Multivariable Function Questions is Here

Recent questions in Multivariable functions

Widersinnby7 2022-11-06

Let $f (x, y) = \frac{y}{1 + x y} - \frac{1 - y \sin (\frac{π x}{y})}{\arctan (x)}$
Compute the limit:
$g (x) = lim_{y \to \infty} f (x, y)$
Wouldn't the value of the limit also depend on $x$ ?

Uriel Hartman 2022-11-05

Multivariable function $F (t) = f (x (t), y (t))$ , and we find derivative of $F$ with respect to $t . . . .$ in our derivative the first member is $\frac{d f}{d x} * \frac{d x}{d t} + . . . . .$ ) Why $\frac{d f}{d x}$ depends on $x$ and $y$ .

klasyvea 2022-11-03

Function $f : R^{4} \to R$ , and we denote the four variables $x, y, z, w$ , are the following statements equivalent?
i) $f$ is continuous
ii) $f |_{x}, f |_{y}, f |_{z}, f |_{w}$ are each continuous
Here, $f |_{x}$ stands for the function attained by fixing the variables $w, y, z$ .

Alexia Avila 2022-11-03

A soft drink can is h centimeters tall and has a radius of r cm. The cost of material in the can is 0.0015 cents per cm^2 and the soda itself costs 0.002 cents per cm^3. The cans are currently 10 cm tall and have a radius of 2 cm. Use calculus to estimate the effect on costs of increasing the radius by 0.5 cm and decreasing the height by 0.7 cm.

Emmanuel Giles 2022-11-02

How to find the extreme values on a specific restriction ,i.e, $f (x, y, z) = x^{2} + y^{2} - z$ on the restriction $2 x - 3 y + z - 6 = 0$ .

Emmy Swanson 2022-10-31

Derivative of multivariable function $f : R^{n} ∖ {0} \to R^{n}, f (x) = g (‖ x ‖) x$
I used the product and chain rule.
$J_{f} (x) = x \cdot g^{'} (‖ x ‖) \cdot \frac{1}{‖ x ‖} \cdot x + g (r) I_{n}$
The solution says it's $J_{f} (x) = g^{'} (r) \frac{1}{‖ x ‖} x x^{T} + g (r) I_{n}$
why do we have $x x^{T}$ ? Why the transpose?

caschaillo7 2022-10-29

How to prove that $f$ is discontinuous at origin using epsilon delta method?
$f (x, y) = {\begin{cases} \frac{x^{3} + y^{3}}{x - y} & x \neq y \\ 0 & x = y \end{cases}$

George Morales 2022-10-28

determine if the following functions are differentiable at $(x, y) = (0, 0)$ .
$f (x, y) = \sqrt{| x y |}$
$g (x, y) = e^{{| x |}^{3} y}$

snaketao0g 2022-10-27

Given a multivariable function $f : R^{n} \to R$ which is both convex and concave, a new function is constructed: $h (x) = f (x) - f (0)$ .
Prove the following:
1. $g (t x) = t \cdot g (x), \forall t \geq 0$
2. $g (x + y) = g (x) + g (y), \forall x, y \in R^{n}$
In addition, based on those two thing I need to prove that $f$ is linear $f (x) = a^{T} x + b$ .

Elise Kelley 2022-10-25

Few functions and I have to study the following aspects:
Continuity in the point (0,0)
If the derivative exists at (0,0)
Continuity of the partial derivatives at (0,0)
Directional derivatives at (0,0)
One of the functions is, for example:
$f (x, y) = {\begin{cases} \frac{x^{2} y^{2}}{\sqrt{(} x^{2} + y^{2})}, & if if (x,y) not (0,0) \\ 0, & if (x,y) = (0,0) \end{cases}$

Aldo Ashley 2022-10-24

How to integrate this multivariable function when one of the variable in the function is in the form of limits:
$\int {[f (x, y)]}_{y = 0}^{y = b} d x$
by ignoring the limits:
$\int f (x, y) d x$
If yes, how can it be justified?

4enevi 2022-10-23

Suppose a multivariable function $f : R^{n} \to R$ is concave and sufficiently smooth. We have:
$‖ f (x) - f (x_{0}) ‖ \leq M ‖ x - x_{0} ‖$ for some positive constant $M$ .
If the $f$ is univariate, we know that $M$ is the absolute value of the slope of the tangent line at $x_{0}$ . But what is it for multivariable case, is there a special name in math given to it?

Ignacio Riggs 2022-10-19

Range of multivariable function $z = e^{x + y} \arctan (\frac{y}{x})$

omgespit9q 2022-10-14

In a function like $f (x, y) = x^{2} + y$ , are $x$ and $y$ independent of each other and are we allowed to pick values for each deliberately?
Say for $1$ for $x$ and $99$ for $y$ ?

Oscar Burton 2022-10-14

Limit of multivariable functions
limit of $\sqrt{x^{2} + y^{2}} \sin (1 / y)$ as $(x, y) \to (0, 0)$
limit of $2 x y / (x^{2} - y + 5 x y)$ as $(x, y) \to (0, 0)$

Emmy Swanson 2022-10-13

If we have some closed form multivariable function say 2-in, 1-out, the cross sections of the graph parallel to the $x z$ and $y z$ plane have equations that are inherently closed form.
But is the same true for the diagonal cross sections? Say for example we have the function $f$ :
$f (x, y) = e^{y} \sin (x)$
If we're given a $\vec{v}$ say, [1, 2] would the cross section of the graph of the plane formed by the $z$ -axis and $\vec{v}$ inherently be closed form as $f$ is on the axes?

Aidyn Crosby 2022-10-08

Fix a natural number $n .$ Let $f, g : R \to R$ be $n$ times differentiable functions. General Leibniz rule states that $n$ th derivative of the product $f g$ is given by $\sum_{k = 0}^{m} (\binom{n}{k}) f^{(n - k)} (x) g^{(k)} (x)$ where $g^{(k)}$ means that $g$ is differentiated $k$ times.
What is a meaning of $β \leq α ?$ ?
Since the formula is for multivariable, I suppose that 𝛽 and 𝛼 are vectors, like $(1, 2)$ in $R^{2} .$ .

Austin Rangel 2022-10-06

We have just started studying functions of several variables and their derivatives and our professor suggested the following problem as food for thought.
Two squares, both with length l=1 intersect in a rectangle that has an area equal with 1/8 . Find the minimum and maximum distance between the centers of the squares.

dannyboi2006tk 2022-09-29

What is the explicit procedure for differentiating multivariable functions with respect to a scalar? For a very simple example, I have a function $f : R^{d} \to R$ and $ϕ (t) = f (x + t v)$ , $v \in R^{d}$ . What are the steps to calculating $ϕ^{'} (t), ϕ^{″} (t), ϕ^{‴} (t)$ , etc?

kenedirkitch 2022-09-27

Use implicit differentiation to find
∂z/∂x and ∂z/∂y.
x^2 + 2y^2 + 5z^2 = 9

The multivariable function is related to regression analysis, which may be a bit challenging for college students as they implement estimation for the various data sets. Keeping all these challenges in mind, we aim to provide you with the answers to the most popular problems and questions where a helpful multivariable function example will be of help. You should also focus on each multivariable question by starting with analysis and proper calculation of each variable as you exchange various practical sets. The purpose is to remove the constraints of single variables as you add any calculations.

Multivariable calculus

Expert Help with Multivariable Function Questions