Estimating maximum error

In summary, to find the maximum possible error in the computed value of T, we use the formula |E| <= (0.5M)(|x-x0| + |y-y0|)^2 where M is the upper bound of the second derivative of T(x,y) with respect to both x and y. By finding the absolute values of the second partial derivatives of T(x,y) and choosing the largest value, we can estimate M to be e^y + e^-y. Substituting this value into the error propagation formula and using the given values of x and y, we can estimate the maximum possible error in the computed value of T to be approximately 0.09.
  • #1
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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 ln2 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| <= (0.5M)(|x-x0| + |y-y0|)^2

The Attempt at a Solution


I really didn't know how to approach this, because I don't understand how to find M (the upper bound). I went ahead and found the partial derivatives with respect to x and y, and the linear approximation, which were...

T(x,y) = 5
Tx(x,y) = 5/2
Ty(x,y) = 3
L(x,y) = (5/2)x + 3y - 3ln2

I'm pretty sure the solution is something along the lines of |E| <= (0.5M)(0.1 + 0.02)^2

My main problem is I don't understand how to find M. Any help is appreciated, thanks.
 
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  • #2

Thank you for your post. I can help you with finding the maximum possible error in the computed value of T.

To start, let's recall the formula for error propagation:

|E| <= (0.5M)(|x-x0| + |y-y0|)^2

where M is the upper bound of the second derivative of T(x,y) with respect to both x and y. In simpler terms, M represents the maximum possible rate of change of the function T(x,y).

To find M, we can use the second partial derivatives of T(x,y) with respect to x and y.

Txx(x,y) = 0
Txy(x,y) = e^y - e^-y
Tyy(x,y) = e^y + e^-y

Now, we can take the absolute value of each of these partial derivatives and find the maximum value:

|Txx(x,y)| = 0
|Txy(x,y)| = e^y + e^-y
|Tyy(x,y)| = e^y + e^-y

Since we are looking for the maximum possible error, we can choose the largest value among these three, which is e^y + e^-y. Therefore, M = e^y + e^-y.

Now, we can substitute this value of M into our error propagation formula:

|E| <= (0.5(e^y + e^-y))(0.1 + 0.02)^2

Since we know that x = 2 and y = ln2, we can plug in these values to get a final estimate of the maximum possible error in the computed value of T:

|E| <= (0.5(e^ln2 + e^-ln2))(0.1 + 0.02)^2 = (0.5(2 + 1/2))(0.12)^2 = 0.09

Therefore, the maximum possible error in the computed value of T is approximately 0.09.

I hope this explanation helps you better understand how to find the maximum possible error in a function. Let me know if you have any further questions.A Scientist
 

What is maximum error?

Maximum error is the largest possible difference between an estimated value and the true value. It is also known as the absolute error.

How is maximum error calculated?

Maximum error is calculated by finding the absolute value of the difference between the estimated value and the true value. This value is then compared to the true value and expressed as a percentage or decimal.

Why is it important to estimate maximum error?

Estimating maximum error allows us to understand the potential accuracy of our measurements and calculations. It helps us determine the reliability and validity of our data.

What factors can affect maximum error?

There are several factors that can affect maximum error, such as the precision of the measuring instrument, human error, and environmental conditions. These factors should be taken into consideration when estimating maximum error.

How can maximum error be minimized?

To minimize maximum error, it is important to use accurate and precise measuring instruments, reduce human error through proper training and procedures, and control environmental conditions. It is also helpful to take multiple measurements and calculate an average to reduce the impact of random error.

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