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Finding the Change in $z$

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

(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 $\Delta z=[ ( 1.1) ^{2}( 1.9) -1] -[(1) ^{2}( 2) -1] =0.299$

EXAMPLE 1Finding the Change in zz

Finding the Change in $z$