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Let's go through the questions one by one:

### 1. Problem 1: Continuity of \( f(x, y) = \frac{x}{x^2 - y} \) at \( y = x^2 \)
- The function \( f(x, y) \) is given as \( f(x, y) = \frac{x}{x^2 - y} \).
- We are asked whether \( f(x, y) \) is continuous at all points where \( y = x^2 \).
  
  **Analysis**: 
  - If \( y = x^2 \), the denominator \( x^2 - y = x^2 - x^2 = 0 \). This makes the function undefined for \( y = x^2 \), which implies that the function cannot be continuous at those points. 
  - Thus, the function is **not continuous** at the points where \( y = x^2 \).

  **Answer**: (b) False

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### 2. Problem 2: Critical Points and Second Partial Derivative Test for \( f(x, y) = x^2 + 4y^2 - 4xy + 2 \)
- The function is given as \( f(x, y) = x^2 + 4y^2 - 4xy + 2 \).
- We are asked whether it has an infinite number of critical points and whether the determinant of the Hessian matrix, \( f_{xx} f_{yy} - f_{xy}^2 \), is non-zero at all critical points.

  **Analysis**:
  - To find critical points, we need to set the partial derivatives \( f_x \) and \( f_y \) equal to zero and solve. This typically yields a finite number of solutions.
  - The determinant of the Hessian matrix, \( D = f_{xx} f_{yy} - f_{xy}^2 \), is used in the second derivative test to determine the nature of critical points. If it is non-zero, the test is conclusive.
  
  Since there is no clear indication of having **infinite critical points**, and from the form of the function, it is most likely that there are **only a finite number of critical points**.
  
  **Answer**: (b) False

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### 3. Problem 3: Level Curve of \( f(x, y) = 4x^2 + y^2 + 1 \)
- The function is given as \( f(x, y) = 4x^2 + y^2 + 1 \).
- The level curve equation is obtained by setting \( f(x, y) = C \), where \( C \) is a constant.

  **Solution**:
  - The level curve is obtained by setting \( f(x, y) = C \), so:
    \[
    4x^2 + y^2 + 1 = C
    \]
    Simplifying, the level curve is:
    \[
    4x^2 + y^2 = C - 1
    \]
    This is the equation of an ellipse for different values of \( C \).

  **Answer**: \( 4x^2 + y^2 = C - 1 \)

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### 4. Problem 4: Rate of Change of Volume of a Right Circular Cylinder
- Given:
  - Radius \( r = 5 \, \text{cm} \)
  - Height \( h = 10 \, \text{cm} \)
  - \( \frac{dr}{dt} = 2 \, \text{cm/sec} \), \( \frac{dh}{dt} = -1 \, \text{cm/sec} \)
  - Find the rate of change of the volume \( V \) of the cylinder.

  **Solution**:
  - The volume of a cylinder is \( V = \pi r^2 h \).
  - Differentiate with respect to time \( t \):
    \[
    \frac{dV}{dt} = \pi (2r \frac{dr}{dt} h + r^2 \frac{dh}{dt})
    \]
  - Substitute the given values:
    \[
    \frac{dV}{dt} = \pi (2(5)(2)(10) + (5)^2(-1)) = \pi (200 - 25) = 175\pi \, \text{cm}^3/\text{sec}
    \]

  **Answer**: \( 175\pi \, \text{cm}^3/\text{sec} \)

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Let me know if you want more details on any of the solutions, or if you have any questions!

Here are 5 related questions:
1. How do you check for continuity of a multivariable function at a point?
2. What is the process to find critical points of a function of two variables?
3. How do level curves relate to the graph of a function?
4. What are the steps for applying the second derivative test using the Hessian determinant?
5. How do you find the rate of change of surface area for a cylinder when both radius and height change?

**Tip**: When solving related rates problems, always differentiate both sides of the equation with respect to time \( t \), and make sure to use the chain rule properly!

1. Is the function f(x, y) = x / (x^2 - y) continuous at all points with y = x^2? 
2. Does the function f(x, y) = x^2 + 4y^2 - 4xy + 2 have an infinite number of critical points, with a non-zero determinant of the Hessian at each? 
3. What is the level curve of the function f(x, y) = 4x^2 + y^2 + 1? 
4. A right circular cylinder has radius r = 5 cm and height h = 10 cm, with dr/dt = 2 cm/sec and dh/dt = -1 cm/sec. What is the rate of change of the volume?

Explore key concepts in multivariable calculus with these solved problems on continuity, critical points, level curves, and related rates. Learn how to apply the second derivative test using the Hessian determinant, find level curves, and compute volume changes for a cylinder with changing dimensions.

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