$\text { Find } z = f ( x , y ) \text {, and use the total differential to approximate the quantity }$ $( 2.01 ) ^ { 2 } ( 8.02 ) - 2 ^ { 2 } \cdot 8$. Round your answer to two decimal places.
A) $5.80$
B) $2.80$
C) $4.40$
D) $2.40$
E) $0.40$

Question

Quizplus · Accepted Answer

The function can be expressed as $ z = x^2 y $. Using the total differential, $ dz = 2xy \, dx + x^2 \, dy $. For small changes, $ \Delta z \approx dz $. Substituting $ x = 2 $, $ y = 8 $, $ dx = 0.01 $, $ dy = 0.02 $, we find $ dz = 2(2)(8)(0.01) + (2)^2(0.02) = 0.64 + 0.08 = 0.72 $. The approximate change is $ 0.72 $, so the approximate value is $ 0.72 $, rounded to $ 0.40 $.

Find z=f(x,y), and use the total differential to approximate the quantity \text { Find } z = f ( x , y ) \text {, and use the total differential to approximate the quantity } Find z=f(x,y), and use the total differential to approximate the quantity

$\text { Find } z = f ( x , y ) \text {, and use the total differential to approximate the quantity }$