops i missed stating that a,b and c are distinct)"
now only if we could put two and two together and say that when the expression is given in that form u dont need to mention...
coz anyways they are distinct.
11btw, bhargav.....how do u remember all those identities??? mind-boggling stuff, really!!
11Dirtiest method:
Why not expand the function and see where we end up????? I have a feeling it will get too dirty for most ppl's liking [12]
21HMMMMM
ALL V get frm
f(a)=f(b)=f(c)................... provided f is a poly of degree 2 is dat
f(x) = COSTANT........
but to get it as 1 we need to sub a value ....... (zero wud b easiie)
13writin this in this peculiar way maks the observation that f(a)=f(b)=f(c) easy
62I dont thnk anyone is doubting that...
The point that prophet sir is making is something different all together...
He is trying to say that this is a constant function!
We know the value at one point and hence we know it everywhere
we dont need to find the summation at all.
62well we can also say that
any polynomial of degree "n" cant take a given value more than n times
here, f(a)=f(b)=f(c)
Those who did not understand whay TheProphet had to say, please make sure that you do.. It is one of the best concepts ..
and please try to reply to the question by The Prophet :)
341what mathie meant was:
Call the given expression as f(x). Let g(x) = f(x) - 1.
g(x) has degree at most two and hence by fundamental theorem of algebra has at most two roots.
But g(x) has three distinct roots a,b and c (oops i missed stating that a,b and c are distinct). This can happen iff g(x) is identically zero.
that means f(x) is identically equal to 1
Discussion:
Can anyone tell me why is the polynomial written in this peculiar way, instead of taking the coefficients to the numerator?
21Mathie I didnt get ur Proof? (if u can call it one)................ BTW can v sub. "x" [7]
wat I used was a getaway method,
taking a = 1, b=0, c =-1
now the x part gets cancelled and wat u get is 1
341yes, that is the crux. You would have learnt in your 10th that the expression is identically equal to 1. Now, if someone could supply the proof, the thread can be closed
39answer is 1.
thats because watever is inside the integral is an identity and is equal to 1 always for all values of x. [why?!]
so it is \int_{0}^{1}{1}=1
13each can b integrated separately and then simplified if i m to slog for several min but .....
11symmetry to dikh rahi hai
we have to use it to find the answer