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anonymous

  • 5 years ago

use newton's method to estimate the requested solution of the equation. start with given value of Xo and then give X2 as the estimated solution. 3x^2+2x-1=0; Xo = 1; Find the right-hand solution.

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  1. amistre64
    • 5 years ago
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    get the derivative, so you know the slope of the line at x0 = 1

  2. amistre64
    • 5 years ago
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    find the value of f(1) so you have a set of coordinates to use for your "equation of the slope" line to find the new "x1" with.

  3. amistre64
    • 5 years ago
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    f(1) = 4; (1,4) we will use this to calibrate our line equation with; now we find the slope at f(1)

  4. anonymous
    • 5 years ago
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    i see

  5. amistre64
    • 5 years ago
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    f'(x) = 6x +2 f'(1) = 8 ; our slope is 8 y = mx+b 4 = 8(1) + b 4 = 8 + (-4) y = 8x -4 find the root :) the x int and that your "new" X value

  6. anonymous
    • 5 years ago
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    with y = to 1 right

  7. amistre64
    • 5 years ago
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    when y = 0 we have our x intercept, (x,0) is the x int..

  8. amistre64
    • 5 years ago
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    0 = 8x -4 x = 4/8 = 1/2

  9. anonymous
    • 5 years ago
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    right. man your awesome

  10. amistre64
    • 5 years ago
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    to keep punching along, we use the NEW x value in our original equation and keep getting closer and closer to our "true" root ;) or farther away depending on the nature of the curve

  11. anonymous
    • 5 years ago
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    i see. hey is there a way to find the critical points of a function on a calculator instead of using calculus

  12. amistre64
    • 5 years ago
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    calculus is the "method" of the math used to find this stuff. calculus is applied to the situation when nothing else can get us there :) So I would say that it depends on the calculator and the functions that it has. But in the end, they rely upon the principles of calculus to get the answer

  13. anonymous
    • 5 years ago
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    true. well maybe since ur smart you can help with a calc problem i have.

  14. amistre64
    • 5 years ago
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    the other maths are really applied calculus :) tell me, is the earth round or flat?

  15. amistre64
    • 5 years ago
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    i might be able to help....my smarts come and go lol

  16. amistre64
    • 5 years ago
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    calculus, in essense, says that in order to figure out a problem you ahve to get close enough to it to "flatten" things out to where we can use our innate understanding of the "flat" world around us :)

  17. anonymous
    • 5 years ago
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    Interesting we'll hv to chat more abt that... i'm trying to Find the function with the given derivative whose graph passes through the point P. r'(t) = sec^2t - 4, P(0,0)

  18. amistre64
    • 5 years ago
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    that would be an integral. or antiderivative.. same thing different jargon. [S] sec^2(t) - 4 dt or \[\int\limits_{}\sec^{2} - 4 dt\]

  19. amistre64
    • 5 years ago
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    just like dressing down into the derivative, we just suit up into the integral... they are opposite movements

  20. amistre64
    • 5 years ago
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    [S] sec^2(t) dt -4 [S] dt does this make sense?

  21. anonymous
    • 5 years ago
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    lol not really. the answer i came up with is sec t - 4t - 4

  22. amistre64
    • 5 years ago
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    what function do you know that dress down to a derivative of sec^2? what function do you know that dress down to a derivative of 1 ?

  23. anonymous
    • 5 years ago
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    x^2

  24. anonymous
    • 5 years ago
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    no

  25. anonymous
    • 5 years ago
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    x

  26. amistre64
    • 5 years ago
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    lets check that:....or it looks like you already did

  27. amistre64
    • 5 years ago
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    x dresses down to 1 d(1x) = 1 that is correct. So what do we dress "1" to be? x right? we just back out of the derivative into the original function right?

  28. amistre64
    • 5 years ago
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    -4 [S] dt -> -4t now for the left part :) sec^2 came from what? do you recall your trig dervivatives?

  29. amistre64
    • 5 years ago
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    I think: d(tan(x)) = sec^2 did I recall correctly?

  30. anonymous
    • 5 years ago
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    i think ur right

  31. anonymous
    • 5 years ago
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    so would the answer be tan t - 4t

  32. amistre64
    • 5 years ago
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    I think im right to, but Ive been known to be mistaken ;) so: [S] sec^2(t) dt suits back up into tan(t) our original equation is of the form: f(x) = tan(t) -4t but that aint it exactly...that just gives us a "family" of curves to choose from...we need to pin this down with a constant

  33. amistre64
    • 5 years ago
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    tell me, can 2 different equations have the exact same derivative? y = 2x^2 + 6 y = 2x^2 -8

  34. anonymous
    • 5 years ago
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    yes

  35. anonymous
    • 5 years ago
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    right?

  36. amistre64
    • 5 years ago
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    that right: so what we have right now is an equation to a curve that is actually floating up and downthe y axis and we need to pin it down with something....... we do that with a constant (C). like this: y = tan(t) -4t +C

  37. amistre64
    • 5 years ago
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    does that make sense?

  38. anonymous
    • 5 years ago
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    so thats the answer

  39. amistre64
    • 5 years ago
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    that is getting to our answer, remember they gave us an initial condition that the curve passes thru the point (0,0) right? so lets use this information to find the actaul value for "C".

  40. anonymous
    • 5 years ago
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    ok

  41. amistre64
    • 5 years ago
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    P(0,0) 0 = tan(0) -4(0) + C solve for C :)

  42. anonymous
    • 5 years ago
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    C would b zero

  43. amistre64
    • 5 years ago
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    thats right, but now we "know" for sure :) so the equation is: r = tan(t) -4t and were done :)

  44. anonymous
    • 5 years ago
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    i like how you make sure i know where the answer is coming from. you would make a great instructor

  45. amistre64
    • 5 years ago
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    thanx; I figure that if I show you whats going on under the problem that you might just be able to use it in another place ;)

  46. anonymous
    • 5 years ago
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    i'm working on this word problem. "From a thin piece of cardboard 40in. by 40in., square corners are cut oout so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? what is the maximum volume rounded to nearest tenth if necessary

  47. amistre64
    • 5 years ago
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    like this right?

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  48. amistre64
    • 5 years ago
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    Volume is to maximized so we need the formula for the volume of a box: Do you know what that would be?

  49. anonymous
    • 5 years ago
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    length x width x height, right?

  50. amistre64
    • 5 years ago
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    thats correct; when we cut out these corners and fold them up, that is our "height" right? lets call that x..ok?

  51. amistre64
    • 5 years ago
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    length and width are equal measurements given the problem, 40 and 40 right? when we remove "x" from on end and "x" from the other; that makes our sides shorter by x+x. length then becomes: 40- 2x and width becomes: 40-2x lets use this in our formula for volume: V = (40-2x)(40-2x)(x) we agree?

  52. anonymous
    • 5 years ago
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    ok

  53. anonymous
    • 5 years ago
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    we can factor out x right

  54. amistre64
    • 5 years ago
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    factor out? now, we need to multiply all these together to find the equation for derive :)

  55. amistre64
    • 5 years ago
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    so far our box is like this:

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  56. anonymous
    • 5 years ago
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    ok

  57. amistre64
    • 5 years ago
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    V = (40-2x)(40-2x)(x) V = 1600x -160x^2 +4x^3 you wanna derive this for me?

  58. anonymous
    • 5 years ago
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    sure i got 1600 - 320x + 12x^2

  59. amistre64
    • 5 years ago
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    thats right... now we need to equal that to 0 to find our max and min.... for x :) 4(3x^2 -80x +400) = 0 3x^2 -80x +400 = 0

  60. amistre64
    • 5 years ago
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    this gives me 2 numbers: x = 40/3 and x = 20/3

  61. amistre64
    • 5 years ago
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    both of these numbers should give us an equal volume of box, but different shapes.

  62. amistre64
    • 5 years ago
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    unless I am mistaken, and I think I am with that last one...

  63. amistre64
    • 5 years ago
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    plug both of these into our forula stuff and see which is bigger :)

  64. anonymous
    • 5 years ago
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    which formula

  65. amistre64
    • 5 years ago
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    V = (40-2(40/3)) (40-2(40/3)) (40/2) V = (40-2(20/3)) (40-2(20/3)) (20/3) we only have one formula for volume that needs a value for x ;)

  66. anonymous
    • 5 years ago
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    20/3 yields a bigger answer

  67. amistre64
    • 5 years ago
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    for 40/3 we get 2014.81 .....then 20/3 it is :)

  68. amistre64
    • 5 years ago
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    yep, for 20/3 we get 7407.41

  69. anonymous
    • 5 years ago
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    i got 4740 for 20/3

  70. amistre64
    • 5 years ago
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    yeah, I noticed that I punched in the wrong thing in the calculator :)

  71. anonymous
    • 5 years ago
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    so the bigger # is the volume?

  72. amistre64
    • 5 years ago
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    yep....when we cut out a corner of "20/3" from each side, we have gotten the maximum volume we can get out of it

  73. anonymous
    • 5 years ago
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    cool

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