Find the equation of the plane tangent to the surface z 9 -

Question

1) Find the equation of the plane tangent to the surface z =9 - x2 - y2 at (1, 2, 4). Graph both the part of the surface including the point and the tangent plane over the rectangle [0, 3] X [0, 3].

2) Use the Solve command to find a solution to the least-squares linear fit for the data (1, 3), (2, 4), (3, 6), (5, 5), and (7, 8).

3) Use the Fit command to find the least-squares linear fit for the data (1, 3), (2, 4), (3, 6), (5, 5), and (7, 8).

4) Use the D and Solve commands to find the relative extrema and saddle points (using the Second Derivative Test) of the function f(x, y, z) = xy + yz + 2xz subject to the constraint x + 2y + z = 1. Use the command FullSimplify, if needed.

5) Use the D and Solve commands to find the relative extrema (using Lagrange Multipliers) of the function f(x, y, z) = xy + yz + 2 xz subject to the constraint x + y + z = 1. Use the command FullSimplify, if needed.

Answer 1

For the following mathematics problem, solve all the questions illustrated below by providing the complete methodical computation part all along with the formulas and graphs if required. Q1. Determine the equation of the plane tangent to the surface z = 9 - x2 - y2 at (1, 2, 4). Graph both the part of the surface comprising the point and the tangent plane above the rectangle [0, 3] X [0, 3]. Q2. Make use of the Solve command to determine a solution to the least-squares linear fit for the data (1, 3), (2, 4), (3, 6), (5, 5) and (7, 8). Q3. Make use of the Fit command to determine the least-squares linear fit for the data (1, 3), (2, 4), (3, 6), (5, 5) and (7, 8). Q4. Make use of the D and Solve commands to determine the relative extrema and saddle points (employing the Second Derivative Test) of the function f(x, y, z) = xy + yz + 2xz subject to the constraint x + 2y + z = 1. Apply the command FullSimplify, if required.