# multivariable

1. ### Multi-variable PDEs

PDEs Solve the equations below : (a) \partial z(x,y) =x^{-1} +y^{-1} . (b) \partial_{x} \partial _{y} Z(x,y)=xy+x+1.
2. ### Multivariable calculus

I am unable to set the limits and reach the conclusive answer and mark the correct option. I am trying to put the the solution in cylindrical coordinates but the answer I am getting is not matching with the options. Please help with the answer and more important the steps to reach such a...
3. ### Multivariable calculus - Limits

Show that the function f(x,y)=y/(x-y) for xâ†’0, yâ†’0, can take any limit. Construct the sequences { f(xn, yn } with (xn,yn)â†’(0, 0) in such way that the lim nâ†’âˆž f(xn,yn) is 3,2,1,0,âˆ’2. Hint: yn=kxn. I am not sure whether I am right, but I did the following: f(x,y) = kxn/(xnâˆ’kxn) =...
4. ### multivariable calculus critical points

Hello everyone, in this question I tried to the critical points. what should be my next step? I could not come up any conclusion :(
5. ### Multivariable Limits

Has anybody idea on what techniques I can apply on these limits? at c and d I did direct substitution since there is no zero at the denominator. Is that correct? a) I converted polar form and denominator became cos^2Theta+Sin^2theta = 1 so that limit exists how about b? no techniques...
6. ### multivariable limit

in this limit: should I go this way (1)- plug in x=1 and simplify the equation and (2) plug y=-1 and simplify the equation or there is some more advanced method that I should apply and I can't see?
7. ### Apply Taylor's multi-variable as the one-variable expansion to the function f not be

Can I apply Taylor's multi-variable as the one-variable expansion to the function f not be in $C^1$ $f(x,y) = \begin{cases}(x^2+y^2) sin\left(\frac{1}{\sqrt{x^2+y^2}}\right)&(x,y) \ne (0,0) &\\0&(x,y)=(0,0)\end{cases}?$
8. ### How to solve Differential Problem?

The productivity of a certain country is given by the function f(x, y) = 45x^(4/5)y^(1/5) when x units of labor and y units of capital are utilized. What is the approximate change in the number of units produced if the amount expended on labor is decreased from 243 to 240 units and the amount...
9. ### Multivariable (maybe) Calculus Help

Hi all - I'm looking for help on how to write an equation. My somewhat blurry memory of high school and college math tells me that this is a calculus problem, but I'm not certain. In (hopefully) simple english, I'm solving for a value equal to the sum of four functions where the variable is a...
10. ### A question about a sufficient condition of differentiability of a multivariable func

I see many times a sufficient condition of differentiability of a multivariable function: But take the following function for example f(x,y) = \left\{ \begin{array}{*{20}{c,l}} \frac{x^2y}{x^2 + y^2}&{\rm{, if}} (x,y) \ne (0,0)\\ 0 & {\rm{, if}} (x,y) = (0,0) \end{array} \right.. It...
11. ### multivariable limit

Hello I have a limit that I am stuck on. It is the limit (x,y) -> (0,0) for sin(x)cos(1/y) Sandwich theorem is screaming at me, but I am not sure on what to do after setting up an inequality to prove the limit is 0. Perhaps I should try using the traditional definition of a limit to prove...
12. ### Finding roots of multivariable equation

Dears, I have a set of two equations each of two variables x and y : f1(x,y) =a1+b1x+c1x^2+...+x^7+a2+b2y+c2y^2+...+y^7+a3xy+b3x^2y^2+...+x^7y^9. f2(x,y) =a4+b4x+c4x^2+...+x^7+a5+b5y+c5y^2+...+y^7+a6xy+b6x^2y^2+...+x^8y^10. These equations are the partial derivatives of one original equation...
13. ### Finding points within a multi-variable calculus function

Consider the following function. fâ€‰(x,â€‰y)â€‰=â€‰[(y+6)â€‰lnâ€‰x]âˆ’xe^2yâˆ’x(yâˆ’5)5fâ€‰(x,â€‰y)â€‰=â€‰[(y+6)â€‰lnâ€‰x]âˆ’xe^2yâˆ’x(yâˆ’5)5 (a) Find â€‰fx(1,â€‰0)fx(1,â€‰0)â€‰. (b) Find â€‰fy(1,â€‰0)fy(1,â€‰0)â€‰. I know I need to take the partial derivatives of ff and evaluate...
14. ### Multivariable Chain rule proof : is it correct?

Hello, I've had a go at proving the chain rule for a composition that maps from R^2 to R to R. There are two images attached. I've used the "arithmetic of limits" rules throughout, that if two limits exist, their product limit and sum limit exist etc. Any thoughts / scrutiny would be...
15. ### Statistical Multivariable Calculus

I am quierisome on a topic a doctor of math once opened my eyes to: applications of calculus to statistics. I am quite fond of both topics so I was attempting to find the area under a normal distribution. After some research, 8 found some clever ways using MultiVariable calculus to obtain some...
16. ### Multivariable Functions: Relative Extrema

Hello, I have some problems with this question: A production function P is given by P=f(m,k)=2.4m^2âˆ’0.1m^3+0.99k^2âˆ’0.06k^3 where l and k are the amounts of labor and capital, respectively, and P is the quantity of output produced. Find the values of m and k that maximize P. Solution...
17. ### Multivariable Regression(with factors) Help

Hi All, I am looking to run a regression analysis on a time study. The study recorded the time it took to service the client, the services provided (could be multiple for the same client), and the size of the client. Larger clients will take longer to service but there are economies of scale...
18. ### Fake multivariable function notation

So let's say I had a multivariable function, random example: f(x,t)=x^2 + x\cdot t + \sin{t} But what if x was actually a function of t itself. Would the following notation be permitted? f(x(t),t)=x^2(t) + x(t)\cdot t + \sin{t} And also, does that imply the following? 1^\circ \...
19. ### Doubt with multivariable function

Hi all, I have a function declared as such: f_{xy}(x,y) = \left\{ \begin{matrix} k & \text{if} & 0 < y < x < 1\\ 0 & & \text{otherwise} \end{matrix}\right . And I have to integrate it. From the fundamental theorem of calculus I know that if the integration limits are a and b with a < b...
20. ### Minimum for multivariable function using nonlinear Gauss Seidel

Let $X_1$...$X_p$ spaces without specific property (i.e., non convex and it's no necessary subset of real numbers $\mathbb{R}$). Let a continuous function $f:X_1\times...\times X_p\rightarrow \mathbb{R}$ such that $f$ has a minimum point $X^*=(X_1^*,...,X_p^*)$ such that...