([SIZE="2"]Moderator's note: thread moved from Calculus and Analysis)
I just need to know if my answer is correct for the following:
Find the z-coordinate of the center of gravity of the region:
(x/(z^3-1))^2 + (y/(z^3+1))^2<=1, 0<=z<=1.
I'm getting 7/16; am I in the ballpark? Thanks...
Given that X is a vector in R^n, what is the derivative of:
\texts{f(X)=|X|^{2}X}
I basically combined the product formula and dot product rule after breaking down |X|^2, which yielded my answer of:
\texts{f'(X)=|X|^{2}v+2(v\bullet X)X}
Where v the direction.
Is this correct/ close?
I've been trying to prove that if the following statement holds for all (x,y)ER^2, f must be a linear function:
f(x,y)-f(0,0)=x*(d/dx)[f(x,y)]+y*(d/dy)[f(x,y)]
It seems to work for any function I plug in, but I'm unable to establish why this always works. Also, when I say (d/dx)[f(x,y)], I...
Read the text; finding a counter example for one function f simply proves that it doesn't hold for that ONE function f; the problem asks to show this is true for ANY nonlinear function.
Prove the following is not true:
Let f : R^2->R^2 be a nonlinear function. For any vectors a,v in R^2;
f(a+v)-f(a)=[Df(a)]v
In terms of my attempt, I've been trying to find some combination of a and v that ensure this fails, but so far the best I've come up with is to start with...
I need to prove that the following is an open subset of R^2:
\left\{(x,y)\inR^{2}|\sqrt{x^2+y^2}<1}
I think the substition r=min{sqrt[x^2+y^2],1-sqrt[x^2+y^2]} works, but I'm stuck on how to take it from that to showing that the distance between X0 and X1 is less that r, and more...
Show that:
...{sin(1/x), x not zero
f(x)={
...{0, x=0
Is not a continuous function using epsilon-delta.
EDIT: I honestly haven't a clue. I figured I could just show that regardless of how small you make delta, there is always a value of f(x) that equals one, but I don't know how to...
I've been working on this for two hours and have had zero luck:
Given:
sum{k=1 to k=oo} [((-1)^(k+1))/k]
Rearrange the terms so the series converges to 5 [lol, I haven't a clue how].
I need to show that:
\lim_{x \to 0} x^{2}sin(1/x)=0
Using the epsilon-delta method. I figured delta=sqrt[epsilon] would make the limit hold, but wanted to be sure. Thanks in advance.
So I've been working on a proof and Wolfram Alpha gives the following partial sum formula for one of the summations in the proof:
http://www.wolframalpha.com/input/?i=sum%5Be%5E%28-%282k%2B1%29%2Fx%29%2F%281%2Be%5E%28-%282k%2B1%29%2Fx%29+%29%5E2+%5D
What does the terminology involving psi...
\sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-\frac{1}{nx}}}{n}
Where 0<x<oo.
I'm looking for a closed form/ closed representation for this series [I was thinking something like a polylogarithm or dirichlet eta function combination might work].
Any ideas or suggestions would be much appreciated.
\sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-(nx)^2}}{n^{1-m}}
Where m is an integer and 0<x<oo. I need a closed form solution, and was thinking something along the lines of a theta-type function, but cannot seem to locate any identities that match. Anyone have a suggestion?