Let's start solving the problems one by one from the assignment. Here, we'll tackle question 1 first.

Question 1

Problem: Find $\frac{\partial}{\partial z}$ at (1,2,3) for $f(x,y,z) = x^2 y z^2$.

Solution: To find $\frac{\partial f}{\partial z}$, we need to differentiate $f(x, y, z) = x^2 y z^2$ with respect to $z$ while treating $x$ and $y$ as constants.

The partial derivative of $f$ with respect to $z$ is: $\frac{\partial}{\partial z}(x^2 y z^2) = x^2 y \cdot 2z$ $\frac{\partial f}{\partial z} = 2x^2 y z$

Now, substituting $x = 1$, $y = 2$, and $z = 3$ into the derivative: $\frac{\partial f}{\partial z} \bigg|_{(1,2,3)} = 2 \cdot 1^2 \cdot 2 \cdot 3 = 12$

So, $\frac{\partial f}{\partial z}$ at the point (1,2,3) is 12.

Let me know if you'd like to proceed to the next question or if you have any questions about this solution! Here are a few related questions you might consider:

1. How would you find the second derivative with respect to $z$ for the function $f(x, y, z)$?
2. Could you solve for $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at the same point?
3. What is the importance of evaluating derivatives at specific points?
4. How would this solution change if $f$ were a function of four variables instead of three?
5. What applications might this type of problem have in fields like engineering or physics?

Tip: When solving for partial derivatives, always remember to treat other variables as constants except the one with respect to which you are differentiating.