自动焊锡机设计三维图纸模型.doc

Errors in Measurement
The Question
Suppose that three variables are measured with percentage error at most ε1, ε2 and ε3 respectively. In
other words, if the measured value of variable number i is xi and exact value of variable number i is xi +∆xi
then
∆xi
100  ≤εi
 xi 
Suppose further that a quantity P is puted by taking the product of the three variables. So the
measured value of P is
P (x1, x2, x3) = x1x2x3
What is the percentage error in this measured value of P ?
The answer
The exact value of P is P (x1 + ∆x1, x2 + ∆x2, x3 + ∆x3). So, the percentage error in P (x1, x2, x3) is
P (x1 + ∆x1, x2 + ∆x2, x3 + ∆x3) − P (x1, x2, x3)
 
100 
 P (x1, x2, x3) 
We can get a much simpler approximate expression for this percentage error, which is good enough for
virtually all applications, by applying
P (x1 + ∆x1, x2 + ∆x2, x3 + ∆x3)
≈ P (x1, x2, x3) + Px1 (x1, x2, x3)∆x1 + Px2 (x1, x2, x3)∆x2 + Px3 (x1, x2, x3)∆x3
The three partial derivatives are
∂
Px x1, x2, x3 x1x2x3 x2x3
1 ( ) = ∂x1 [ ] =
∂
Px x1, x2, x3 x1x2x3 x1x3
2 ( ) = ∂x2 [ ] =
∂
Px x1, x2, x3 x1x2x3 x1x2
3 ( ) = ∂x3 [ ] =
So
P (x1 + ∆x1, x2 + ∆x2, x3 + ∆x3) ≈ P (x1, x2, x3) + x2x3∆x1 + x1x3∆x2 + x1x2∆x3
and the percentage error in P is
P