Multivariable Definition and 524 Threads

Homework Statement (Squeeze Principle) Suppose f, g, and h are real-valued functions on a neighborhood of a (perhaps not including the point a itself). Suppose f(x) ≤ g(x) ≤ h(x) for all x and limx→a f(x) = l = limx→a h(x). Prove that limx→a g(x) = l. (Hint: Given ε > 0, show that there is δ...

Homework Statement Hi, I have to evaluate the following limit: \lim_{(x,y) \to (1,0)}\frac{x*y-y}{(x-1)^2+y^2} Homework Equations I'm pretty sure I have to use the squeeze theorem. The Attempt at a Solution Well, I'm pretty sure it has something to do with the fact that the top...

Homework Statement Problem 1: If A is an m x n matrix and Ax = 0 for all x ε ℝ^n, prove that A = O. If A and B are m x n matrices and Ax = Bx for all x ε ℝ^n, prove that A = B. (O is the 0 matrix, x is the vector x, and 0 is the 0 vector.)2. The attempt at a solution First off, I understand...

Hi. I borrowed many multivariable Calculus book so that I can choose one for the next semester. The one I liked most is Multivariable Calculus by Ron Larson. It is full of graphics and colours, somthing that is essential to understand functions of two variables. The only thing is that it does...

I just got to a point in multivariable calculus where I realize I can solve problems in assignments and tests but have no actual idea of what I'm doing. So I started thinking about stuff and came up with a few questions: 1. Is picturing the derivative as the slope of the tangent line to a...

Hello! I am wondering if it is possible to establish a relationship between three sets of points (x,y,z) by interpolating. Basically i need a function that takes x and y and gives me a z that matches the following points: 130 472 5 130...

Find the volume of the region inside both the sphere x^2+y^2+z^2= 4 and the cylinder (x-1)+y^2=1using cylindrical coordinates. I was thinking the inner limits would go from +(2-r) to - (2-r) the middle intervals would go from 0 to 1 and the outer limits 0 to 2pi is my approach correct

Homework Statement Let f:R^2 to R^2 be defined by f(r,theta) = (rcos(theta), rsin(theta)) Sketch the image under f of the set S = (1,2) X (0,pi) (The open brackets should be closed brackets but I am on a foreign keyboard and can't figure out how to get closed brackets). Homework...

Homework Statement I'm Having a hard time grasping what seems to be a simple topic-the parametrization and vector representation of curves...As a result it's making finding unit Tangent vectors and the length of Curves a nightmare :( Please does anyone have any good resources they would...

My Calc III teacher this fall gives take home exams because he insists that there isn't enough time to cover all the material in Calc III, so he needs every class hour to teach, thus he gives take-home tests. Knowing people who took his calc III class last fall, and him curving an A to be 70%, I...

I have to do a calculus project by researching somewhere it's applied and writing about it and explaining it thoroughly. I was thinking multivariable calculus / vector calculus was used in satellite orbit math, but I could not find enough on this. Any ideas on where else it's applied that...

Homework Statement Given f(x,y) = xy(x+2y-6), let D be the region in the plane between the hyperbola xy = 4 (let this be g) and the line x+2y-6 = 0 (let this be h). Find the maximum and minimum values of f(x,y) on D. The Attempt at a Solution I first found the critical points for f(x,y)...

Homework Statement Find all critical points of u(x,y)=(x-y)(x^2+y^2-1) Homework Equations - The Attempt at a Solution Partial differentials: ux=3x^2+y^2-1-2xy=0 uy=-x^2-3y^2+2xy+1=0 I know the critical points are the solutions to the above two equations. But how do...

I have a symbolic function of three variables which I'm trying to numerically integrate wrt a single variable. Consider (syms x y a) and the function f(x,y,z). Here are some things I've tried, without success: 1. >>int(f(x,y,z),x,a,b) which analytically integrates f wrt x from a to b but...

I don't know how to calculate the surface area after setting everything up. I have tried both MAPLE 15 program and wolfram alpha, but I can't find the answer. I have the function: f(x,y)= x*(y^2)*e^-((x^2+y^2)/4) The form I found for surface are was the square root of (sum of squares of...

Homework Statement Here is the question with the solution: http://dl.dropbox.com/u/64325990/MATH%20253/midterm%202.PNG I don't understand how to do a. How did they know to use x=0 and y=0 for Fx(0,0) and Fy(0,0)? I understand that this is implicit differentiation using Chain Rule where...

Homework Statement Here is the problem with the solution: http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because...

Homework Statement For example in: lim (x,y) -> (0,0) [(xy^2)/(x^2+y^4)] This limit does not exist (according to textbook), but if you use squeeze theorem since y^2<(x^2+y^4) y^2/(x^2+y^4) <= 1 and therefore 0 <= (xy^2)/(x^2+y^4) <= x as x--> 0 so lim (x,y) -> (0,0)...

f(x,y) = ke^(-x-y), 0<x<∞ , 0<y<∞ , and x<y Find k to make this a PDF. So I set up: ∫(from 0 to ∞)∫(from 0 to ∞) [ke^(-x-y)]dxdy. Is this integral right? I also need to find the marginals of X and Y but my k has to be right first.

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Homework Statement the base of an aquarium with given volume V is made of slate and the sides are made of glass. if slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. Homework Equations The Attempt...

Homework Statement Given: Concentration of Fluid = F(x,y,z) = 2x^2 + 4y^4 + 2*x^2*z^2 at point (-1,1,1) Found Grad(F(x,y,z)) = <-8,16,4> ----If you start to move in the direction of Grad(F) at a speed of 8, how fast is the concentration changing?Homework Equations Already found the gradient...

Homework Statement What is the maximum possible volume of a rectangular box inscribed in a hemisphere of radius R? Assume that one face of the box lies in the planar base of the hemisphere. NOTE: For this problem, we're not allowed to use Lagrange multipliers, since we technically haven't...

Homework Statement Hi everyone! I'm pretty good with multivariable limits, but this one has me stumped: Find the limit or show that it does not exist: \underset{\left(x,y,z\right)\rightarrow\left(1,-1,1\right)}{\lim}\frac{yz+xz+xy}{1+xyz} Homework Equations The Attempt at a Solution I could...

Homework Statement Im righting this down for my roommates since he's having tons of trouble trying to figure this out and I can't answer it. also sorry for having to hotlink it. http://i.imgur.com/afShz.jpg the equation is on the image since its very difficult to type it all out...

Is the definition of an odd/even function in multiple variables what I would expect it to be, ie f(-x,-y)=-f(x,y) Thanks!

Integral of cos(8x^2)? multivariable calc Homework Statement Homework Equations none The Attempt at a Solution So i reversed the order... and couldn't figure out how to integrate cos(8x^2). I looked at my trig identities, tried on my graphing calc, and even wolfram alpha...

Homework Statement Find the limit of: lim (x,y) ->(0,0) (x^2*y)/(x^2 + y^4)Homework Equations x=rcosθ y=rsinθThe Attempt at a Solution lim r->0 for all steps L = (rcosθ)^2*(rsinθ)/[(rcosθ)^2 + (rsinθ)^4] L = r^3 (cosθ)^2 (sinθ) / [ r^2 * (cosθ)^2 + r^4 * (sinθ)^4] L = r (cos^2 * sin ) / (cos^2...

Homework Statement lim(x,y)->(1,0) of ln((1+y^2)/(x^2+xy)) Homework Equations The Attempt at a Solution Used two paths, x=1 y=0 both gave my lim=0 so I tried x=rsin y=rcos, in attempt to use ε-δ to prove it. got to ln((1+r^2sin^2)/(r^2cos(cos+sin))) not sure where to go...

Homework Statement there is a straight river 40m wide. the water speed on the river can be expressed as the function f(x)=3sin(∏x/40) If a boat is to cross the river from point a to point b at a constant velocity of 5 m/s determine the angle the boater should head. Homework Equations...

Homework Statement I have to find this integral: \int \delta (( \frac{p^{2}}{2m} + Cz ) - E ) p^{2} dp dz where E, m, and C can be considered to be constants. Homework Equations I'm semi-familiar with delta functions, i.e. i know that: \int \delta (x - a) dx = 1 and that you can usually...

Homework Statement Hi, Find the line L of the intersection of the two planes x+y+z=1 z-2y+3z=1 What I did was use Gaussian reduction on the augmented matrix. It was easy [x y z] = [3/2 -1/2 0] + z[-5/2 3/2 1] or in equation form or whatever it's called x = 3/2 -5/2 z y = -1/2...

I'm looking for any kind of reference on a multivariable generalization of a (confluent) hypergeometric function. In particular, Horns list is a list of 34 two-variable hypergeometric functions, 20 of which are confluent. Then one of these has the following series expansion: \Phi_2(\beta...

Homework Statement Vector calculus It can be shown that the area of the surface described by the vector valued function r(s; t) between the limits a ≤ s≤ b and c ≤ t ≤ d is given by A=∫(from a to b) ∫(from c to d ) ‖(∂r/∂s)×(∂r/∂t)‖ dtdsFind the surface area of the bowl described by r(s; t)...

Homework Statement Calculate the limit: lim_{(x,y)\rightarrow(0,0)}=\frac{4\sqrt{x^{14}+y^{14}}}{(x^{2}+y^{2})^{3}} The Attempt at a Solution I think I'm supposed to use algebra for this but I can't seem to figure anything useful out. I've found this...

Homework Statement find the critical point of, x^(2)+y^(2)+2x^(-1)y^(-1) Homework Equations none The Attempt at a Solution Ok so first we differentiate the function such that fx = 0 and fy=0 doing this yields, fx = 2x-2x^(-2)y^(-1) fy = 2y-2x^(-1)y^(-2) both set to...

Homework Statement Locate the critical point of the function, e^-(x^(2)+y^(2)+2x) Homework Equations none The Attempt at a Solution Ok first step differentiate the function and set it to zero for both fx and fy, Fx(x,y) = (-2x-2)e^(-x^(2)-y^(2)-2x) Fy =...

Homework Statement problem in attachment Homework Equations The Attempt at a Solution I can get f'(x) as sqrt(1 + (sinx)^2) and derive that to get the second derivative but as far as that I don't really get the concept behind this question will y be another function I have...

Hi, I'm trying to wrap my head around epsilon/delta proofs for multivariable limits and it turns out I became stuck on an easy one! The limit is: \lim_{(x,y) \to (1,1)}\frac{xy}{x+y} Obviously, the result is 1/2, but I'm unable to prove it! Any hints? Thank you!

I think i found the solution to my problem but i was hoping to have someone check to make sure i did not make a mistake. \xi = x - ct...... (1) u(t,x) = v(t,\xi)......(2) Taking the derivative d[u(t,x) = v(t,\xi)] \frac{\partial u}{\partial t}dt + \frac{\partial u}{\partial x}dx =...

Homework Statement z = f (x^2 + y^2) prove using multi var chain rule that y * dz/dx - x * dz/dy = 0 Homework Equations The Attempt at a Solution honestly i just need to no how to start it then I am sure i could figure the rest out so i would find dz/dx and dz/dy then...

Homework Statement Find and classify all relative extrema and saddle points of the function f(x; y) = xy - x^3 - y^2. Homework Equations D = fxx *fyy -fxy^2 The Attempt at a Solution I got D < 0 where D = -1 and fxx = 0, when x=0 and y=0. However I am unsure as to the conclusion I should...

Homework Statement Let f be a differentiable function of one variable, and let z = f(x + 2y). Show that 2∂z/∂x − ∂z/∂y = 0 Homework Equations Multi-variable chain rule The Attempt at a Solution I have no idea where to start with this, any advice would be greatly appreciated...

Homework Statement See attachment Homework Equations The Attempt at a Solution I'm not sure how to determine which points are co linear or which point lies in between the two. My book doesn't discuss how to do this at all. but PQ→ = <1,-1,2>, PR→ = <3,-3,6> I believe I found...

I'm taking multi-var calculus next semester and I found out that we are using this terrible excuse for a book: https://www.amazon.com/dp/0470131586/?tag=pfamazon01-20 Anyway, I have been studying ahead using my huge copy of Stuart, which my university uses for Calc I and II. I asked my...

Homework Statement Hello all, I have been working on a 3rd order taylor series, but the formula I have does not seem to get me the right answer. The formula I was given is for a taylor polynomial about point (a,b) is: P_3=f(a,b) +\left( f_{1}(a,b)x+f_{2}(a,b)y\right)...

Homework Statement If u=x^3 + 3xy + y^3, determine du/dx on the ellipse 2x^2+3y^2=1 2. The attempt at a solution Imagine I just use partial derivative somehow, but I'm not sure what the question is asking by on the ellipse. I have a feeling its just something simple that I'm...

Hi there. In the multivariate case it is proven that if X~Nm(U, S) and Y = a + CX where C is an invertible n x n matrix, and S is the covariance matrix then: Y ~ Nm(a + CU, CSC') I am trying to apply this proof to a similar problem. If we have X1...Xk with Xi ~ Nm(Ui, Si) where i = 1...k...

Homework Statement Find extreme values for f(x, y, z)=x+y+z subject to: x^4+y^4+z^4=c>0 Therefore, since c>0, exclude the origin. Let L(x, y, z, \lambda{)}=x+y+z+\lambda{(}x^4+y^4+z^4-c) and thus L_1(x, y, z, \lambda{)}=1+4x^3\lambda L_2(x, y, z, \lambda{)}=1+4y^3\lambda L_3(x, y...

Suppose I have F(x,y) and y=y(t) and x=x(t) Therefore, Ft = Fx*xt + Fy*yt. Right? Can I write Ftt = (Fxx*xt + Fyy*yt)*xt + Fx*xtt + (Fxx*xt + Fyy*yt)*yt + Fy*ytt ? Basically I'm trying to figure out the second derivative by chain rule.