Tangent Planes and Linear Approximations — Question 8

Question 8 – Solution

Strategy. Apply the defining small-error condition first at zero displacement and then on each coordinate axis.

Step 1: Constant term For $A$ to approximate $f$ at the base point, it must agree there. Since $A(a,b)=c$, $$\boxed{c=f(a,b)}.$$ Equivalently, the remainder must vanish when the displacement is zero.

Step 2: The $x$ coefficient The approximation condition along $(h,0)$ is $$f(a+h,b)=f(a,b)+ph+o(|h|).$$ Subtract $f(a,b)$, divide by $h\ne 0$, and let $h\to 0$. Since $o(|h|)/h\to 0$ from both sides, $$p=\lim_{h\to 0}\frac{f(a+h,b)-f(a,b)}h =\boxed{f_x(a,b)}.$$

Step 3: The $y$ coefficient The same argument along $(0,k)$ yields $$q=\lim_{k\to 0}\frac{f(a,b+k)-f(a,b)}k =\boxed{f_y(a,b)}.$$

Conclusion Every affine first-order approximation must therefore be $$\boxed{L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)}.$$ All three coefficients are forced, proving uniqueness.