Write the level surface 5=3x+5y^2+sqrt(z) as the graph of a function f(x,y).

Answer 1

Write the equation 5 = 3x + 5y² + √z to isolate z:

√z = 5 − 3x − 5y².

For z = 0, sketch and consider 5 − 3x − 5y² = 0 or 5y² = 5 − 3x or y² = 1 − 0.6x in the x-y plane.

For y = 0, sketch and consider 5 − 3x = √z or z = 9x²  − 30x + 25 the x-z plane.

For x = 0, sketch and consider √z = 5 − 5y² or z = 25y⁴  − 50y² + 25 in the y-z plane.

Use a graphing calculator to graph the 3 equations in bold type above. Visualize the 3 graphs pieced together in their respective planes to form a 3-dimensional skeleton of √z = 5 − 3x − 5y².

3-dimensional space is formed and governed by 2 walls and a floor coming together in a corner of your 

house. The bottom line of the wall behind you is the x-axis.  The bottom line of the wall on your left side is the y-axis. The vertical line in the corner to your left and rear is the z-axis. The floor beneath your feet is the x-y plane. The wall behind you is the x-z plane. The wall to your left side is the y-z plane.