So, I've got 5ish days to decide which of these institutions (UChicago, Columbia) to attend next year. At UChicago, I'd like to double major in Physics and Economics. At Columbia, I am accepted at the engineering school and I would major in Applied Physics and minor in Economics.
The main...
Homework Statement
A uniform round tabletop of diameter 4.0 m and mass 50.0 kg rests on massless, evenly spaced legs of length 1.0 m and spacing 3.0 m. A carpenter sits on the edge of the table. What is the maximum mass of the carpenter such that the table remains upright? Assume that the force...
Homework Statement
A spaceborne energy storage device consists of two equal masses connected by a tether and rotating about their center of mass. Additional energy is stored by reeling in the tether; no external forces are applied. Initially the device has kinetic energy E and rotates at...
Homework Statement
Evaluate \displaystyle \int_C y^2dx + x^2dy for the path C: the boundary of the region lying between the graphs of \displaystyle y=x and \displaystyle y=\frac{x^2}{4}.
Homework Equations
The catch is that you can't use Green's Theorem.
The Attempt at a Solution...
Homework Statement
Given the integral: \int_1^2\int_{\frac{3}{\sqrt{x}}}^{\sqrt{3}x}{{(x^2+y^2)}^{\frac{3}{2}}}dy \; dx. Convert to polar and evaluate.
Homework Equations
r=\sqrt{(x^2+y^2)}
The Attempt at a Solution
Ok, I've gotten bounds on \theta, \frac{\pi}{6} \le \theta \le...
Homework Statement
Find the dimensions of the rectangular box of largest volume that can be inscribed in a sphere of radius 1.
Homework Equations
v=w*l*h, Set the partials equal to 0, then solve a system, etc.
The Attempt at a Solution
I'm really just unsure of the constraints that...
Homework Statement
Given some plane, 3x+2y-z=6, and a point (2,3,6)
Find a line in the plane passing through that point.
Homework Equations
The Attempt at a Solution
I tried finding the vector perpendicular to the plane, <3,2,-1>, but I'm not sure what to do with it.
Homework Statement
Why are the following not contradictory?
r=\sqrt{x^2+y^2}
\frac{\partial r}{\partial x}=\frac{x}{\sqrt{x^2+y^2}}=\frac{x}{r}=cos{\theta}
and
r=\frac{x}{cos{\theta}}
\frac{\partial r}{\partial x}=\frac{1}{cos{\theta}}
Homework Equations
The Attempt at a...
Homework Statement
Let f(x,y)=2x+3y.
Let \epsilon be any positive number. Show that there is a disk with center (1,1) such that whenever P is in that disk, |f(P)-5|< \epsilon. (Give \delta as a function of \epsilon.)
Homework Equations
None.
The Attempt at a Solution
Um, I tried...
I am a 10th grader in high school and I applied to a prestigious prep school in New England this spring. They were accepting late applications and would then place students on a waiting list. The only way off was attrition, for which they expected at least a few spots to open. The said they...
Homework Statement
Show that if Lim|\frac{a_{n+1}}{a_{n}}| = L > 1, then {a_{n}\rightarrow \infty as n\rightarrow\infty
Also, from that, deduce that a_{n} does not approach 0 as n \rightarrow \infty .
Homework Equations
The book suggests showing some number r>1 such that for some...
[SOLVED] SAT Subject Test in Physics
Homework Statement
Assume that every projectile fired from a toy cannon experiences force F along the length of the barrel. If a projectile of mass M leaves the barrel with speed, V, at what speed will a projectile of mass 2M leave the barrel with...
Homework Statement
a) If c is a number and \sum a_{n} from n=1 to infinity is convergent to L, show that \sum ca_{n} from n=1 to infinity is convergent to cL, using the precise definition of a sequence.
b)If \sum a_{n} from n=1 to infinity and \sum b_{n} from n=1 to infinity are convergent...
[SOLVED] Convergent Series Identities
Homework Statement
a) If c is a number and \sum a_{n} from n=1 to infinity is convergent to L, show that \sum ca_{n} from n=1 to infinity is convergent to cL, using the precise definition of a sequence.
b)If \sum a_{n} from n=1 to infinity and \sum...
Homework Statement
Show that each limit is a definite integral.
Lim (n \rightarrow \infty) of \sum \frac{n}{n^{2}+i^{2}} from i=1 to n
Homework Equations
Lim (n \rightarrow \infty) of \sum f(c)\Delta X from i=1 to n
The Attempt at a Solution
I can't really get this started, so...
Homework Statement
Prove by induction. The sum from 1 to n of \frac{1}{\sqrt{i}} \geq \sqrt{n}.
Homework Equations
none.
The Attempt at a Solution
I verified the base case, and all of that, I just can't get to anything useful. I tried expanding the sum and adding the...
Homework Statement
Prove that 2^{x} \geq x^{2}
\forall x\geq5
Homework Equations
2^{k} \geq x^{k}
The Attempt at a Solution
I multiplied both sides by 2, then added to factor the RHS, and eventually got to the point where I just need to prove 2^{n} \geq...
Homework Statement
Prove by Induction that x^{k}-1=(x-1)(x^{k-1}+x^{k-2}+...+1)
Homework Equations
None? I need to prove x^{k+1}-1=(x-1)(x^{k}+x^{k-1}+...+1) by assuming this is true: x^{k}-1=(x-1)(x^{k-1}+x^{k-2}+...+1)
The Attempt at a Solution...
So I got rejected from both programs I had lined up and applied to. I have a few options, but was really hoping you guys would have many more.
I could take two classes online from ArtOfProblemSolving.com, they would help me prepare for math olympiads and contests, and I would certainly enjoy...
So, I am a sophomore at a Math/Science Magnet school and I am considering applying to colleges a year early and relinquishing my right to obtain a high school diploma. I scored 2300+ on the SAT I. I also have 800s on the necessary SAT IIs, so the standardized testing crap is out of the way. I...
Homework Statement
\int\frac{\sqrt{x^{2}+1}}{x^{2}} dx
Homework Equations
The Attempt at a Solution
I tried a Trig Substitution with x=tan(u) and ended up with \int (csc(u))^{2} * sec(u) du
From here I am kind of stuck. I tried a few different integration by parts methods, but they got...
Homework Statement
Find the length of the spiral of r=1/theta for theta\geq2 \pi
Homework Equations
\int\sqrt{r^{2}+r'^{2}}
The Attempt at a Solution
I thought of the formula for polar arc length, which is the integral of the square root of the sum of the square of r and the square...
Homework Statement
Does the integral from 0 to Infinity of \int\frac{1}{\sqrt x \sqrt{x+1}\sqrt{x+2}}dx converge or diverge?
Homework Equations
None.
The Attempt at a Solution
I tried to integrate it, but I haven't even been able to do that so I couldn't then evaluate the limit of...
[SOLVED] Simple Integration Problem
Homework Statement
Integrate: \int\frac{dx}{3\sin x+2}
Homework Equations
....none
The Attempt at a Solution
I tried to do this by u-sub, and that was unsuccessful, then after that I tried a trig. sub with 3sinx=2tan(theta)^2, and that got very...
If given two arbitrary points on a parabola, is there anyway to determine an explicit equation for that parabola. I know there are multiple possible parabolas, but how can I find just one of them?
This is a problem from Terence Tao's book, and I cannot locate the solution anywhere, so I thought I'd post it here.
1. Homework Statement
X^4+131=3y^4, Show that the equation has no solutions when x and y are integers.
Homework Equations
None? Not sure.
The Attempt at a...
What methods are there to show that three points are, or aren't collinear?
I know the standard, check the slopes stuff, but what other ways are there, I think there are some more obscure methods for geometric shapes right?
Thanks very much.
OK I have a need to print a textbook out, mainly because I cannot locate it for cheap anywhere. It is 690 pages and would just need to be spiral bound or something. It doesn't have to be fancy paper or anything crazy, just black and white, double sided, the normal stuff.
I was thinking maybe...
Hey,
I'm interested in Distance Education, specifically for Linear Algebra. I know about the standard EPGY, and JHU, etc. I know there must be better, or at least other options out there and I was wondering if anyone has experience with them. Also is Linear algebra a course you can take...
I am finishing up a Calc. I course @ columbia university and I'm planning on taking Calc II back @ school (High school - 10th grade). I was wondering if it is possible to take Linear Algebra without having taken courses in Calc. II and Multi-Variable. In Calc I, I've done fairly well...