Analysis and Calculus

how can i find a formulation for a curve that connects 2 fixed points on the horizontal axis, has a fixed arc length L, and encloses the maximum area??

Hi anybody... I have a problem. Someone (or most) should find it really easy, but I'm having trouble. Here it is. f(x)= x^(1/3) (or the cube root of x) need to find f'(x) using limit as h->0 of (f(x+h) - f(x))/h Now, I already know what the answer is: (1/(3(x^(2/3))), but how do I get that using the difference quotient (above). Obviously, my algebra skills are rusty, but here is what I did to get the wrong answer: ((x+h)^(1/3)) - (x^(1/3)) ( times the conjugate) ------------------------ h ((x+h)^(2/3)) + (x^(2/3)) ------------------------- ((x+h)^(2/3)) + (x^(2/3))…

take the inverse square law for gravity(or electric charges for that matter) [math]\vec{F}=G\frac{m_1m_2}{r^2}[/math]. r2 has to be a dot product(if it wasn't, it would make gravity undefined for rxr is 0). mass is a scalar and so is the gravitational constant. so, how do we get a vector from a bunch of scalars?

Here is the question: f(x)= (x^2-36)/(x-6) for 0<x<6 and 2x for 6<x<12 How should f(6) be defined in order for f to be continuous? I originally said 'no limit' but apparantly that is wrong. The answer is actually 12... which is fine for the second equation because 2x=2*6=12... but I don't understand how the first equation could equal 12? If I replace x with 6 in the first equation I get 0/0 could someone help me out with this?

I have a question about improper integrals. Say you have a function with an asymptote at, say x=5. But you only have to integrate until x=4. Is is still considered an improper integral?

differential forms are basic I still have the thin red book of Flanders bought decades ago and recall there were places in Flanders where he didnt explain enough or draw enough pictures. apparently Flanders is still the usual book on Forms for physics students now there is a free download book by Dave Bachman http://pzacad.pitzer.edu/%7Edbachman/ it looks good, first glance----might be more helpful in some cases than Flanders and other people with more exposure to it seem to think so: Tom M has been working thru it and has led a couple of his students thru it. I printed out most of it (not chapter 1 which is very beginner) might read in it and work some…

I'm kindof wondering this... this is the question (Extra Credit)... The following function is continuous at one and only one point. x=0. Prove it's continutity... f(x)=0 when x is rational & fx^2 when x is irrational. To prove a function is continuous at a point n... 1) f(n) must exist 2) [math]\lim_{x \to n}{f(x)}[/math] must exist. 3) [math]\lim_{x \to n}{f(x)}=f(n)[/math] f(0)=0. [math]\lim_{x \to 0}{f(x)}=0[/math] I have no idea how to prove this, but I know it's true because the question says that it's continous, and f(0) has to equal the limit, so the limit must be 0, but I don't know how to do this limit... I know how to do the li…

I am trying to find the set A such that For r > 0 let A ={w, w = exp (1/z) where 0<|z|<r}.

[math]f(x) = e^x[/math] I keep getting the silly answer of just X, when I know it is [math]e^x[/math]

are you following it? if not, you should.

Can anybody help with this? I'm trying to do function spaces, and bumped up against the Lebesgue integral. My books give a pat on the head, and tell me not to worry over much about it, then proceed to tell me it's "zero almost everywhere". That, as a definition, is about as much use as a rubber hammer. I am very familiar with the Reimann integral, but can anybody gentle me through this?

what is the radius of convergence of the sum (n=0 to infinity) of z^(n!) ?

Can someone find an example of a sequence such that it doesn't have a limit, but it has a lim sup? and find the lim sup? I thought about the sequence of all rational numbers in the interval [0,1], but not sure if that's a correct example.

We're going over basic differentiation, and the problem I had is to find the derivative of f(x)= x^(1/3). I have to use the definition of a derivative, but I'm not sure what to do when I begin with two cube roots in the numerator. Can anyone help?

Ok, I'm taking Calculus I. My teacher is not very good (he's a grad student and not very good at teaching. He knows what he's talking about, but not good at explaining). This was his problem: [math]\lim_{n\to 8} \frac{e^x-e^8}{\sqrt{x+1}-3}[/math] Now, direct substitution says [math]\frac{e^8-e^8}{\sqrt{8+1}-3}=\frac{0}{\sqrt{9}-3}=\frac{0}{3-3}=\frac{0}{0}[/math] So, we tried to get the 0 out of the denominator bu multiplying by the congigate. [math]\frac{(e^x-e^8)(\sqrt{x+1}+3)}{(\sqrt{x+1}-3)(\sqrt{x+1}+3)}[/math] [math]\frac{(e^x-e^8)(\sqrt{x+1}+3)}{x+1-9}[/math] [math]\frac{(e^x-e^8)(\sqrt{x+1}+3)}{x-8}[/math] So try direct subsitiutio…

Stereographic projection on complex plane -------------------------------------------------------------------------------- Let V be a circle lying in S. Then there is a unique plane P in R^3 such that p /\ S = V ( /\ = intersection). Recall from analytuc geomerty that P = { (x_1,x_2,x_3) : x_1 b_1 + x_2 b_2 + x_3 b_3 = L, where L is a real number}. Where ( b_1,b_2,b_3) is a vector orthogonal to P . It can be assumed that (b_1)^2 + (b_2)^2 + (b_3)^2=1. Use this information to show that if V contains the point N then its seteographic projection on the complex plane is a straight line. Otherwise, V projects onto a circle in complex plane. N = (0,0,1) the north po…

Solve e^(xy)(ycosx-sinx)+xe^(xy)cosxdy/dx=0 I've checked this is exact and found that H(x,y)=e^(xy)(y^2cosy-cosy)/(1+y^2)+y^2/(1+y^2)C(y) As the answer is simply e^(xy)cosx=C, I guess I made a mistake before differentiating H(x,y) to get C(y) What's wrong with my H(x,y)?

Hi! I have an assignment to find the derivatives of two funtions (1/t and the square root of t). I know how to find derivatives, (a couple of ways, actually) I've just never been taught something with the name 'the first principle of calculus.' How does he want me to find them? I probably already know the method, just not by that name. I saw Dave's lessons, but for some reason they don't display properly on my computer. ( portions of the text are bold and missing) Sorry to ask for something I know is already posted, but I can't get it to display properly. Thanks! Maria

Can someone help me with this problem. I need any hints to help me get started. I am assuming limit points of cisx are -1- i and 1+i I am not sure if this is correct. And if it, I don't know how to show that it is dense, I know the def of a dense subset, but I don't know how to apply it here. Please help me. Show that {cis K : K is a non-negative integer) is dense in T = { z in C ( Complex space) : |z|= 1}. For which values of theta is {cis ( k*theta) : k is non-negative integer} dense in T?

Hi, I've just started working through volume one of Tom M Apostol's Calculus. The first chapter gives a brief history of the method of exhaustion and attempts to show how this is used to show that the area of a parabolic segment is exactly 1/3 of the area of the rectangle enclosing it. However when it comes to explaining equations 1.1 and 1.2, I dont understand how (n-1)^2 comes into equation 1.2. I have attatched the parabolic segment and explanation to the problem. I know this will be some fundamental gap in my understanding of basic math but i cant follow through and understand how equations 1.1 and 1.2 differ. Thank you in advance for any help. parabolic segm…

Please can anybody help on this. Use (beta,delta)definition of a limit to prove that lim (x+y/x^2+y^2+1) = 0 (x,y)-(0,0)

Can somebody tell me what type is this differential equation: y''-f(x)y'-sin(y + h(x))=0 I am interested more at numerical solution.

I am going over Differential Forms, because people are reformulating Electromagnetic Theory using this approach. When I did it, it was all from the point of view of Vector Calculus, but some things are quite clumsy, long and inelegant in vectors, not to mention un-intuitive. I was hoping Differential Forms would be a nice approach that may simplify alot of almost intractable problems: For instance Green functions applied to QM... Anybody knowledgeable, or interested, or even both? Thanks in advance

In a beehive, each cell is a regular prism, open at one end with a trihedral angle at the other end, it is believed that bees form their cells in such a way as to minimize the surface area for a given volume, thus using the least amount of wax in cell construction examination of these cells has shown that the measure of the apex angle is amazingly consistent based on the geometry of the cell, it can be shown that the surface area is given by A = 6sh -1.5s^2 cot X + ((3 square root 3s^2)/2sinX) where s is the length of sides of the hexagon and h is the height , and they are constants. Use the optimisation theory to determine the minimum surface area of the cell in terms …

Ok, so i just started hyperbolic trig at uni and although its sweet, i mean just the basics with a different name but im having trouble with a couple of q's sinh(2x)= 2sinh(x)cosh(x), for all xE|R d/dx(tanh^2(3x) = 6sech^2(3x)tanh(3x), xE|R 8sin^4(x)= 3-4cos(2x) + cos(4x), xE|R i need the working, actually even just a method or explnation would be much appreciated, Cheers