How to solve these two questions of calculus?

1.

(2 marks) You and your colleagues spend the evening at the beach building sand castles. Later, you end up building a perfect dome with height f(x, y) , given by f(x, y) =1/77-22-22. (a) Use the total differential approximation of f at (5,6) to estimate the height of the the dome at (21, y1 ) = (5.1, 6.2). 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 (a , y) by the value at (x, y) of its tangent plane at a point close to (x, y, f (x, y)).Considerthe surface S with equation z = f(:1:, y) = m2 + y2 + 4. a) Which surface is a sketch of S in R3 b) Which of the following points lies on the surface 5'. (select all that apply) 2, 2, 12) 1,2,11) 2,-119) ( ( ' (-3:1,13) ( (1: _2! 9) ( 2, 3, 17) c) Find an equation for the tangent plane to S at P(\\/_, \\/i, 8) and enter it in the box below, :IE d) We can observe that tangent plane at the point P passes through the origin. Write down the coordinates of another point Q on the surface S , different from P, which also has the property that that the tangent plane to S at Q passes through the origin. Enteryour point in theform (1, 2, 3) :i B L."