微分与误差传播

The differential is the linear part of the story

Error propagation for measured quantities

Relative errors and the logarithmic trick

What the differential cannot promise

For $f(x, y) = x^2 y$, what is the differential $df$ at the point $(2, 3)$?

A rectangle has sides measured as $x = 2$ and $y = 3$ with possible errors $|dx| \le 0.02$ and $|dy| \le 0.01$. What is the maximum error the differential predicts for the area $A = xy$?

Quantity $W = xy$ is computed from measurements with relative errors of $2$ percent in $x$ and $3$ percent in $y$. What does the differential say about the relative error in $W$?

For $W = xy$ at $(2, 3)$, use the differential to estimate the new value after $dx = 0.02$ and $dy = -0.01$. Give the estimate, as a decimal like 6.04.

The side of a cube is measured as $5$ with relative error $1$ percent. Using the power rule for relative errors, give the maximum relative error in the volume, in percent, as a plain number.