The level surface of the function f(x,y,z)=(x2+y2)^(-1/2) are a) Circles centered at the origin b) spheres centered at the origin c) cylinders around the z-axis d) upper halves of spheres centered at the origin I had chosen a) asmy answer, but it turns out the answer is c). Could someone help me understand why?

Kiana Arias

Kiana Arias

Answered question

2022-09-04

The level surface of the function f ( x , y , z ) = ( x 2 + y 2 ) 1 / 2 are
a) Circles centered at the origin b) spheres centered at the origin c) cylinders around the z-axis d) upper halves of spheres centered at the origin
I had chosen a) asmy answer, but it turns out the answer is c). Could someone help me understand why?

Answer & Explanation

Cameron Benitez

Cameron Benitez

Beginner2022-09-05Added 17 answers

Step 1
The answer is c because the function only depends on x and y. You can move freely on the z direction without changing the value of f.
Christina Matthews

Christina Matthews

Beginner2022-09-06Added 16 answers

Step 1
f ( x , y , z ) = K 1 x 2 + y 2 = K x 2 + y 2 = 1 K
Therefore the level surfaces of f have equation x 2 + y 2 = 1 K , which is indeed a circle in R 2 or a cylinder in R 3 . Since f is defined on a subset of R 3 , it is necessarily a cylinder here.
If the function was f ( x , y ) = 1 x 2 + y 2 , , your answer would be correct.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?