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Newton's Method/Trapezoidal Rule

  1. Sep 22, 2012 #1
    My professor had asked that I solve the follwing problem using Newton's Method and the Trapezoidal Rule:

    "A company modeled the demand curve for its product (in dollars) by the equation

    p = (800000e^(- x / 5000)) / (x + 20000).

    Use Newton's Method to estimate the sales level (x I'm pretty sure) when the selling price is $16. Then find the approximate consumer surplus for this level."

    I'm currently stuck on how Newton's Method is meant to find x when p = 16.

    The derivative of p is:

    - (160e^(- x / 5000) * (x + 25000)) / (x + 20000) ^ 2
     
    Last edited: Sep 22, 2012
  2. jcsd
  3. Sep 22, 2012 #2

    To get Newton's Method started, you need an initial guess for x.
    What is the value of p when x is zero? If this is close to 16, this value of x might be a good initial guess for newton's method. What is the initial error in p for this value of x? What is the Newton Method formula?
     
  4. Sep 22, 2012 #3
    x is never equal to zero. That's the thing. Unless I'm mistaken.

    http://www.wolframalpha.com/input/?i=y+=+(800000e^(-x/5000))/(x+20000)
     
  5. Sep 22, 2012 #4
  6. Sep 22, 2012 #5
    p = 40 when x = 0
     
  7. Sep 22, 2012 #6
    Good. Now, the problem you are trying to solve is:

    f(x) = (800000e^(- x / 5000)) / (x + 20000) -16 =0

    Your initial guess is x =0, and at x = 0, f(x) = 40 -16 = 34

    Your Newton method equation is:

    xn+1 = xn-f(xn)/f '(xn)

    with x1 =0 and f(x1)= 34

    Now find f ' (x1) , and then x2

    If that's not close enough to satisfy f(x) = 0, do another iteration.
     
  8. Sep 23, 2012 #7
    I see now what to do now. Thanks bunches : D
     
  9. Sep 23, 2012 #8

    Ray Vickson

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    In problems of this type it is always a good idea to "scale" the problem properly (even if you use a computer!). Instead of x it would be better to use, say y = x/5000, so your equation becomes p = 800,000 exp(-y)/(5000 y + 20,000) = 160*exp(-y)/(y+4).

    Now, for p = 16 you need to solve 10*exp(-y) = y+4, or 10*exp(y)-y-4 = 0. Using f(y) = 10*exp(-y) - y - 4 instead of 160*exp(-y)/(y+4)-16 is a lot easier in Newton's method, since the derivative is a lot easier.

    RGV
     
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