Recitation 5

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.