## anonymous one year ago hey @parthkohli

1. anonymous

when i say 30 i mean 40

2. ParthKohli

what does "reducing the length of the curve" mean? I mean you can draw the shortest possible path from (10,10) to (30,0) that also happens to be a straight line.

3. anonymous

yea but it needs to be smooth

4. anonymous

so polynomial

5. ParthKohli

what does that mean? a linear function is a polynomial too. what exactly do you mean by smooth?

6. anonymous

mena no rough edges and its curve

7. anonymous

like continuous function

8. ParthKohli

ahhh, better.

9. anonymous

is there function that looks like the black line i drew

10. ParthKohli

so you need a polynomial function that makes a smooth transition and is also the shortest

11. anonymous

so it has curve to smothly attach to another track but at the smae shorter length

12. anonymous

yes

13. anonymous

but at the same time *

14. ParthKohli

what's the context of this question? it's really specific

15. anonymous

i have done the question let me send the link

16. anonymous

but what i am trying to figure out is the shorter length than this

17. ParthKohli

sorry, my OS stopped working.

18. anonymous

so on xy plane i want a function that give me length <27.37

19. anonymous

|dw:1443516744812:dw|

20. anonymous

ok

21. ParthKohli

You're permitted to use calculators right?$g(x) = ax^4 + bx^3 + cx^2 + dx + e$$10^4 a + 10^3 b + 10^2 c + 10 d + e = 10$$30^4 a + 30^3 b + 30^2 c + 30 d + e=0$$\int_{10}^{30} g(x)dx=200$$4a(10)^3 + 3b(10)^2 + 2c(10) + d(10) =0$Now the last one is tricky.

22. anonymous

yess

23. ParthKohli

You know how to calculate arc length using integrals right?

24. anonymous

yea but i use wolfram alpha normaly u mind using it sending the link plz

25. ParthKohli

Because you really are looking to minimise arc-length given the above constraints.

26. anonymous

ok wait do i do the same steps as i did for my previous function find a,b,c,d

27. anonymous

and then put them in an expression and use arc length to get length

28. ParthKohli

I think Wolfram is gonna have some good fun with this one.

29. anonymous

heheh

30. ParthKohli

Here, you have a, b, c, d and e.

31. anonymous

32. anonymous

thank you so much for ur help

33. ParthKohli

I'm curious to see how you'll input so much into Wolfram. It's scary.

34. anonymous

what if i find an expression then too

35. ParthKohli

sorry, the last equation is$4a(10^3) + 3b(10^2 ) + 2c(10) + d = \color{red}1$

36. anonymous

ok

37. ParthKohli

wow the expression is way too long... you'll have to be subscribe to pro

38. anonymous

i have subscribe pro

39. ParthKohli

wow, cool.

40. baru

@ParthKohli for arguments sake, (specifics not necessary) i would write the function for arc length squared lets call it L=f(a b c d e) now what?to minimize, we need to substitute variables in terms of each other so that they represent their inter-dependencies right? how will you do that? you have four other equations ? or am i way off, and talking crap?

41. ParthKohli

yes - we have four equations. in the arc length function, we'll eliminate four variables and will be left with a function that is in terms of only one variable.

42. baru

is that a general thing, as in if there are n unknowns and (n-1) equations we can bring it down to 1 variable?

43. ParthKohli

I suppose. Let's take a smaller example.$f(x,y,z) = x+y-z$$x=y+1$$z = -x$$f(x,y,z) =x+(x-1) - (-x) = 3x - 1$

44. ParthKohli

Three variables, two equations. Now we can minimise the function with one-variable very easily.

45. baru

ok. thanks :)

46. baru

@ParthKohli one other thing, i get why you have picked an equation with 5 unknowns since we have 5 pieces of information, but the equation could just as easily been $ax^7 + bx^6 +cx^5 +dx +e$ or something else equally arbitrary so why specifically have you chosen the one that you have?

47. ParthKohli

that is a very good observation, one that I did give some thought to but skipped. unfortunately, it may be impossible to find *the* polynomial that satisfies the above constraints because the degree may go as high as... anything we want it to reach. the degree-seven polynomial you wrote may not be the best degree-seven polynomial to satisfy the conditions because it's missing some terms and unless we're sure while solving for the seventh-degree that those terms turn out to be zero, we'll never be sure. so choose a degree-four polynomial adds certainty to what we get - we can be sure that we'll get the best fourth-degree polynomial. of course there are polynomials much better than that one, but we can only go as far as we can look.

48. ParthKohli

ultimately, we're looking for a polynomial that resembles a straight-line the most in the region x = 10 to x = 30, so we can add as many terms as we like, but we can't go that far - can we?

49. baru

ah..makes sense. thanks a lot!!

50. ParthKohli

by the way, we still can approach this problem through multivariable calculus for polynomials that have more terms than five. that way, we can get an even closer polynomial.

51. anonymous

hey its not making sense

52. ParthKohli

thanks for the question - I really had to think about it to come up with an answer.

53. anonymous

5 variables how do i get values for 5 unknown variable

54. ParthKohli

plug it into WolframAlpha

55. anonymous

i only have 4 equation

56. anonymous

i found a ,b, c ,d but not e

57. ParthKohli

were you able to find a, b, c, d in terms of e?

58. anonymous

$0.0001852x ^{4}-0.01246x ^{3}+0.2076x ^{2}-0.01555x+e$

59. anonymous
60. ParthKohli

wait what? you were able to uniquely determine a, b, c, d?!

61. anonymous

from here

62. ParthKohli

it's not a 4x4 equation so

63. anonymous

yea i thought so help plz

64. ParthKohli

plug the stuff into Wolfram and it should give you all other variables in terms of one variable

65. baru

@ayeshaafzal221 where did you find this question?

66. anonymous

it didnt

67. anonymous

its my practise question for exams

68. ParthKohli

what did it return?

69. ParthKohli

you don't have a Wolfram pro subscription do you?

70. anonymous

71. anonymous

even on my ipad not working

72. ParthKohli

there's this data entry or something feature which you can use

73. anonymous

ok

74. anonymous

or u know the values i have found cant i just sub it into any equation and get the e value

75. ParthKohli

no, you can't do that. it's not a 4x4 equation so you most likely input the wrong equations.

76. anonymous

yea true i am gonna go old fashion , and do em manually

77. baru

@ParthKohli , when you say multi-variable calculus approach: are you talking about Lagrange multipliers?(i've just reached this topic)

78. ParthKohli

Obviously. Yes.

79. baru

alright..thanks

80. anonymous

@ParthKohli srry i am disturbing u again when u say we have five variables and one of them was minimum arc length what do u mean?