# Search results

1. ### Polynomials question

I think this it: You can show that any real solution of ##x^4+ax^3+bx^2+ax+1=0## is also a solution of ##x^4-ax^3+bx^2-ax+1=0##. Then you can combine these two equations and get that this solution must be a solution of ##x^4+bx^2+1=0##. But this is great, because you can show from here that...
2. ### Polynomials question

Here is another way to think about it: If ##x^4+ax^3+bx^2+ax+1## has at least one real root, then that means its graph must intersect the x-axis. Since this is a 4th degree equation with a positive leading coefficient the tails of the graph both go off towards positive infinity. Therefore, if...
3. ### Spinning Egg Problem

I'm trying to understand the spinning egg phenomenon. That is, why does a hardboiled egg (or any solid ellipsoid) "stand on end" when spun at high velocities. Upon searching the web I found one site written by physicist Rod Cross in Sydney who tried to give a simple, intuitive explanation...
4. ### Area of a circle by integration

That's probably what you call it. Notice that the power reducing formulas are really just rewritten versions of double angle formulas - so I tend to call them all a "double angle formula" because I try to memorize as few formulas as possible.
5. ### At what points on this curve is the tangent line horizontal?

Wolfram Alpha gave the following graph when asking for the tangent line to y=(x^2)/(2x+5) at x=-5. If your graph doesn't look like this then you have either told us the wrong function or mis-entered something into your calculator...
6. ### Area of a circle by integration

Your problem was in your substitution, you successfully substituted for ##x## but you did not substitute for " ## dx ## ". If you do this you will get that: $$dx = -r \sin (\theta ) d\theta$$ Which will give the integral: $$\int_0^{\pi / 2} r^2 \sin^2 \theta$$ which you can solve using...
7. ### Differentiation with different variables

I found the expression \frac{d}{d V} \int_0^t{V(\tau)}d\tau a little surprising. I'm not sure how to interpret the \frac{d}{d V}. I would have expected to see \frac{d}{d t} \int_0^t{V(\tau)}d\tau instead. In that case it is exactly the fundamental theorem of calculus and the solution is...
8. ### Limit of factorial functions

I approached it a little differently than you - here was my thinking: in the numerator distribute the 2's so that you get two products of only even numbers: ##\frac{(2*4*6*...*2n)^2}{2n!}## This would allow you to cancel out the even terms in the denominator and get...
9. ### Cardinality of infinite sequences of real numbers

Mathematicians do occasionally discuss "transfinite sequences" or sequences with an arbitrary index that may the uncountable. If the sequence is of real numbers and an uncountable number of the terms are nonzero, then the sum of the sequence necessarily diverges.
10. ### Motion of a Spinning Coin

Thanks for the response - I really appreciate it. let me see if I can clarify (or have clarified) some of these comments: It sounds to me that you believe the axis about which an object will spin is most strongly determined by the stability of its axes. I was going to approach this...

17. ### Cardinality of infinite sequences of real numbers

Great point, jbunniii, we should definitely make sure to deal with multiple expansions. (By the way, I'm sorry that I have edited and updated my post multiple times. I'm still trying to figure out how to use the Tex features properly.)
18. ### Cardinality of infinite sequences of real numbers

I would start by thinking about why the cardinality of ##[0,1)^2## is equal to the cardinality ##[0,1)##. To do this you think realize that any element of ##[0,1)^2## can be written as ##(x,y)## where ##x## and ##y## have infinite decimal expansions ##x = a_1 a_2 a_3 ...## and ##y = b_1 b_2...
19. ### Motion of a Spinning Coin

I have several questions about the motion of spinning coins. I won't get to them all in this post, but hopefully some of you can help get me started. (Note: I am a mathematician, not a physicist - that might help color your responses.) First I want to think about spinning an idealized penny...