Spheres, Ellipsoids & Level Surfaces - Calculus 3

1. Consider the family of level surfaces $f(x,y,z) = x^2 + y^2 + 4z^2 + 4z = c$ for various constants $c$. For what value of $c$ does this level surface have minimum volume?
   Answer: $c = -1$ Completing the square in $z$: $4z^2 + 4z = 4(z^2 + z) = 4((z + \frac{1}{2})^2 - \frac{1}{4}) = 4(z + \frac{1}{2})^2 - 1$. So the equation becomes $x^2 + y^2 + 4(z + \frac{1}{2})^2 - 1 = c$, or $x^2 + y^2 + 4(z + \frac{1}{2})^2 = c + 1$. This represents an ellipsoid when $c + 1 > 0$, i.e., $c > -1$. The volume of the ellipsoid is proportional to $(c+1)^{3/2}$, which is minimized when $c+1$ is as small as possible while still positive. As $c \to -1^+$, the ellipsoid shrinks to a point, so $c = -1$ gives the minimum volume (which is 0).
2. Consider the level surface $xyz = 8$. At the point $(2, 2, 2)$, what is the equation of the tangent plane to this surface?
   Answer: $x + y + z = 6$ For $F(x,y,z) = xyz - 8 = 0$, we have $\nabla F = (yz, xz, xy)$. At $(2,2,2)$: $\nabla F(2,2,2) = (4, 4, 4)$. The tangent plane equation is $4(x-2) + 4(y-2) + 4(z-2) = 0$, which simplifies to $4x + 4y + 4z = 24$, or dividing by 4: $x + y + z = 6$. Choice B is equivalent but not in simplest form, choice C is the unsimplified version, and choice D represents a different geometric relationship.
3. Consider the family of level surfaces $F(x,y,z) = x^2 + y^2 - z^2 = k$ for different values of $k$. Which statement about the geometric nature of these surfaces is correct?
   Answer: For $k > 0$, the surfaces are hyperboloids of one sheet; for $k < 0$, they are hyperboloids of two sheets The equation $x^2 + y^2 - z^2 = k$ can be rewritten as $\frac{x^2}{k} + \frac{y^2}{k} - \frac{z^2}{k} = 1$ for $k > 0$, which is a hyperboloid of one sheet. For $k < 0$, we get $\frac{x^2}{|k|} + \frac{y^2}{|k|} - \frac{z^2}{|k|} = -1$, or equivalently $\frac{z^2}{|k|} - \frac{x^2}{|k|} - \frac{y^2}{|k|} = 1$, which is a hyperboloid of two sheets. Choice B reverses this relationship incorrectly.

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