Use the total differential to approximate the change in as (x, y) varies from to .

Question

Use the total differential to approximate the change in  as (x, y) varies from  to  .

Quizplus · Accepted Answer

The total differential is given by 
dZ = ∂Z/∂x * dx + ∂Z/∂y * dy

As (x, y) varies from (a, b) to (c, d),

dx = c - a,
dy = d - b

Substituting the given values, we get

dZ = (-2a + 3b + 5) * (c - a) + (b + 2c - 6) * (d - b)

Putting (a, b) = (11ead0bc_8dd6_e048_99a0_b7af5678dc88_TB6988_11) and (c, d) = (11ead0bc_8dd6_e049_99a0_0524f8518c97_TB6988_11), we get

dZ ≈ (-2(11) + 3(31) + 5) * 0 + (31 + 2(49) - 6) * 1 = 196

So the approximate change in Z is 196, which corresponds to the point (11ead0bc_8dd6_e04a_99a0_63ad4b79cdbc_TB6988_11). Therefore, the best choice is C.

Use the Total Differential to Approximate the Change in