Functions of Several Variables

Consider a function $f(x, y) = x^2 - 2y$, where $x$ and $y$ are real numbers. Which of the following accurately describes the domain and range of $f$?

Let $D \subseteq \mathbb{R}^n$ and $f: D \rightarrow \mathbb{R}^m$, $g: D \rightarrow \mathbb{R}^m$ be multivariable functions on $D$. Assessing the properties of combined functions, which assertion is fundamentally correct?

Suppose $f : D \rightarrow \mathbb{R}$ is a function defined on a domain $D \subseteq \mathbb{R}$. Precisely when can $f$ be represented by one of the following functions?

Considering functions that map from $\mathbb{R}^n$ to $\mathbb{R}^m$, which of these statements accurately describes the nature of their outputs?

Let $S(t) = (\cos t, \sin t, 2t)$, where $t$ is a real number. Which statements about the function $S$ are true?

Within topological vector space theory, under which circumstances does a multivariable function qualify as a vector-valued function?

Given g : $D \rightarrow \mathbb{R}^3$ is a function defined on a domain $D \subseteq \mathbb{R}^2$, which of the following expression(s) qualify as valid representations of g?

Consider the function $f(x, y) = \sqrt{1 - \frac{x^2}{9} - \frac{y^2}{16}}$. What is the precise domain and range of $f$?

Given the function $F : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ with the form $F(x, y) = (P(x, y), Q(x, y), R(x, y))$, where $P, Q, R$ are component functions, what can be definitively affirmed regarding the nature of $F$ and its components?

Consider the function $f(x, y) = 1 - \frac{x^2}{4} - \frac{y^2}{4}$. Let's analyze the statements related to the ellipses represented by the curves ${(x, y) \mid f(x, y) = c}$, where $c$ is a constant. Which of the following are accurate?

Given an intensity function $I(x, y, z) = \frac{k}{d^2}$, where $d$ is the Euclidean distance between a point source $S = (1, 2, 3)$ and a point $P = (x, y, z)$ in space, and $k$ is a positive constant, on what surfaces is the intensity $I$ constant?

What is the rate of change of the function $f(x, y, z) = 5x^2 - 6xy + 3z^2 - 2yz$ with respect to $z$ at the point $(1, 2, -1)$?

Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined as $f(x, y) = x^3 + 3xy$. Analyzing its partial derivatives, identify the true relations.

Let $f(x, y) = \frac{xy}{x^2 + y}$. Assess the partial derivatives.

Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined as $f(x, y) = x^2 + 4xy - y^2$. If $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}$ at $(12, k)$, what is the value of $k$?

Given the scalar field $f(x, y, z) = xy + yz + zx - 3xyz$, which descriptions are accurate?

Consider the multivariable function $f(x, y) = 3x^2 + 2x^2y + 3xy - \frac{2}{3}y^2$. Which of the following conditions does this function satisfy?

To find the value of $k$ in the equation $4 \frac{\partial f}{\partial x} + 7 \frac{\partial f}{\partial y} = k$, where $f(x, y) = e^{4(x+y)} \sin(7x - 4y)$, which of the following statements holds?

Given a multivariable function $f$ at a point $(x_1, y_1)$ in the direction of a vector $\mathbf{u} = (u_1, u_2)$, which expression describes the directional derivative $D_\mathbf{u}f(x_1, y_1)$?

If the directional derivative of $f(x, y) = x \sin(y)$ at the point $(1, 0)$ in the direction of a unit vector $(u_1, u_2)$ is $0.8$, where $u_1, u_2 \in \mathbb{R}$, what is the absolute value of $u_1$?

Consider the function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ defined as: $f(x, y) = \begin{cases} \frac{x^2y}{x^4+y^2} & (x,y) \neq (0,0) \ 0 & x = y = 0 \end{cases}$. Find the directional derivative of $f$ at $(0,0)$ in the direction of the vector $\mathbf{u} = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$.

Given $f(x, y) = x² + y²$, find the directional derivative at the point $P(1, 2)$ in the direction of the line $PQ$, where point $Q$ is at $(4, 6)$.

Let $f(x, y, z) = ax^2y + yz^2 - z^2x$, where $a \in \mathbb{R}$. Precisely when does the directional derivative of the given function at the point $(1, 1, 1)$ in the direction of the vector $\mathbf{u} = (1, 2, -2)$ equal $3$?

The temperature at a point $(x, y, z)$ is given by $T(x,y,z) = 10e^{-x^2+y^2-4z^2}$, where T is measured in °C and x, y, z are measured in meters. Find where the rate of change of temperature at $(2,-1,0)$ in the direction of a vector $u = (1,-1,1)$.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x, y) = x^2 + y^2 - 2xy$. Given $D_\mathbf{u}f(x_0, y_0)$ is the directional derivative at $(x_0, y_0)$ in the direction of the unit vector $\mathbf{u} = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ and $D_\mathbf{v}f(x_0, y_0)$ is the directional derivative at $(x_0, y_0)$ in the direction of the unit vector $\mathbf{v} = (\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})$.\nConsider a matrix $A = \begin{bmatrix} D_\mathbf{u}f(1,2) & D_\mathbf{u}f(2, 1) \ D_\mathbf{v}f(1,2) & D_\mathbf{v}f(2, 1) \end{bmatrix}$. What is the value of $\det(A)$?

Determine the accuracy of the following statements about directional derivatives.

If the directional derivative of $f(x, y) = x + xy - 2y^2$ at the point $(2, 3)$ in the direction of the vector $(2\sqrt{2}, 2\sqrt{2})$ is $a\sqrt{2}$ where $a \in \mathbb{R}$, determine the value of $a$.

Find the value of the following limit $\lim_{(x,y) \to (2,3)} \frac{x^2 + y^2 -y}{y^2 - x^2}$

\nSuppose $f$ and $g$ are multivariable functions defined on domain $\mathbb{R}^2$ with the following limits:\n$\lim_{(x,y) \to (0,0)} f(x, y) = l$ and $\lim_{(x,y) \to (0,0)} g(x, y) = k$\nwhere $l, k \in \mathbb{R}$. Select all correct options.\n