Differentials — Question 9

Question 9 – Solution

Strategy. Reconstruct the function first, then separate its linear and nonlinear increment terms.

Step 1: Reconstruct Integrating $f_x=y^2+2x$ gives $$f=xy^2+x^2+C(y).$$ Since $f_y=2xy+C'(y)=2xy+e^y$, $C=e^y+C_0$. The initial value gives $1+C_0=5$, so $$\boxed{f=xy^2+x^2+e^y+4}.$$

Step 2: Differential estimate At $(0,0)$, $$f_x=0,\qquad f_y=1.$$ For $dx=0.02$, $dy=-0.01$, $$f(0.02,-0.01)\approx 5+df=5-0.01=\boxed{4.99}.$$

Step 3: Exact comparison Exactly, $$f=0.02(0.01)^2+(0.02)^2+e^{-0.01}+4\approx 4.99045183.$$ The estimate error is about $\boxed{0.00045183}$, while the second-order correction is $dx^2+dy^2/2=0.00045$. Thus the second-order estimate is $4.99045$, consistent with the exact value.