Question

F(x,y)=√x²+y²+2 a. Draw the section determined by the plane x=0. [1] Show clear work for all solutions. Don't make any mistakes. Good luck! 1. Givea f(x,y)=√x²+y2²

+2. a. Draw the section determined by the plane x=0. [1] b. Draw the section determined by the plane y=0. [1] c. Draw the section (level carve) determined by z0​=3. [1] d. Sketch the graph of z=f(x,y) along with the contour curve from the level curve of z0​=3. [2] e. From your gnph and uing proper set notatios, what is the range of f(x,y) ? [1] OVER

Answer 1

The problem involves drawing sections of a 3D surface represented by the function f(x,y) for planes x=0 and y=0, and a level curve for z=3. The graphs show curves on the y-z and x-z planes, both rising from the point (0,2), and a level curve circle in the x-y plane. The range of f(x,y) is determined to be [2,∞).

Step-by-step explanation:

The student has been asked to consider the function f(x,y)=√x²+y²+2 and draw sections for different planes and a contour curve. Let's address each part of the question:

• x=0 plane: Inserting x=0 into the function gives f(0,y)=√y²+2. This is a vertical cross-section of the 3D surface, a slice along the y-z plane, which will reveal a curve that rises from the point (0,2) upwards in symmetry on either side of the y-axis.
• y=0 plane: Inserting y=0 gives f(x,0)=√x²+2, an identical curve to the x=0 plane but mirrored over the x-axis in the x-z plane.
• Level curve for z0=3: To find this level curve, set f(x,y) equal to 3: √x²+y²+2=3. Solving for y, we get the circle y=√1-x².
• To sketch the graph of z=f(x,y), imagine a 3D surface where every point (x,y,z) satisfies z=√x²+y²+2. This surface is a cone with its tip at (0,0,2) and opens upwards.
• Range of f(x,y): By observing the graph, we can infer that the range of f(x,y) is [2,∞), as the smallest value f(x,y) can take is 2, when x and y are both 0, and it increases without limit as x or y increases.