Final Practice T1 Q7 - Looking for a detailed worked solution with theoretical explanation

This question has many parts. Please scroll down and try to answer all sub-questions until you get to "END OF THIS QUESTION" You and your mates spend the afternoon at the beach building sand castles. Eventually, you end up building a perfect dome with height f(x, y), given by f(x, y) = 169 -x2 -72. (a) With respect to some Cartesian coordinates, the top of the dome has height f(x], y1 ), where (1, y1) = (2.2, 3.8). Use the total differential approximation of f at (2, 4) to estimate the height of the top of the dome. Approximated height = (you should give the answer as a rational number.) (b) Complete the following statement about the geometric meaning of total differential approximation. We use the total differential approximation to approximate the value of a function at one point (EC, y) O by the value at (x, y) of its tangent plane at a point close to (x, y, f (x, y)) END OF THIS