Differentials — Question 2

Question 2 – Solution

Strategy. Let $f(x,y)=x^2\sqrt y$ and linearize its change from $(4,9)$.

Step 1: Base data $$f(4,9)=16(3)=48,\qquad dx=0.02,\qquad dy=-0.03.$$ $$f_x=2x\sqrt y,\qquad f_y=\frac{x^2}{2\sqrt y}.$$ At $(4,9)$, $f_x=24$ and $f_y=8/3$.

Step 2: Differential $$df=24(0.02)+\frac 83(-0.03)=0.48-0.08=0.40.$$

Step 3: Estimate $$\boxed{(4.02)^2\sqrt{8.97}\approx 48+0.40=48.40}.$$ The differential keeps terms linear in $dx,dy$ and neglects quadratic and higher-order increment effects.