Analysis and Calculus

Given that: [math] f: [0,\infty)\rightarrow R[/math],such that [math]f(x)=\sqrt{x+1}[/math] prove that: [math] lim_{x\to 0}\sqrt{x+1} = 1[/math] using the ε-δ definition

I guess everybody should know how to calculate the falling time at approximately uniform acceleration. Here's a problem. Say Milky masses a trillion (10^12) solar and there is nothing else around and you release a brick at a distance of 3 million lightyears. How long does it take to fall 0.5 million light years? It's elementary. Type this into google: sqrt(9 million (light year)^3/(G*mass of sun)) I've canceled 10^12 in the numerator and in the denominator. Otherwise it is just (approx.) uniform acceleration GM/r^2 and distance fallen = (1/2)a t^2. the point is that google calculator is a remarkable tool because it handles all the units for you. It kn…

Suppose that ; 1)[math] f: R\rightarrow R^2[/math] such that : [math]f(x) =(2x+1,x^2)[/math] 2)The Euclidian norm of a vector [math] v=(u_{1}, u_{2})[/math] is defined as : [math] ||v||_{Eu} =\sqrt{ u_{1}^2 +u_{2}^2}[/math] 3) The maxnorm of a vector [math] v=(u_{1},u_{2})[/math] is defined as : [math] ||v||_{max} = max( |u_{1}|,|u_{2}|)[/math] Where [math] u_{1},u_{2}[/math] belong to the real Nos R Then prove : [math] \lim_{x\to 0} f(x) = (1,0)[/math] ,with respect to both norms

prove that the sequence ln n diverges to infinity

Prove that the the sequence [math] n^2[/math] diverges to infinity

prove using the ε-definition,that : ln n is not a Cauchy sequence

I have attached a pdf of output from MAPLE attempting to evaluate an integral of an exponential function containing a complex irrational trig function as the exponent. MAPLE could not evaluate it when integrating with respect to the hangle phi. I then transformed the integral via varible substitutions to hopefully get it in a form that might be executable, but no luck there either. Any suggestions? This integral is a portion of a solution of a radiation wave emanating in the vicinity of source that is configured as a disc. integral output.pdf

Hi to all, I was wondering if any of the scienceforum.net sub forums have any URL addresses written specifically to access RSS Feeds? psychlone.

Suppose that the sequence [math] S_{n}[/math] diverges to [math]+\infty[/math]. Now if [math]S_{n}[/math] does not diverge to [math]+\infty[/math] , does that imply that [math]S_{n}[/math] goes to a limit s

Hi. In 3 dimensional Euclidean space with the usual metric, d=[(delta x)^2+(delta y)^2+(delta z)^2]^1/2, I'm trying to figure out the average distance between nearest neighbors in a randomly distributed sample of particles. My best initial guess for the average distance from any given particle to its nearest neighbor is d_nearest neighbor_mean=(volume/n)^1/3 where n particles are randomly distributed in a 3 dimensional volume. The question originated when I wondered what was the average distance between stars in the solar neighborhood. atlasoftheuniverse.com gives 35 stars (including the Sun) within 12.5 light-years, and the above formula yields 6.16 ly as the avg di…

I presented a professor with the following proof: Prove that the empty set is closed. Proof : By definition : [math]\emptyset[/math] is closed <=> cl[math]\emptyset\subseteq\emptyset[/math] <=> ([math]x\in[/math] cl [math]\emptyset[/math]=>[math]x\in\emptyset[/math]) cl [math]\emptyset[/math] is the closure of the empty set,and B(x,r) is a ball of radius r round x But ,by definition again [math]x\in[/math]cl [math]\emptyset[/math] <=> for all r>0 ,B(x,r)[math]\cap\emptyset\neq\emptyset[/math]....................................................................1 But , B(x,r)[math]\cap\emptyset =\emptyset[/math] => (…

is the empty set bounded from above? has it got a supremum (l.u.b)? Whether it has or not how can we prove that??

Why is [math] \lim_{t\to0} \frac{\sin(t) }{t} = 1 [/math]

Hello. In maple: the signum function is signum(x)=x/|x| the derivate is d/dx {signum(x)}=signum(1,x) what is: signum(1,1,0) signum(1,x,0) signum(0,x,0) ????

For my thesis I need to solve many differential equations non linear, second order by using maple.... For example figure adjoint using dsolve command, the solutions are very extensives and very bad. there is a suggest for to solve the differential equations by using maple? there is some methods in maple to find the aproximate solution? thanks

any suggestions to solve the follows differential equations by using maple??? see adjoint figure

Note: This isn't graded work, it's the basics.. Use the Chain Rule to get dw/dt when, w = cos(x-y) x = t^2 y = sin t I know dw/dt = (dw/dx)(dx/dt)+(dw/dy)(dy/dt) Thus far I have, dw/dt = (-sin(x)-cos(y))(2t)+(cos(x)+sin(y))(cos(t)) Substituting in x and y.. dw/dt = (-sin(t^2)-cos(sint)(2t))+(cos(t^2)+sin(sint)(cost)) I feel like I have done something wrong (I don't have the solution to this problem)..but I know it doesn't look correct. I'm thinking that perhaps I made a mistake with the derivatives and it should be something closer to.. dw/dt = ((cos(x))(2t))+((sin(y)(cost)) dw/dt = ((cos(t^2))(2t))+((sin(sint))(cost)) But again, I'm cluel…

Hi everyone, I have been trying to practice calculus for the benfit of my quantum mechanics studies and I am stuck with a problem so thought you people might have some advice to give.Here goes; [math] \frac{d sin\theta}{d(\theta)cos(\theta)} sin(\theta)sin(\phi)[/math] Now we have to operate the [math] \frac{d sin(\theta)}{d(\theta)cos(\theta)} [/math] onto function [math] sin(\theta)sin(\phi)[/math]. The operator can expressed into [math] cos^2(\theta)-sin^2(\theta) [/math] after diffentiation. Now my question is that can I simply multiply this by [math] sin(\theta)sin(\phi)[/math] or will this be wrong? It would have been much easier if the expres…

The definition of the partial derivative of [math]f[/math] with respect to [math]x[/math] is [math]\frac{\partial f}{\partial x} = \frac{f(x+\mathrm{d}x,y,z)-f(x,y,z)}{\mathrm{d}x}[/math] right? While [math]\mathrm{d}x[/math] has a meaning, an infinitesimal change in [math]x[/math], what does [math]\partial x[/math] mean?

Simple question. Is rolling a six-sided dice that has 5 sides with a 1 and 1 side with a 6 random? Not a 1-6 dice. Or is rolling two dice random?

I am currently reading about the divergence and I am curious what is gained by reformulating stuff like Gauss law from integral form into differential form? The expressions become shorter and nicer, but surely there must be another reason?

What is the value of x for these equations? a. Find all solutions for this equation: (e^2x)-(6e^x)+32/4=0 b.(e^x)-24e^(-x)=-2 Use common logs to solve for x in terms of y for the following: c.y=10^x+10^(-x) ----------------- 18 I don't know even where to start. . . I'd apprecite all the help possible.

Is it possible to express [math]\sin(4x)[/math] in terms of [math]\sin(x),\sin(2x),\sin(3x)[/math] only ??

CAN somebody,please write the definition for the linear independence of the following functions?? [math] e^x,e^{2x}[/math]

Today's calculus warmup question has stumped everyone in the class, including our teacher. Here's the question: [math]\lim_{x\to 0} x^{\sin x}[/math] We can't think of any safe way to evaluate the limit. We tried rewriting the problem to this: [math]y = \lim_{x\to 0} x^{\sin x}[/math] and then changing to this: [math]\ln y = \lim_{x\to 0}\sin x \ln x[/math] but in the end it didn't help. Any ideas? [hide]The answer turns out to be 1, but we have no idea to get it.[/hide]