Section 11.1 Problem 8: Find and sketch the domain of the function. f(x,y)=\sqrt{y}+\sqrt{25-x^2-y^2}
Comment: Easy to see the domain is the intersection of y\geq 0 (upper half plane) and 25-x^2-y^2\geq 0 (a disc centered at the origin with radius 5).
Section 11.1 Problem 17: Sketch the graph of the function. f(x,y)=9-x^2-9y^2
Comment: The graph of the function is given by z=f(x,y)=9-x^2-9y^2. Thus -x^2-9y^2=z-9 which is an elliptic paraboloid.
Section 11.1 Problem 25: Make a rough sketch of a contour map for the function whose graph is shown. (Refer to page 623 of the textbook)
Comment: To sketch the contour map, just imagine you are taking MRI(magnetic resonance imaging) on the graph along the z-axis and read off the sections periodically.
Section 11.1 Problem 48: Describe the level surfaces of the function. f(x,y,z)=x^2+3y^2+5z^2.
Solution: The level surface is given by k=x^2+3y^2+5z^2 For k<0, the level surface is empty. For k=0, it is a single point of the origin. For k > 0, the level surface is an ellipsoid centered at the origin. Moreover, as k increases, the ellipsoid becomes bigger and bigger.