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Use the fact that (125)^1/3 = 5 and the tangent line approximation to estimate (123)^1/3

  • Math -

    Let the function be y = x^(1/3)
    so we know (125, 5) lies on it, we need (123, ?)

    dy/dx = (1/3)x^(-2/3)
    at (125,5) , dy/dx = 1/75

    the tangent equation is y = (1/75)x + b
    but (125,5) is on it
    5 = (1/75)(125) + b
    5 = 5/3 + b
    b = 10/3

    tangent line at (125,5) is y = (1/75)x + 10/3

    so when x = 123
    y = 123/75 + 10/3 = 4.9733

    (calculator answer for 123^(1/3) is 4.97319

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