Using differentials to estimate the maximum possible error

In summary: B]In summary, the student is trying to solve for the volume of a rectangular box with a square base and height, but is unsure of how to proceed.
  • #1
whatphysics
29
2

Homework Statement


Consider a closed rectangular box with a square base of side 3cm and height 5cm. If the side is measured with an error at most 0.02cm and the height is measured with an error at most 0.01cm, use differentials to estimate the maximum possible error in computing the volume of the box.

Homework Equations


Volume of the box = [x][/2] * [y][/2] , x base side, y height

The Attempt at a Solution


▽f(x,y) = x^2y + 2xy , x^2 * ((1+y))

and i am unsure of how i can move on from here! do i sub in the x and y values 3 and 5? or the error 0.02 and 0.01 respectively?

thank you![/B]
 
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  • #2
whatphysics said:

Homework Statement


Consider a closed rectangular box with a square base of side 3cm and height 5cm. If the side is measured with an error at most 0.02cm and the height is measured with an error at most 0.01cm, use differentials to estimate the maximum possible error in computing the volume of the box.

Homework Equations


Volume of the box = [x][/2] * [y][/2] , x base side, y height
What does this mean?

The Attempt at a Solution


▽f(x,y) = x^2y + 2xy , x^2 * ((1+y))
Not sure what you've done here either.

You want to express the differential df in terms of dx and dy.

and i am unsure of how i can move on from here! do i sub in the x and y values 3 and 5? or the error 0.02 and 0.01 respectively?

thank you![/B]
 
  • #3
whatphysics said:

Homework Statement


Consider a closed rectangular box with a square base of side 3cm and height 5cm. If the side is measured with an error at most 0.02cm and the height is measured with an error at most 0.01cm, use differentials to estimate the maximum possible error in computing the volume of the box.

Homework Equations


Volume of the box = [x][/2] * [y][/2] , x base side, y height
I'm as mystified as vela is by what you have here. It's given that the box has a square base. The volume of a rectangular box is the area of the base times the height.
whatphysics said:

The Attempt at a Solution


▽f(x,y) = x^2y + 2xy , x^2 * ((1+y))[/B]
?
Where does this come from?
You don't want the gradient -- you want the differential of the volume, dV.
whatphysics said:
and i am unsure of how i can move on from here! do i sub in the x and y values 3 and 5? or the error 0.02 and 0.01 respectively?
 

1. What is the purpose of using differentials to estimate maximum possible error?

The use of differentials allows for a more precise estimation of the maximum possible error in a given mathematical calculation. It takes into account the small changes in variables and allows for a more accurate prediction of the potential error.

2. How do you calculate the differential of a function?

The differential of a function is calculated by taking the derivative of the function with respect to the variable of interest. This can be done using various techniques such as the power rule, product rule, or chain rule.

3. Can differentials be used for all types of functions?

Yes, differentials can be used for all types of functions, including polynomial, exponential, trigonometric, and logarithmic functions. The method of calculating the differential may vary depending on the type of function.

4. How accurate is the estimation of maximum possible error using differentials?

The accuracy of the estimation depends on the precision of the calculation and the value of the differential. In general, the smaller the differential, the more accurate the estimation will be.

5. Can differentials be used to find the minimum possible error?

No, differentials are only used to estimate the maximum possible error. To find the minimum possible error, other methods such as optimization techniques may be used.

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