To determine which of the given equations represent cylinders, we first need to review the general definition of a cylinder in mathematics.

A cylinder is a surface that extends infinitely in one direction (along a specific axis), and its cross-sections in planes perpendicular to that axis are curves (typically circles, ellipses, parabolas, or other conic sections).

Now let's analyze each equation:

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A. $4y^2 + z^2 = 4$

This equation describes a curve in the $yz$-plane, specifically an ellipse. Since $x$ does not appear in the equation, this curve is extruded along the $x$-axis, forming a cylinder.
Conclusion: A is a cylinder.

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B. $x^2 + 2y^2 - 3z = -1$

This equation involves all three variables $x$, $y$, and $z$. It does not represent a surface where one variable is absent, which would be necessary for a cylindrical shape.
Conclusion: B is not a cylinder.

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

This equation involves all three variables $x$, $y$, and $z$, and represents an ellipsoid, not a cylinder, as there is no "missing" variable to indicate extrusion along one axis.
Conclusion: C is not a cylinder.

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D. $y = 3x^2 - 2$

This is a parabola in the $xy$-plane. Since $z$ is missing, this parabola extends infinitely along the $z$-axis, forming a parabolic cylinder.
Conclusion: D is a cylinder.

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E. $4x^2 - y^2 + 4z^2 = 1$

This equation involves all three variables $x$, $y$, and $z$. It represents a hyperboloid, not a cylinder, as all three variables are involved in a non-trivial way.
Conclusion: E is not a cylinder.

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F. $-( \cos(-3y) ) = -(z + 3)$

This equation can be simplified to $\cos(3y) = z + 3$, which represents a non-linear relationship between $y$ and $z$. Since all variables are involved and there is no clear extrusion along any axis, this is not a cylinder.
Conclusion: F is not a cylinder.

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Final Answer:

The equations that represent cylinders are:
A. $4y^2 + z^2 = 4$
D. $y = 3x^2 - 2$

Would you like more details or have any questions?

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Related Questions:

1. What are the different types of cylinders in mathematics?
2. How can you distinguish a hyperboloid from a cylinder?
3. What is the difference between an elliptic cylinder and a parabolic cylinder?
4. Can a cone be considered a cylinder in any special case?
5. How do cylinders appear in higher-dimensional spaces?

Tip:

When identifying cylinders, look for an equation where one variable is missing, indicating that the shape extends infinitely along the axis of the missing variable.