$\text { Given } f ( x , y ) = 16 - x ^ { 2 } - y ^ { 2 } \text {, use the total differential to approximate } \Delta z \text { at } ( 2,3 )$ towards $( 2.2,1.04 ) \text {. Round your answer to two decimal places. }$
A)10.96
B)-6.04
C)49.96
D)31.92
E)33.92

Question

Quizplus · Accepted Answer

The total differential is given by:$$dz = -2xdx - 2ydy$$So, at the point (2,3), the total differential is:$$dz = -2(2)dx - 2(3)dy = -4dx - 6dy$$To approximate Δz at (2,3) towards (2.2,1.04), we use the following formula:$$\Delta z \approx dz = -4\Delta x - 6\Delta y$$where Δx = 2.2 - 2 = 0.2 and Δy = 1.04 - 3 = -1.96.Substituting these values into the formula, we get:$$\Delta z \approx -4(0.2) - 6(-1.96) = 10.96$$Therefore, the correct answer is A) 10.96.