Wait, wah...? Partial Fractions and Derivative Proof, I think..?

http://books.google.com/books?id=x1q...hat%22&f=false

72a. I have no frigging clue where to start.

isn't this a Riemann summation? It's been a longtime since I've done that, but it looks really similar.

Just bumping this post up to say no. @Locked, you have any ideas?

Where's @Noah when you need him?

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Originally Posted by Corn

Just bumping this post up to say no. @Locked, you have any ideas?

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Other than it being a seemingly useless proof, not really.

http://mathurl.com/9x5uq5a.png

My friend proved it though. I'll tell him to post here.

By useless, I mean that it doesn't make anything easier really. You could do the second problem (72b) without proving anything and just using regular partial fractions. Though that ruins the spirit of the problem ;p

Full proof: http://mathurl.com/9mnetnt.png

Additional resource:

http://mathurl.com/943lpmy.png

Basically just do the same thing you do for a normal partial fraction problem, just with 10x more symbols and a little bit of reasoning. There's a sketchy part in expressing the product from m=1 to j-1 when j can very well be 1. I can easily do a piece-wise solution (when j = 1, when j = n, and otherwise), but you get the idea of the proof anyway. And by "two of the three terms", I mean either the A1 term remains, or the An term remains, or Aj remains where 1 < j < n. Better wording should say "all but one term get eliminated" instead.


EDIT: Long time no see, guys.

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Originally Posted by ImagineAll

Where's @Noah when you need him?

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Holy sh`it he hasn't posted since 2010.

It's been that long?

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Originally Posted by Kalovale

Full proof: http://mathurl.com/9mnetnt.png

Additional resource:

http://mathurl.com/943lpmy.png

Basically just do the same thing you do for a normal partial fraction problem, just with 10x more symbols and a little bit of reasoning. There's a sketchy part in expressing the product from m=1 to j-1 when j can very well be 1. I can easily do a piece-wise solution (when j = 1, when j = n, and otherwise), but you get the idea of the proof anyway. And by "two of the three terms", I mean either the A1 term remains, or the An term remains, or Aj remains where 1 < j < n.


EDIT: Long time no see, guys.

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0.o im in 10th grade and i think my brain just exploded. THIS IS MATH ISNT IT?? THIER ARE NO NUMBERS??

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Originally Posted by Yugidude

0.o im in 10th grade and i think my brain just exploded. THIS IS MATH ISNT IT?? THIER ARE NO NUMBERS??

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Math stops having a lot of numbers at around the Calculus level and is replaced with symbols.

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Originally Posted by Locked

Math stops having a lot of numbers at around the Calculus level and is replaced with symbols.

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great... just when i was starting to understand math. :f7:

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Originally Posted by Yugidude

great... just when i was starting to understand math. :f7:

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Symbols represent numbers.

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Originally Posted by Yugidude

great... just when i was starting to understand math. :f7:

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Here's an English translation:

- Step 1: Clear the denominator by (1) multiplying each numerator (A1, A2, ..., Aj, ..., An) with every (x - aj) except for the one directly below it and (2) summing them together.
- Step 2: Notice that each term of this pattern: Aj * (x - a[everything else except j]), thus the only term that is left when plugging in x = aj is the one that does not have (x - aj) in it. Incidentally, this is the Aj term.
- Step 3: So P(aj) = Aj * (aj - a1) * (aj - a2) * ... * (aj - aj-1) * (aj - aj+1) * ... * (aj - an)
- Step 4: Take the derivative of Q(x), using product rule after re-ordering Q(x) into (x - aj) * everything else. Q'(x) = (x - aj)' * everything else + (x - aj) * (everything else)'
- Step 5: The second part is 0 when plugging in x = aj, so Q'(aj) = everything else = (aj - a1) * (aj - a2) * ... * (aj - aj-1) * (aj - aj+1) * ... * (aj - an)
- Step 6: P(aj)/Q'(aj) = Aj

I can't see the page for some reason, but if you were trying to isolate Q'(x) I think the 2nd method is a better proof, academically speaking. Also in terms of clarity.

Hadriel

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Originally Posted by hadriel

I can't see the page for some reason, but if you were trying to isolate Q'(x) I think the 2nd method is a better proof, academically speaking. Also in terms of clarity.

Hadriel

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The second one is a proof of a statement in the first one.

Ah ok I see why that was necessary now.

Hadriel

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Originally Posted by Yugidude

0.o im in 10th grade and i think my brain just exploded. THIS IS MATH ISNT IT?? THIER ARE NO NUMBERS??

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Bro, I am in calculus and I had to spend like half an hour trying to process this.

Thanks Kalovale.

I feel like we need more math questions in here.

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Originally Posted by Locked

I feel like we need more math questions in here.

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In this thread?

I don't have that fancy math-writing program, sorry if it's sloppy :(

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Originally Posted by Question

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. < what does this mean? I thought it was there if there were more than one local min/max. If an answer does not exist, enter DNE.)

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f(x,y) = xy + 64/x + 64/y

Local Maximum values(s) : DNE [correct]

Local Minimum values(s) : 48 Fixed it; works now

Saddle points(s) (x, y, f) = DNE [correct]

I think I'm putting the answer in the wrong form..?

First derivatives:
fx = y - 64/x^2; fy = x - 64/y^2

Second derivatives:
fxx = 128/x^3; fyy = 128/y^3; fxy = 1

Set first derivatives = 0, substitute:
y = 64/x^2; x = 64/y^2
x = 64/(64/x^2)^2 = x^4/64

Solve for x:
x(x^3 - 64) = 0
x = 0, x = 4

Function is undefined at x = 0.

Substituting x = 4 into fx gives y = 4.

D(x,y) = fxx(x,y)*fyy(x,y) - fxy(x,y)
D(4,4) = (2)(2) - 1 = 3

Since D(4,4) > 0 and fxx(4,4) > 0, f(4,4) is a local minimum. Therefore, there are no local maximum or saddle points.

WebAssign is not accepting (4,4) as an answer. I have tried (4,4,48) [for (x,y,f) format?], f(4,4), f(4,4)=48. Is there a standard form for answering that I missed?


EDIT: 48 worked. Guess value(s) should have given it away, since it wasn't asking for a point.

I'll probably be looking for more math help next semester and post a few questions. :f3:

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Originally Posted by Marksman Bryan

In this thread?

I don't have that fancy math-writing program, sorry if it's sloppy :(

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It ain't fancy. It's just LaTeX.