Recipe

Estimate propagated error with the total differential

Use when: x and y are measurements with known uncertainties dx, dy, and you need to know how much that uncertainty propagates into a computed quantity z=f(x,y)

Goal: an estimate dz of how far the computed quantity could be off, given the input uncertainties

Taught in Linear approximation and differentials · extends Find a tangent plane and use it for linear approximation

1. 1

   Same as the tangent-plane recipe: find f_x, f_y, and substitute the known base values (x_0,y_0).

   Example

   For a cylinder's volume V=\pi r^2h, measured at r=5, h=10: V_r=2\pi rh \Rightarrow V_r(5,10)=100\pi, and V_h=\pi r^2 \Rightarrow V_h(5,10)=25\pi.

   Why: nothing about having uncertainties instead of a target point changes how the partials themselves are found.

2. 2

   Read the problem for how much each measurement could be off, and use those two numbers directly as dx and dy.

   Example

   If r is measured to within dr=0.1 and h to within dh=0.2, those numbers ARE the differentials. They are not a second measured point to subtract from the first.

   Why: the total differential is built from a stated tolerance on the inputs, unlike the base recipe's last step, which measured the gap to an actual second point.

3. 3

   Multiply each partial by its matching uncertainty and add the two products.

   Example

   dV=V_r\,dr+V_h\,dh=(100\pi)(0.1)+(25\pi)(0.2)=10\pi+5\pi=15\pi\approx47.12.

   Why: this is the tangent plane's change in height for a step of size dx in x and dy in y. It is the same linear model as before, just read as an error bound instead of a new value.

4. 4

   Reread the question to see which form of error it wants.

   Example

   Base volume V(5,10)=\pi(5)^2(10)=250\pi\approx785.40. Absolute error dV\approx47.12. Relative error dV/V\approx47.12/785.40\approx0.06, so about 6\%.

   • it asks for absolute error — report dz itself, in the same units as z.
   • it asks for relative or percent error — divide dz by the base value f(x_0,y_0), times 100 for a percentage.

   Why: absolute and relative error answer different questions. One is an engineering tolerance in real units, the other a percentage that stays meaningful even if the units change.