Final answer:
To estimate the change in z when x decreases from 4 to 3.9 and y increases by 0.3 from 2, we use the small increments formula resulting in an estimated change of ∆z = 7.1.
Step-by-step explanation:
To estimate the change in z = xy - 10x + 17y when x decreases from 4 to 3.9 and y increases by 0.3 from 2, we can use the formula for the small increments in x and y. This formula involves taking the partial derivatives of z with respect to x and y, then multiplying them by the changes in x and y, respectively.
First, the partial derivative of z with respect to x is ∂z/∂x = y - 10, and with respect to y is ∂z/∂y = x + 17. Substituting x = 4 and y = 2 into these expressions, we get ∂z/∂x = 2 - 10 = -8 and ∂z/∂y = 4 + 17 = 21. The change in x is -0.1 (decrease from 4 to 3.9) and in y is 0.3 (increase from 2 to 2.3).
Now, we can calculate the estimated change in z (∆z) by multiplying these derivatives by the respective changes in x and y:
∆z ≈ (∂z/∂x)∆x + (∂z/∂y)∆y
∆z ≈ (-8)(-0.1) + (21)(0.3)
∆z ≈ 0.8 + 6.3
∆z ≈ 7.1