Calculus 4 Question

Question

For the function f(x, y) = sqrt(9 - x^2 - y^2),

a. Sketch the level curves f(x, y) = c the values c = 0, 1, 2, 3, justifying your conclusions. You should sketch them all on the same graph in the xy-plane to within one decimal place of accuracy. You can sketch by hand or use an online grapher (such as GeoGebra).

b. What is the domain and range of f? Show all your work.

c. Find the equation of the tangent plane (in exact form) to f for the inputs x = 1, y = 2. Show your work, and quote any formulas used.

d. Using an online 3D grapher, graph f and the tangent plane you obtained in part

(c). Use a suitable scales and viewing angle to show your tangent plane is

correct.

Answer 1

z = f(x,y) = √(9 - x² - y²)

z² = 9 - x² - y²

So, x² + y² + z² = 9 (Graph is the sphere of radius 3 with center (0,0,0))

z = f(x,y) = √(9 - x² - y²) is the upper hemisphere with radius 3 and center (0,0,0).

Domain = {(x,y) l -3 ≤ x ≤3 and -3 ≤ y ≤ 3}

Range = {z l 0 ≤ z ≤ 3}

For c = 3, the level curve f(x,y) = 3 is the point (0,0,3)

For c = 0, 1, and 2, the level curves are circles with centers on the z-axis parallel to the xy-plane.