To classify whether the given equations represent cylinders, we can use the traces (cross-sections) method, which involves analyzing the behavior of the equation in different planes to check if one of the variables is independent of the others (which would indicate a cylindrical shape). Let's go through each option.

A. $y = x^2 - 1$

This equation describes a parabola in the $x$-$y$ plane. There is no $z$-dependence, which means the equation describes a surface that extends infinitely along the $z$-axis. Thus, this is a parabolic cylinder.

Conclusion: A is a cylinder.

B. $-( \cos(-y)) = -(z+1)$

Simplifying this equation: $\cos(y) = z + 1 \quad \text{or} \quad z = \cos(y) - 1$ This describes a cosine wave in the $y$-$z$ plane, with no dependence on $x$. Therefore, it extends infinitely along the $x$-axis, and we get a cosine cylindrical surface.

Conclusion: B is a cylinder.

C. $4x^2 - 4y^2 + 4z^2 = 3$

This is a quadratic equation involving all three variables $x$, $y$, and $z$. It doesn't fit the form of a cylinder because all variables are coupled. This equation resembles a hyperboloid.

Conclusion: C is not a cylinder.

D. $y^2 + 4z^2 = 5$

This describes an ellipse in the $y$-$z$ plane, with no dependence on $x$, so the shape extends infinitely along the $x$-axis. Thus, this is an elliptic cylinder.

Conclusion: D is a cylinder.

E. $3x^2 + 3y^2 - 4z = 2$

Rewriting: $3x^2 + 3y^2 = 4z + 2$ This equation has all three variables involved and describes a paraboloid rather than a cylinder.

Conclusion: E is not a cylinder.

F. $x^2 + 5y^2 + 4z^2 = 1$

This is an equation of an ellipsoid, involving all three variables. There is no infinite extension in one of the directions, so it is not a cylinder.

Conclusion: F is not a cylinder.

G. None of the above

This option can be disregarded since we have found cylinders in A, B, and D.

---

Final Answer:

• A. $y = x^2 - 1$
• B. $-( \cos(-y)) = -(z+1)$
• D. $y^2 + 4z^2 = 5$

Would you like more details or clarifications on any of these?

---

Here are 5 related questions to expand on this topic:

1. What is the general equation for a parabolic cylinder?
2. How do you distinguish between a hyperboloid and a hyperbolic cylinder?
3. How can you determine if an equation represents a cylindrical surface just by looking at it?
4. What are the key differences between a cylinder and a prism?
5. How do cylindrical coordinates relate to these types of surfaces?

Tip: When identifying cylinders, look for the variable that is "missing" from the equation or appears as a constant, as this indicates the axis along which the surface extends infinitely.