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The document you've uploaded contains several math problems, mainly related to vector calculus, solid geometry, and multivariable calculus. Here's a brief breakdown of the questions:

1. **Parametric Equation of a Line**:
   - Given the parametric form of the line:
     \[
     x = 12 - 5t, \quad y = 2 - t, \quad z = 3t
     \]
     The task is to find its general symmetric and vector form and represent it graphically.

2. **Volume of a Solid**:
   - Given the region defined by the inequalities \( x^2 + y^2 \leq 1 \), \( 0 \leq y \leq x \), and \( 0 \leq z \leq 5 \), calculate the volume of this solid.

3. **Local Extrema of a Function**:
   - Determine and classify the local extrema of the function:
     \[
     f(x, y) = x^3 + y^3 - 3y
     \]

4. **Continuity of a Function**:
   - Analyze the continuity of the function:
     \[
     f(x, y) = 
     \begin{cases} 
     \frac{3xy}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\
     1 & \text{if } (x, y) = (0, 0)
     \end{cases}
     \]

5. **Partial and Mixed Derivatives**:
   - Calculate partial and mixed derivatives for the following functions:
     \[
     f(x, y) = 3x^2y^3 - 6x^2 - 9y - 9
     \]
     \[
     f(x, y) = \sin(xy^3) + 3x\cos(y)
     \]
     \[
     f(x, y) = e^{-xy^2 + y^3x^4}
     \]

6. **Mass of a Lamina**:
   - Find the mass of a lamina occupying the region \( D: (x, y) \in \mathbb{R}^2, -1 \leq x \leq 2, 0 \leq y \leq 1 \) with surface density \( \delta(x, y) = x^2 + xy^3 \).

Would you like to solve any specific problem from this list in detail? Let me know which one you'd like to start with.

Here are some follow-up questions:

1. How do you find the symmetric form of a line given its parametric equation?
2. What method can be used to calculate the volume of a solid defined by inequalities in 3D space?
3. How can we classify the local extrema using the second derivative test?
4. What steps are involved in determining the continuity of a piecewise function?
5. How do you approach finding partial and mixed derivatives of a given function?

**Tip**: When calculating the volume of a solid, double or triple integrals are often used to compute the total volume based on the bounds of the region.

Dada la ecuación de la recta en forma paramétrica x=12-5ty=  2-tz     =  3t
Obtenga la ecuación general simétrica y vectorial
Represéntelo gráficamente
Trace el siguiente sólido. Calcule el volumen del mismo.
W1= xyzϵ R3: x24+y24≤1;0≤y≤x;0≤z≤5
Determine y clasifique los extremos locales de la siguiente función:
fxy=x3+y3-3y
Analice la continuidad de la siguiente función:
fxy= 3xyx2+y2;(xy)≠(00)1         ;xy=(00)
Calcule las derivadas parciales y mixtas de las funciones:
fxy=3x2y3-6x2-9y-9
fxy=senxy3+3xcos(y)
fxy= e-xy2+y3x4
Encuentre la masa de la lamina que ocupa la región
D: xyϵ R2; -1≤x≤2;0≤y≤1 y la densidad superficial de masa viene dada por la función:
δxy=x2+xy3

Solve complex problems involving parametric equations, volume of solids, local extrema, and continuity in this Math II Final Exam for Chemical Engineering. Topics include vector calculus, partial derivatives, and mass calculation of a lamina.

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