Differentials — Question 1

Question 1 – Solution

Strategy. Use $dz=f_x\,dx+f_y\,dy$ and evaluate coefficients before applying the increments.

Step 1: Partial derivatives $$f_x=2xe^y+\frac 1{x+y},\qquad f_y=x^2e^y+\frac 1{x+y}.$$ Thus $$\boxed{dz=\left(2xe^y+\frac 1{x+y}\right)dx+ \left(x^2e^y+\frac 1{x+y}\right)dy}.$$

Step 2: Evaluate At $(1,1)$, $$dz=(2e+\tfrac 12)(-0.02)+(e+\tfrac 12)(0.03).$$ Combining terms, $$dz=-0.04e-0.01+0.03e+0.015 =\boxed{0.005-0.01e\approx-0.02218}.$$

Step 3: Contributions The $x$ change contributes $-(0.04e+0.01)$; the $y$ change contributes $0.03e+0.015$. Their signs oppose, and the $x$ contribution has larger magnitude.