Tangent Planes and Linear Approximations — Question 3

Question 3 – Solution

Strategy. Read the constant and slope coefficients from the standard linearization form, then add terms that vanish to first order.

Step 1: Recover the data Comparing with $$L=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$$ at $(a,b)=(2,-1)$ gives $$\boxed{f(2,-1)=7},\qquad \boxed{f_x(2,-1)=3},\qquad \boxed{f_y(2,-1)=-4}.$$

Step 2: Change along the path The input changes are $\Delta x=t$ and $\Delta y=2t$. Hence $$\Delta f\approx 3t-4(2t)=\boxed{-5t}.$$

Step 3: Construct an example One choice is $$\boxed{f(x,y)=7+3(x-2)-4(y+1)+(x-2)^2+(y+1)^2}.$$ The two quadratic terms and their first partial derivatives vanish at $(2,-1)$, so they do not alter the tangent plane.

Verification Substitution gives $f(2,-1)=7$. Differentiation gives $f_x=3+2(x-2)$ and $f_y=-4+2(y+1)$, which reproduce the recovered slopes at the base point.