Analysis and Calculus

Can the following two definitions of sqrt(x) be considered as equivalent?? 1) if [math]x\geq 0[/math] ,then ([math]\sqrt x=y[/math]) if ([math]y^2=x\wedge y\geq 0[/math]) 2) if [math]x\geq 0[/math] ,then ([math]\sqrt x=y[/math]) iff ([math]y^2=x\wedge y\geq 0[/math])

Can the following definition of the absolute value be considered as correct?? ([math]x\geq 0\Longrightarrow |x|=x[/math]) and (x<0[math]\Longrightarrow |x|=-x[/math])

Suppose I have a small (infinitesimal) quantity [math]dy[/math] and another small quantity [math]dx[/math] and they are related by [math]dy = k \cdot dx[/math]. Will that automatically imply that [math]\frac{dy}{dx}=k[/math] is the derivative [math]\left(\frac{\mathrm{d}y}{\mathrm{d}x}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)- f(x)}{\Delta x}\right)[/math] of [math]x[/math] with respect to [math]y[/math]? I have seen several examples of such things occuring in engineering textbooks, such as electrical relations between the charge on a capacitor and the voltage across it. (I can't remember the details, and my notes are safely stored in the basement).

Hello, I am suck on this question and when through 1 tutor in person and 2 online tutors with no success (they could not get the solution, no joke) can someone help me out. The question is... Find the interval(s) in [ 0 , 2 ] on which the following function is increasing and those on which it is decreasing. f(t) = - sin t - cos t Any help would be great thanks! Tacobell

I ran into this problem here [math]\int {1}{x^2 + 7x + 10}dx[/math] and the only way I can think of to solve this kinda thing is to do a substitution and get [math]u^-1[/math] but in this case I have no idea how to turn the substitute in the du. Is this a trig substitution problem?

Solve for t: y0e-kt = y0 / 2 y0 > 0, k >0 I have no idea how to approach this problem. I don't even know what category this falls under so I'm having trouble googling info this type of problem.

prove without using contradiction.that the empty set is the subset of every set

Let : [math]f_{n}(x) =\frac{x}{x^2+n}[/math] be a sequence of functions in the Real Nos. I can prove point wise convergence to the zero function as n goes to infinity . But is there a uniform convergence ?? If there is, can anyone prove it ,please??

How do we prove that: [math](1+x)^h<1 +xh [/math] ,where : 0<h<1 and [math]x\neq 0[/math] x>-1

Prove whether or not the function [math] f(x)=\frac{1}{x+1}[/math] is uniformly continuous in [math](0,\infty)[/math]

I think every mathematical forum at the very beginning of its existence should produce a definition of the mathematical proof and then according to that definition every mathematical proof should be written down. Also a check process should be mention ,hence to avoid useless duals between the members of the forum whether a proof is correct or not. I think the above should be stipulated as the basic rules of each and every mathematical forum

Hello, if the arc-length of an archimdes spiral (polar formula is r = a * theta) is s(theta) = 1/2 a (theta * sqrt (1 + thetaˆ2) + asinh theta) what is the formula calculate theta as a function of S? Best regards, Oren

Hello, this is my first post. I have a question of related rates that was giving me trouble, could someone help me, it is... There is an isosceles triangle with the sides increasing at a rate of 2 ft/min, the base is 19 feet, what is the rate of change of area when the height is 8 ft? Someone said to me take the derivative with respect to a variable other than t, this did not make sense to me because you want rate of growth per unit of time. Tacobell

Suppose that ; 1)[math] f: R^2\rightarrow R[/math] such that : [math]f(x,y) =xy[/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,y)\to(1,1)} f(x,y) = 1[/math] ,with respect to both norms

How do we -prove that: The axiom (theorem ) of the sequence of the nested interval in real Nos implies that every monotone sequence in real nos ,bounded from above, has a limit

[math](\frac{du}{dy})^2=A+Be^{2u}+C \sqrt{D+Ee^{4u}}[/math] where A,B,C,D,E are nonzero

While reading the Feynman lectures, I stumbled upon a passage which I do not quite understand. In Vol II, 3-9, Feynman is telling us about Stokes' theorem. He writes: [math] \oint \vec{C} \cdot d\vec{s} = C_x(1) \Delta x + C_y(2)\Delta y -C_x(3) \Delta x - C_y(4)\Delta y [/math] and he looks at [math] [C_x(1) -C_x(3)]\Delta x [/math] and writes: "You might think that to our approximation the difference is zero. That is true to the first approximation. We can be more accurate, however, and take into account the rate of change of [math]C_x[/math]. If we do, we may write [math] C_x(3)=C_x(1)+\frac{\partial C_x}{\partial y} \Delta y [/math] "If we incl…

Okay, so i've manager to lump a mathematical model into the one mega equation but i've come up against a slight problem, getting the derivative(not essential but makes my work a lot easier). see, the first derivative is a function of the second derivative which is a function of the third which is a function of the fourth which is... you get the picture. as i have a program that can go through them automatically, i got the 1000th derivative and it was still a function of the 1001th derivative. now, i did encounter such stuff in my math classes, but only there. i've never had to deal with one in real life never mind this one thats so complex. how is it you …

Hello, I want something productive to do over the Christmas break and would like to learn calculus. I have a pretty good understanding of Algebra, but know nothing about calculus. I am wondering where I should begin. Are there any books or lessons you people would recommend I purchase? I learn pretty quickly, and would like to be very proficient in calculus by the time the winter break is over. What I am basically looking for is a starting point and some good study materials. Can anyone help me?

I noticed some unexpected behavior in the real-valued f(x)=(1+x)^1/x, as a function of real numbers, when plotting it on wolfram alpha. I inputed: plot (1+x)^1/x from x=-0.0000001 to x=0.0000001 and saw that it unexpectedly seemed to oscillate near zero. I took a closer look with: plot (1+x)^1/x from x=-0.00000000001 to x=0.00000000001 and saw that it definitely seems to oscillate near zero. My original rough graph on paper using a hand calculator suggested the curve was smooth near zero, and even windows calculator's 32 decimal places were unable to reveal the oscillation when I manually calculated many different values near zero. I don't think f(x) i…

I have a flashlight angled at [math]\phi[/math] from zenith (the z-axis). the flashlight can be rotated around the z-axis so the beam forms a cone (angled at [math]\phi[/math] from zenith). Moreover, the bulb in the flashlight can also be angled [math]\phi[/math], so the resulting angle from zenith can vary from [math]0[/math] to [math]2\phi[/math] degrees. The question is, how do I transform a cartesian coordinate representation in space, into the two axes? So, for instance, [math](x,y,z)[/math] becomes [math](r,\theta_1,\theta_2)[/math] where [math]\theta_1[/math] is the angle of the flashlight itself. and [math]\theta_2[/math] the angle of the bulb. [math]r[/m…

Manipulate the equations: 1. (w/v) = (g/(v-w)) to get 1/v + 1/w = 1/g 2. (w - g) = f²(1/(v-g)) into 1/v + 1/w = 1/g Please give me solutions so that I can compare them to the other similar questions. Thank you. Merged post follows: Consecutive posts mergedAnyone...?

Prove that the set S = {x: a<x<b } is open in R (=Real Nos ) and not in [math]R^2[/math]

So I get that it is "elegant" and "advanced" and stuff. How exactly do you graph ANYTHING with "i" in it though. Isn't it by definition, and "i"maginary number?!?!? Um... doesn't that mean that this equation is basically null? I don't see any unicorns in Einstein's Relativity equation...

Given that: 1) A function [math] f: D_{f}\subset R\rightarrow R[/math] is uniformly continuous iff for any pair of sequences {[math]x_{n}[/math]} {[math]y_{n}[/math]} in [math] D_{f}[/math] ( = domain of f), [math]lim_{n\to\infty}|x_{n}-y_{n}| = 0[/math][math]\Longrightarrow lim_{n\to\infty}|f(x_{n})-f(y_{n})| = 0[/math], Determine if the function f(x) = [math]x^2[/math] is uniformly continuous or not.