Finding the Change in \(z\)

1. Find the change \(\Delta z\) in \(z=f(x,y)=x^{2}y-1\) from \((x_{0},y_{0})\) to \((x_{0}+\Delta x, y_{0}+\Delta y)\).
2. Use the answer to calculate the change in \(z\) from \((1,2)\) to \((1.1,1.9)\).

Solution (a) \[ \begin{eqnarray*} \Delta z&=&f(x_{0}+\Delta x, y_{0}+\Delta\;y)-f(x_{0},y_{0})\nonumber\\ &=& [ (x_{0}+\Delta x)^{2}(y_{0}+\Delta y)-1]-[ x_{0}^{2}\,y_{0}-1] \end{eqnarray*} \]

(b) Let \((x_{0},y_{0}) =(1,2)\) and \((x_{0}+\Delta x,y_{0}+\Delta y) =( 1.1,1.9)\). Then \[ \Delta z=[ ( 1.1) ^{2}( 1.9) -1] -[(1) ^{2}( 2) -1] =0.299 \]