📘 multivariable calculus

1. **Problem:** Evaluate $f(1,1)$ for $f(x,y) = x^2 e^{3xy}$. 2. **Formula:** The function is given as $f(x,y) = x^2 e^{3xy}$.

1. **Problem statement:** Compute the volume under the surface defined by the function $$f(s,t) = e^s \sqrt{t^3} = e^s t^{3/2}$$ over the region in the first quadrant bounded by th

1. **Problem Statement:** Identify the local maxima and minima of the function $f(x,y)$ from the given contour plot data, list their coordinates and function values, and determine

1. **Problem Statement:** We have a vector function $$\mathbf{w} = \begin{pmatrix} 4t \\ r \\ s \end{pmatrix}$$ where $$t = \tan^{-1}\left(\frac{u}{v}\right), s = 2uv, r = \ln\left

1. **Stating the problem:** We have a vector function $$\mathbf{w} = \begin{pmatrix}4t r s \\ 2r \\ r^2 + s^2 \end{pmatrix}$$ where $$t = \tan^{-1}\left(\frac{u}{v}\right), s = 2uv

1. **State the problem:** We are given the function $$f(x,y) = x^2 \arctan\left(\frac{y}{x}\right)$$ and need to find the second partial derivatives $$\frac{\partial^2 f}{\partial

1. **Problem Statement:** Given the vector function $$\mathbf{w} = \begin{bmatrix} 4t \\ r \\ s \end{bmatrix}$$ where $$t = \tan^{-1}\left(\frac{u}{v}\right), s = 2uv, r = \ln\left

1. **Stating the problem:** We have a vector function $$\mathbf{w} = \begin{pmatrix} 4t \\ r \\ s \end{pmatrix}$$ where $$t = \tan^{-1}\left(\frac{u}{v}\right),\quad s = 2uv,\quad

1. **Problem Statement:** Find the volume of the solid bounded by the planes \(E_1: y = x^2\), \(E_2: y + z = 16\), and \(E_3: z = 0\). 2. **Boundaries:**

1. **Problem:** Determine the domain of the function $f(x,y) = \sqrt{x + y - 1}$. 2. **Formula and rules:** The square root function $\sqrt{z}$ is defined only for $z \geq 0$. Ther

1. **Stating the problem:** We are given a system of three equations in four variables $(x,y,z,t)$:

1. **State the problem:** Find the directional derivative of the function $f(x,y) = x^3 - y^3$ at the point $(4,3)$ in the direction of the vector $\mathbf{v} = \frac{\sqrt{2}}{2}(

1. **State the problem:** Find the gradient of the function $f(x,y) = \cos(x^2 + y^2)$ at the point $(3, -4)$. 2. **Recall the gradient formula:** The gradient of a function $f(x,y

1. **Problem Statement:** Find the volume of the solid region $D$ bounded below by the cone $z=\sqrt{x^2+y^2}$ and above by the plane $z=1$ using spherical coordinates.

1. **Problem statement:** Sketch the region of integration for the integral $$\int_0^1 \int_0^{\sqrt{1+x^2}} f(x,y) \, dy \, dx$$

1. **Problem Statement:** We are given a solid region $Q$ bounded by:

1. **State the problem:** We want to find the limit of the function $$f(x,y) = \begin{cases} \frac{x^3 - y^3}{x + y} & \text{if } x + y \neq 0 \\ 0 & \text{if } x + y = 0 \end{case

1. **State the problem:** Find the volume of the solid tetrahedron enclosed by the plane $2x + y + z = 4$ and the coordinate planes $x=0$, $y=0$, and $z=0$ using a triple integral.

1. **Problem Statement:** We are given two spheres and bounding planes:

1. **Problem statement:** We need to evaluate the triple integral $$\iiint_Q \frac{\sqrt[3]{x^2 + y^2 + z^2}}{2} \, dV$$ using spherical coordinates. 2. **Recall spherical coordina

1. The problem is to understand the concept of limits in multivariable calculus. 2. The limit of a function $f(x,y)$ as $(x,y)$ approaches a point $(a,b)$ is defined as $$\lim_{(x,