decide whether the level surfaces of each function are concentric circular cylinders, concentric spheres…

Explanation:

Step1: Recall level - surface definition

Set $f(x,y,z)=k$.

Step2: Analyze function 1

For $f(x,y,z)=\ln(\sqrt{y^{2}+z^{2}})=k$, we have $\sqrt{y^{2}+z^{2}} = e^{k}$. Squaring both sides gives $y^{2}+z^{2}=e^{2k}$, which represents concentric circular cylinders.

Step3: Analyze function 2

For $f(x,y,z)=e^{-(x^{2}+y^{2}+z^{2})}=k$, then $x^{2}+y^{2}+z^{2}=-\ln k$. This represents concentric spheres.

Step4: Analyze function 3

For $f(x,y,z)=3x^{2}+y^{2}-z = k$, we can rewrite it as $z = 3x^{2}+y^{2}-k$, which is an elliptical paraboloid.

Step5: Analyze function 4

For $f(x,y,z)=\sin(\sqrt{4(x^{2}+y^{2}+z^{2})})=k$, then $\sqrt{4(x^{2}+y^{2}+z^{2})}=\arcsin k + 2n\pi$ or $\sqrt{4(x^{2}+y^{2}+z^{2})}=\pi-\arcsin k + 2n\pi$. Squaring gives $x^{2}+y^{2}+z^{2}=\frac{(\arcsin k + 2n\pi)^{2}}{4}$ or $x^{2}+y^{2}+z^{2}=\frac{(\pi - \arcsin k+2n\pi)^{2}}{4}$, representing concentric spheres.

Step6: Analyze function 5

For $f(x,y,z)=\cos(6x + y+z)=k$, then $6x + y + z=\arccos k+2n\pi$ or $6x + y + z = 2\pi-\arccos k + 2n\pi$, which represents parallel planes.

Step7: Analyze function 6

For $f(x,y,z)=2x^{2}-y = k$, we can rewrite it as $y = 2x^{2}-k$, which is a parabolic cylinder.

Step8: Analyze function 7

For $f(x,y,z)=9y^{2}-6x^{2}-z = k$, we can rewrite it as $z=9y^{2}-6x^{2}-k$, which is a hyperbolic paraboloid.

Step9: Analyze function 8

For $f(x,y,z)=x + y-8z = k$, we can rewrite it as $z=\frac{1}{8}(x + y - k)$, which represents parallel planes.

Answer:

1. Concentric circular cylinders
2. Concentric spheres
3. Elliptical paraboloid
4. Concentric spheres
5. Parallel planes
6. Parabolic cylinder
7. Hyperbolic paraboloid
8. Parallel planes