Posted on 15 December 2016 - 10:36 AM
Utsuho wrote
Purpkey wrote...
I see nothing wrong here, and anyways it's going to be up to him to answer my questions or not.
I didnt say anything was wrong, I simply stated your type of question is drifting away from the actual name of the thread which is math and science. I understand where your coming from and since this is still within his field he may answer this.
Last edited on 15 December 2016 - 10:42 AM by Utsuho
Purpkey wrote
Utsuho wrote...
I didnt say anything was wrong, I simply stated your type of question is drifting away from the actual name of the thread which is math and science. I understand where your coming from and since this is still within his field he may answer this.
I'm curious to see what he thinks. Besides, if someone is taking their time and effort to solve math/science problems, why not try to add something more philosophical in the middle? The last thing I would want are a constant flow of math/science problems, it just gets intolerable.
I don't think the title of the thread is accurate now, I can see this including other topics as well.
Last edited on 15 December 2016 - 11:03 AM by *deleted-49709
Automatically DeletedPosted on 15 December 2016 - 11:38 AM
AlbertEinstein wrote
Metarus wrote...
For generality, let's take a,b be integers and f(x) = y = (x-a)/(x-b). Solve for x.
yx-by = x-a —> yx-x = by-a —> x = f^(-1)(x) = b(x-(a/b))/(x-1).
By substitution into f(x), f(f^(-1))(x) = x, as desired.
Thus, this is discontinuous at x = 1 and has a zero at x = (a/b). So in your case, f^(-1)(x) = (4x+3)/(x-1).
Posted on 15 December 2016 - 11:40 AM
Alpha527 wrote
AlbertEinstein wrote...
That's exactly what I just did there. lol
Posted on 15 December 2016 - 11:43 AM
Utsuho wrote
Few Questions: (No need to rush on these, I'm in no hurry and I wanna hear your most well-designed opinion on these)
- At the current rate of advancement in various technological fields (Primarily Medicine, Physics, Environmental Science and Engineering) do you think the human race will be in a safe spot for the foreseeable future (In other words, will we be able to get past the current issues of environmental damage, fossil fuels, lack of investment in the space program, etc.) I know you focus on math/science but I see what you think. Personally I believe it's very possible if we can have a permanent habitable presence on Mars by 2050, but given the current direction of the space program thanks to the Obama Administration I have my doubts.
- Do you think the Theory of Everything is possible? Broad question interpret it as you wish.
- What is your best scientific interpretation on what happens to the human mind after death? (Factoring out religious possibilities)
- This one is more basic and personal: Can you help me out with understanding Hess's Law and Bond Enthalpy? The process of handling those confuses me. I can be more specific on this if you want.
Posted on 15 December 2016 - 11:54 AM
Purpkey wrote
Can you also explain the correct way to divide by zero?
1.) Infinite pi? I'm assuming you mean how pi's decimal representation never terminates.
The source of this lies in the fact that pi is irrational (there are proofs that I will not present here). By definition, a rational number can be represented as a ratio of who integers. Thus, an irrational number is a number that cannot be represented as the ratio of two integers.
Recall that any terminating or repeated decimal can be represented as a fraction (the latter is less direct… see http://mathcentral.uregina.ca/QQ/database/QQ.09.06/h/lil1.html). Thus, irrational numbers simply cannot have a terminating or repeating decimal form. That's why pi is "infinite."
2.) Division by zero is interesting. Another person on the forums gave a good analogy: how can you divide some number of things into zero people? I'm not quite sure. But let's look at this from an analytical viewpoint.
Let's study the function f(x) = 1/x. Starting from x=1, let's move towards x=0. Notice that f get larger and larger as we do this, eventually diverging to infinity. So from this we could say 1/0 ~ infinity. OK, that's a little weird. Going back to the analogy, we would divide that number of things into an infinite number of ways to divide into zero people. That doesn't make sense physically. I'm supposed to divide three candy bars into an infinite number of pieces to give to no people?
But let's go back tot he function. What if we approach x=0 from x=-1? We then approach negative infinity. So we're getting TWO infinitely spaced answers for 1/0. So there's just not really a good way to define what 1/0 is. You can only approximate and limit fractions that are relatively infinity, say 1/10^(100)–> a really big number, and a relative infinity.
Posted on 15 December 2016 - 11:55 AM
_Happykitten_ wrote
Good question!
Last edited on 15 December 2016 - 01:26 PM by Bestaquiio
What is the (n+1)th derivative of x→(x^n)ln(x)?Consider this as a warm up before we can start having some real fun ;)
Posted on 15 December 2016 - 01:43 PM
Tanthegreat wrote
This is supposed to be an easy problem but for some reason I am unable to solve it
Let's first look at a simpler situation.
Consider all five-element subsets of the set A={1,2,…10}. Let's do the same task on this.
First of all, how many five-element subsets are there? Let's use a combination: 10C5 = 252.
Let's see how many there are for having least element 1. If 1 is the least element, then there are nine numbers left to choose from for the remaining four elements of the subset. Thus, 9C4 = 126. Now, for 2 being the least element, we have eight elements to choose from (since we have already covered the ones with one being the least): 8C4=70. Next, of course, we have 7C4=35 for 3, and so on.
Notice the relation that 9C4+8C4+7C4+6C4+5C4+4C4 = 10C4 = 252. Recall that the arithmetic mean is the sum of all the elements divided by the total number of elements. Thus, there are 126 sets with 1 being the least element, 70 with 2, and so on.
We add them all up: 126*1+70*2+35*3+15*4+5*5+1*6 = 462. Thus, the fraction we want is 462/252 = 11/6. Thus, p-q = 11-6 = 5.
Let's now go the problem with larger numbers. There are (2015 C 1000) of these 1000-element subsets.
Again, let's start with 1. There are 2014 elements to choose from for the remaining 999 elements of the subset. Thus, (2014 C 999). Then, for 2, we have 2013 elements for the 999-element subset: (2013 C 999). The process goes on.
Thus, to find p, we must evaluate the sum: sum(n=1–>1016) n [(2015-n) C 999]. Now we have to use properties of binomial coefficients.
Let's take 2015-n = m. Then, sum(m=999–>2014) (2015-m) (m C 999). There is an identity such that sum (j=k –> n) (n+1-j) ((j-1) C (k-1)) = ((n+1) C (k+1)). Take m = r -1. Then, we have sum(r=1000–>2015) (2016-r) ((r-1) C 999) = (2016 C 1001).
Thus, our ratio p/q = (2016 C 1001)/(2015 C 1000) = (2016! 1000! 1015!)/(2015! 1001! 1015!) = 2016/1001 = (2^5*3^2*7)/(7*11*13) = (2^5*3^2)/(11*13)= 288/143. Hence, p=288, q=143. Then, p-q = 145.
Let's hope I didn't make a silly mistake along the way.
Posted on 15 December 2016 - 02:10 PM
Bestaquiio wrote
Consider this as a warm up before we can start having some real fun ;)
This is a page for homework help, but I guess I'll accept your challenge.
I've been doing calculus for a very long time now, FYI.
After some scratch work, I propose that the solution is n!/x.
Let's prove this by induction.
Suppose n=1. Then, d^2/dx^2 (xlnx) = d/dx(lnx+1) = 1/x = 1!/x, as desired.
Next, suppose for a given k, we have d^(k+1)/dx^(k+1) (x^k lnx) = k!/x. Consider now d^(k+2)/dx^(k+2) (x^(k+1) lnx).
This is equivalent to (d^(k+1)/dx^(k+1))(d/dx)(x^(k+1) lnx) = (d^(k+1)/dx^(k+1))((k+1)x^k lnx + x^k). Clearly, d^(k+1)/dx^(k+1) (x^k) = 0.
Thus, we have (k+1) (d^(k+1)/dx^(k+1)) (x^k lnx) = (k+1) (k!/x), by the induction hypothesis. This, of course, is (k+1)!/x.
This completes the proof.
QED
Posted on 15 December 2016 - 02:21 PM
Convince me God does not exist. I'm your Typical American who thinks the world is 4000 years old.Explain how matter originated. The universe can't come from nowhere. What caused the big bang?
Posted on 15 December 2016 - 02:55 PM
Jinxful wrote
Explain how matter originated. The universe can't come from nowhere. What caused the big bang?
1.) I can't do that. And that's not the typical American, thankfully.
2.) Science does has unanswered questions.
Posted on 15 December 2016 - 04:03 PM
AlbertEinstein wrote
Jinxful wrote...
1.) I can't do that. And that's not the typical American, thankfully.
2.) Science does has unanswered questions.
Posted on 15 December 2016 - 05:56 PM
Just some basic algebra 2Solve:
Root 3 Root x+1 = Root 3x-5
(The root x+1 is under the root that contains the 3)
Posted on 15 December 2016 - 06:02 PM
How do you find a parallel slope to another by only using the original slope and the x-intercept and so the line would pass through a specific point?Posted on 15 December 2016 - 08:49 PM
oops just thought of it, and thats why i went check but I forgot to say that the proof needed to be direct not by inductionPosted on 15 December 2016 - 09:24 PM
Automatically DeletedPosted on 15 December 2016 - 09:25 PM
What is 10*9*8*7*6*5*4*3*2*1?Posted on 15 December 2016 - 10:43 PM
Automatically Deleted