Question

The general equation for propagation of uncertainty for independent variables is given in

Appendix B-1 of the text (equation B-1). Consider the function F = 5x3 + 4xy, where x = 2.00 ± 0.01

and y = 3.00 ± 0.02, and x and y are independent variables. Calculate F including its uncertainty.

Calculate the uncertainty in F using equation B-1 by deriving the equations for the partial derivatives

and then evaluating the derivatives for the values of x and y given. Report F rounded to the correct

number of significant figures.

Answer 1

Since the equation in Appendix B-1 is not mentioned, the following equation (1) is used.

The square of uncertainty of a function of two independent variable is given by

-----------------------------(1)

where is the uncertainty in x and is the uncertainty in y.

----------------------------------------------------------(2)

Taking partial derivative w.r.t x,

--------------------------------------------------------(3)

Taking partial derivative w.r.y y,

----------------------------------------------------------------------(4)

Substituting and in equations (3) and (4), we get

----------------------------------------------------------------------(5)

and

------------------------------------------------------------------------(6)

We have and ---------------------------(7)

Substituting equations (7), (6) and (5) in (1), we get

The uncertainty in F

The value of at and from equation (2) is

Therefore