Let's analyze and solve some of the questions from the image you uploaded. The questions involve partial derivatives, limits, critical points, gradients, and integration. I’ll help you solve the questions step by step.

Part (h):

The expression of $x + y = 3$ in powers of $(x-1)$ and $(y-1)$ is asking for a series expansion. This typically suggests using a Taylor series expansion around the point $x = 1$ and $y = 1$.

Part (i):

The function $z = xy f\left(\frac{x}{y}\right)$ asks for $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y}$. To solve this:

• First, calculate $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.
• Then substitute those into the given expression.

Part (j):

The function $f(x, y) = x^2 - 2y^2 + 1$ asks for the extreme values of $f$. This involves finding the critical points:

1. Compute $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$.
2. Set these equal to zero and solve for $x$ and $y$.
3. Determine whether these points correspond to maxima, minima, or saddle points using the second derivative test.

Part (k):

Given $z = x^3 - xy + y^3$ and $x = r \cos \theta$, $y = r \sin \theta$, you're asked to find $\frac{\partial z}{\partial r}$. This involves applying the chain rule for polar coordinates.

Part (l):

The function $f(x, y) = \frac{2xy}{x^2 + y^2}$ asks for the limit as $(x, y) \to (0, 0)$. Here, you can attempt to solve this using polar coordinates by substituting $x = r \cos \theta$ and $y = r \sin \theta$, then taking the limit as $r \to 0$.

Section 2:

The second section contains additional questions, some of which require computing partial derivatives and gradients, and solving integrals using iterated integration.

• (a): Differentiate $z = x^2 \sin(3x + y^3)$ with respect to $x$, then evaluate at $\left( \frac{\pi}{3}, 0 \right)$.
• (b): Find $\nabla f(x, y)$ where $f(x, y) = x^2 y + y^3$, i.e., calculate both partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$.
• (c): Find the gradient of $f^n$ (this likely refers to a function raised to the power $n$).
• (d): Find the critical points of $f(x, y) = 8x^3 - 24xy + y^3$.
• (e): Evaluate the double integral $\int \int_R x^2 y^5 \, dA$, where $R$ is a rectangular region, using iterated integration.
• (f): Express work as a line integral. This could involve applying vector calculus concepts such as the gradient or a potential function.

Would you like me to begin solving specific parts, or would you prefer detailed steps on a particular question?

5 Follow-up Questions:

1. Would you like help expanding the function in (h) using a Taylor series?
2. Should I guide you through solving the partial derivatives in question (i)?
3. Do you need assistance finding critical points for the function in (j)?
4. Should I explain how to handle limits using polar coordinates for (l)?
5. Would you like help setting up and solving the double integral in part (e)?

Tip:

When working with multivariable functions, always remember that partial derivatives treat all other variables as constants when differentiating with respect to one specific variable.