Which of the following are considered cylinders
?

Hint : Try to do this as if it were on the exam. Use the traces (or cross-sections) method to draw a picture to help you classify.
(Click all that apply to get credit.)

A. y=x2−1


B. −(cos(−y))=−(z+1)


C. 4x2−4y2+4z2=3


D. y2+4z2=5


E. 3x2+3y2−4z=2


F. x2+5y2+4z2=1


G. None of the above.


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.

Which of the following are considered cylinders?

Hint: Try to do this as if it were on the exam. Use the traces (or cross-sections) method to draw a picture to help you classify.
(Click all that apply to get credit.)

A. y=x^2−1

B. −(cos(−y))=−(z+1)

C. 4x^2−4y^2+4z^2=3

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

E. 3x^2+3y^2−4z=2

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

G. None of the above.

Learn how to classify surfaces as cylinders using the traces (cross-sections) method. This problem analyzes several equations and identifies parabolic, cosine, and elliptic cylinders. Explore the detailed solutions for each equation, designed for Grade 11-12 and college-level students.

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