Using differentials to estimate the maximum possible error in computed product

[navalava]

Homework Statement

Four positive numbers, each less than 40, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed product that might result from the rounding.

Homework Equations

dz=(dz/dx)dx+(dz/dy)dy

The Attempt at a Solution

I know that the solution to this problem is 4(40)^3(0.05)=12800 and I know how to apply the formula for differential to get that answer.
But I'm having trouble understanding WHY this answer is correct. I tried out this question using numbers closest to 40 that I could get when rounded to the first decimal place.
I used 39.85 for the four numbers. When rounded off, this gives us 39.9.
So I computed the error that would result like so: (39.9)^4 - (39.85)^4= 12680.37959
However, the correct answer is supposed to be 12800. I chose the closest possible numbers to 40 that I could, and I used the maximum possible error from rounding to the first decimal place (0.05). I would really appreciate it if someone could help me figure out what I'm doing wrong. Thanks!

 

[Mark44]

Homework Statement

Four positive numbers, each less than 40, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed product that might result from the rounding.

Homework Equations

dz=(dz/dx)dx+(dz/dy)dy

The Attempt at a Solution

I know that the solution to this problem is 4(40)^3(0.05)=12800 and I know how to apply the formula for differential to get that answer.
But I'm having trouble understanding WHY this answer is correct. I tried out this question using numbers closest to 40 that I could get when rounded to the first decimal place.
I used 39.85 for the four numbers. When rounded off, this gives us 39.9.
So I computed the error that would result like so: (39.9)^4 - (39.85)^4= 12680.37959
However, the correct answer is supposed to be 12800. I chose the closest possible numbers to 40 that I could, and I used the maximum possible error from rounding to the first decimal place (0.05). I would really appreciate it if someone could help me figure out what I'm doing wrong. Thanks!

Your relevant equation is pretty relevant.

If P = xyzw

you can approximate $\Delta P$ by the total differential of P, dP.

 

[Bacle]

Actually, there is nothing necessarily wrong, since the answer given is just an upper-bound , and not necessarily the absolute maximum error. And, as mark44 suggested, "you're not playing with a full differential" when you write dz=(dz/dx)dx+(dz/dy)dy.

 

[navalava]

Ohh okay, that would make sense if the differential gives only the upper bound and not the absolute maximum error. But in that case is the differential considered reliable to give the actual maximum error? I'm pretty sure that they are used in practical applications as well, not just theoretical, in which case a rough estimate may not be accurate enough.

 

[Mark44]

Ohh okay, that would make sense if the differential gives only the upper bound and not the absolute maximum error. But in that case is the differential considered reliable to give the actual maximum error? I'm pretty sure that they are used in practical applications as well, not just theoretical, in which case a rough estimate may not be accurate enough.

I'm not sure I understand your question, but I'll answer what I think you're asking as well as I can.

Let P = x*y*z*w, be the product of the four numbers, with 0 < x, y, z, w < 40.

Let P_est = x_r * y_r * z_r *w_r, be the product of the four numbers rounded to the nearest tenth.

The exact value of the error is P - P_est = $\Delta P$, and this error could be positive or negative, or even zero if rounding each of the four numbers results in no change to any of them.

It's very seldom that we're interested in the exact error, in part because we usually can't calculate it exactly, but a good estimate (not a rough estimate) usually suffices. If we can calculate a good estimate for |P - P_est|, then we have upper and lower bounds for the error.

|P - P_est| = |$\Delta P| \approx$ |dP| = |P_xdx + P_ydy + P_zdz + P_wdw|
$\leq P_x|\Delta x| + P_y|\Delta y| + P_z|\Delta z| + P_w|\Delta w|$

If $\Delta x$ and the other errors are small in comparison to the partial derivatives (all of which are positive, since we're dealing with positive numbers), the error in our calculation will also be small.