Estimating Maximum Error (Multivariable)

[d-rock]

Estimating Maximum Error, Odd Question? (Please Help Today!)

Homework Statement

Suppose that T is to be found from the formula T = x(e^y + e^-y), where x and y are found to be 2 and ln(2) with maximum possible errors of |dx| = 0.1 and |dy| = 0.02. Estimate the maximum possible error in the computed value of T.

Homework Equations

|E| <= (1/2) * M * (|x - xo| + |y - yo|) ^ 2

The Attempt at a Solution

dT/dx = (e^y + e^-y) and dT/dy = x(e^y - e^-y)

d2T/dx2 = 0, d2T/dy2 = x(e^y + e^-y) @ (2, Ln(2)) = 2(2 + 1/2) = 5

and lastly d2T/dxy = (e^y - e^-y) @ (2, Ln(2)) = 2 - 1/2 = 1.5


So I believe the M maximum is 5, for the second derivative with respect to y.

Then, I used the equation:

|E| <= (1/2) * 5 * (|dx| + |dy|)^2 = 2.5 * (0.1 + 0.02)^2 = 0.036


So my answer was 0.036. But the book's answer is 0.31. So I am unsure what went wrong or if I overlooked something...?

 

[lanedance]

I'm not exctaly sure what you doing? Is that a quadratic apporimation to the error

Why not use a linear approximation to the function at the point by finding the gradient of the function there? (which you already have, but then use it to estimate the error...)

dependent on the function, for small dx & dy, generally the linear term will be largest (unless its an extremum) with higher order terms decreasing in magnitude.

 

[d-rock]

I don't know what you mean?

The error formula in my book (which is surrounded by Linearization, etc.) has

|E| <= (1/2)*M*(|x-xo| + |y-yo|)^2

where M is the largest possible value of the second derivatives at the point given. Then I just assumed that (x - xo) and (y-yo) are the dx and dy values they gave me, but I'm guessing it's wrong.

So, what linear approximation?