anonymous
  • anonymous
Part A, Session 6: Calculating Derivatives At 34:04 the professor is claiming that the gap is so tiny. This is not mathematical prove, is it? It looks smaller, but don't we need to proof mathematically that the gap (1-cos(theta) ) is smaller than length theta?
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  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
|dw:1332895319706:dw|
anonymous
  • anonymous
|dw:1332896219366:dw| You would need to finish the proof using mathematical induction. Show true for the first case...n=1|dw:1332896419560:dw| Assume true for n=k Prove true for n=k+1
anonymous
  • anonymous
Nothing wrong with his logic...he did not of course allude to a formal proof...as you stated....he indicated a more "logical acceptance" as shown with the first few cases I indicated....

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anonymous
  • anonymous
I graphed \[1 - \cos (\Pi/x) - (\Pi/x)\] If what you are saying is correct, I will have to get a function which is always negative, right? then I saw that after 7 it becomes positive. You can graph it here http://www.coolmath.com/graphit/ and then you will get something like this: |dw:1332920038619:dw| then I tried this: \[1 - \sin (\Pi / 9) - (\Pi / 9) \] I got ~ 0.3089 so we should not be able to prove what you claimed. Let me know if I am wrong. I tried to prove it but I could not. :( then I found counter example.
anonymous
  • anonymous
I did not know how to edit my post, but thank you so much for the reply.
anonymous
  • anonymous
Hi alirez,...you must look at the interval...the domain from 0 to 1 and in the proof, where n is greater than or equal to 1.
anonymous
  • anonymous
Consider values in the first quadrant.
anonymous
  • anonymous
I found my mistake, I graphed 1−sin(Π/9)−(Π/9) and not 1−cos(Π/9)−(Π/9) so I tried again and you are right. I consider that x is greater than zero but I was graphing a wrong function. Now I will try to prove this: if n=1, 1−cos(Π) < (Π) then I have to prove n=k+1, 1 - cos(Π/(k+1)) < (Π/(k+1)) seems hard but give me some time to prove it. Thanks
anonymous
  • anonymous
these are ways I tried and I could not get answer 1) first step I will take this: 1−cos(Π/K) < (Π/K) now if I change K to K+1 cos(Π/K+1) is increasing, therefore 1- cos(Π/K+1) is decreasing. so I can conclude that: 1- cos(Π/K+1) < 1−cos(Π/K) < (Π/K) then: 1- cos(Π/K+1) < (Π/K) I stop here and I will start something else considering (Π/K), then we add a one to K. we will get (Π/K+1). (Π/K+1) is decreasing, therefore (Π/K+1) < (Π/K) So I have to prove (Π/K+1) > 1−cos(Π/K) which I tried few ways and I could not prove it. 2) tried all trigonometric formula, I had in mind and could get nothing 3) I tried to prove 1−cos(Π/k)<(Π/K) [1−cos(Π/x)] * [1+ cos(Π/x)] <(Π/x) * [1+ cos(Π/x)] 1- (cos(Π/x))^2 < (Π/x) + (Π/x) * cos(Π/x) (sin(Π/x))^2 < (Π/x) + (Π/x) * cos(Π/x) Once I read that sin (x), x approaches 0, then sin(x)=x (not sure where it is coming from) if I get that in mind I can prove this for when x is getting higher we know that sin (x) > (sin (x))^2 if sin (x)>0 and here sin (x) is greater than zero so we know that sin(Π/x) < (Π/x) and we can say (sin(Π/x))^2 < (Π/x) and then we know that (Π/x) * cos(Π/x) is a positive number so adding a positive value to greater side it will just make it greater. so, (sin(Π/x))^2 < (Π/x) + (Π/x) * cos(Π/x)
anonymous
  • anonymous
|dw:1333690096652:dw|
anonymous
  • anonymous
i dont agree to the above graph it will be ^^
anonymous
  • anonymous
Rohangrr what is that graph. why you don't agree with that?
anonymous
  • anonymous
@PROSS Could you help me to prove it using induction?
anonymous
  • anonymous
Hi Alireza, Start by looking at my first post. Show (1-cos(pi/1) is less than or equal to pi/1 is true (Note: The sum of the terms on the left will be less than or equal to the sum of the terms on the right.) Assume that (1-cos(pi/1))+(1-cos(pi/2))+...+(1-cos(pi/k)) is less than or equal to pi/1+pi/2+...pi/k is true. Prove that (1-cos(pi/1))+(1-cos(pi/2))+...+(1-cos(pi/k))+(1-cos(pi/(k+1)) is less than or equal to pi/1+pi/2+...pi/k+pi/(k+1) is true. then substitute... pi/1+pi/2+...+pi/k +(1-cos(pi/k+1) is less than or equal to pi/1+pi/2+...+pi/k+pi/(k+1). Then subtract equivalent values on both sides.... The leaves: (1-cos(pi/(k+1)) is less than or equal to pi/(k+1) Which is what were wanting to prove.
anonymous
  • anonymous
I have to digest what you just explained. Thanks for reply. :)

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