Analysis and Calculus

Lim[sin(x)/x] as x ->0 Note-it is floor function

So I know there's a specific way to derive the probability of something in an infinite set with some kind of arbitrary label of 0. Normally it would seem like the chances of picking any element in an infinite set would be 1/infinity, but you can't divide something by infinity, so how does probability and permutation get around that? I mean there's infinite area of matter to occupy, but if it had 0 chance of existing in a particular location, how could all this be here? Or like with the domain of a function that extends indefinitely but that models probability as y values being more likely at certain x values, like a probability bell curve, how is this modeled? What else …

In another thread (now locked) "Dr.Rocket" and others offer a useful perspective on the use of differentials. For example, it is concluded that dx/dy (in usual mathematics) is not a ratio. I have seen differentials defined to be real numbers - really just del(x), del(y) in which case they can be a ratio, but this is not what I want to ask about. I see many cases in the physical sciences, including the earth sciences and especially in thermodynamics, where differentials are used in what I'll call "casual" (or short cut) derivations. I'll include an example below. My particular interest is not to rain on anyone's parade; the contexts in which I find these deri…

Hello I have the limit lim (x^9 * y) / (x^6 + y^2)^2 (x,y)---> (0,0) when I use polar the final result is limit = lim (r^6 cos^9 (theta) sin (theta) ) / (r^4 cos^6 (theta) + sin^2 (theta)) r--->0 and substituting r = 0 , it will give zero * I tried it on wolfram alpha and it gave zero http://www.wolframalpha.com/input/?i=limit+%28x^9+*y%29%2F%28x^6+%2By^2%29^2++as++%28x%2Cy%29+--%3E+%280%2C0%29 but when I use cartezean and try the path y = mx^3 the result turns to be m/(1+m^2)^2 which depends on m so what is it ?!

Just wondering, but suppose [math]f(x)=\dfrac{a(x)}{b(x)}[/math]. If both [math]a(x)[/math] and [math]b(x)[/math] are each represented by their own infinite series, can I condense the function for both series into one sum and call it [math]f(x)[/math]?

HOWDY! I had a question in my homework which i needed to double check with my answer . All i would require is the correct final solution . Please help me out. Q. Find the inverse Laplace transform of the function F(s) = e^-2s (s - 3) _________________ (s - 1) (s^2 - 6s + 13) [ {e^-(2s)} (s-3) ] / [ (s-1) (s^2 - 6s + 13) ]

While reviewing calculus, I noticed that James Stewart's Calculus 3rd ed., chap 14, problems plus (at the end of the chapter), p.965, prob#2 asks the student to show that [latex]\zeta(2)=\sum_{i=1}^\infty\frac 1{i^2}=\int_0^1 dx\int_0^1 dy\frac1{1-xy}=I_\circ[/latex] by rotating the integration region by [latex]-\frac {\pi}4[/latex]. Mathworld also evaluates the same integral with the same rotation, http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html . Mathworld refers to Simmons (1992), which is Simmons, G. F. "Euler's Formula [latex]\sum_{k=1}^\infty\frac 1{k^2}=\frac {\pi^2}6[/latex] by Double Integration." Ch. B. 24 in Calculus Gems: Brief Lives and Memorabl…

Hello How do you show that there are no whole numbers [math]n[/math], [math]m[/math], [math]n\times m\neq 0[/math] such that [math]m\sqrt{2}+n\sqrt{2}[/math] is a rational number. Thanks!

So I know the general approximation formula using the limit of the function with n numbers of DeltaX times the height that changes over n intervals of delta x, but what if I want to be infinitely accurate or get the exact amount of area and not just some approximation? Would I use an infinitely small deltaX? How?

Before I start, just want to say that this is not homework. I'm studying single-variable calculus and I sometimes have trouble getting certain things. (btw, might have more questions after this one) So. Let's say I have a function [math]f(x)[/math]. I'm aware that one can show it is continuous at a certain point, but how do I prove that it's continuous on an interval, say [math][a, b][/math]?

So say I have a bag of 10 marbles, 1 red, blue, green, yellow, purple, orange, magenta, cyan, brown and gray, and there's a 1/i change of me picking a green one at any given second. What is the likelihood I will pick a green number as a percentage of 10? How much time could pass before I would actually be able to pick a green one? Actually, let's just say I have a bag of 10 marbles, they're all green and the probability of picking one is 1/i. What is the percentage I'd pick a green one using that probability?

not sure if this belongs here, However, i was wondering with: 3+i/1-i = 3+3i+i-1/2 ; in the process of solving it, Where does the '2' come from..(in the denominator).

-(23/27) * integral of (x+6)/( (x^2)+3x+9 ) dx = -(23/27) * integral of (x+(3/2))/( (x^2)+3x+9 ) dx - (23/27) * integral of (9/2)/((x+(3/2))^2) + (27/4)). I don't know how this is done, primarily, everything else is the same, except for the numerator, where the left hand side is (x+6), which is equal to the right hand side of (x+(3/2)-(9/2)). How is this possible?

The Semjase equation for pi ln(.5-.75^.5)= -a+i*pi Anyone care to prove or disprove?

Hello These notes were taken from a lecture I had the other day: [math]\sum\limits_{k=0}^m \binom{m}{k}x^k + \sum\limits_{k=0}^m \binom{m}{k}x^{k+1}[/math] [math]= \sum\limits_{k=0}^m \binom{m}{k}x^k + \sum\limits_{k}^{m+k} \binom{m}{k}x^{k(k+1)}[/math] [math]= \binom{m}{0}x^0 + \sum\limits_{k=1}^m (\binom{m}{k} + \binom{m}{k-1})x^k + \binom{m}{m}x^{m+1}[/math] [math]= 1 + \sum\limits_{k=1}^m \binom{m+1}{k}x^k + x^{n+1}[/math] [math]= \sum\limits_{k=0}^{m+1} \binom{m+1}{k}x^k[/math] Can someone please explain to me by which rules these steps are possible, because it is all greek to me at the moment. The steps are part of a proof for the Binomial Theorem. …

If two expressions are exactly equivalent, then are their integrals exactly equivalent? I was trying to work out, without using integration by parts (trying to avoid infinite series and all that) [math] \int sin(x) cos(x) dx [/math] So naturally, I consulted my double angle formulae, and saw that [math] sin(2x) = 2 sin(x) cos(x) [/math] which obviously implies that, [math] sin(x) cos(x) = \tfrac{1}{2} sin(2x) [/math] The integral of the RHS is an easy one, so I just did, [math] \frac{1}{2} \int sin(2x) dx = -\tfrac{1}{4} cos(2x) + c [/math] and so assumed that, [math] \int sin(x) cos(x) dx = -\tfrac{1}{4} cos(2x) + c [/mat…

Can anyone help me to find this limit? See the attached picture. Thank you.

Hi guys, I'm working on some preparatory work for Fall term and I've come across this question in which I require some help. I'm using the 3rd order Sallen-Key low-pass filter as in: http://sim.okawa-den...y3Lowkeisan.htm and I want to find the differential equation relating the output voltage (Vout) and input voltage (Vin). Currently what I've done is split the circuit into 3 nodes from where I am think deducing the output voltage using kirchoffs circuit laws. What I have for the above circuit is: Va = (Va-Vin)/C1 + (Va-Vb)/R2 + (Va-Vout)/C2 = 0 Vb = (Vb-Va)/R2 + (Vb-Vout)/C2 + (Vb-Vc)/R3 = 0 Vc = (Vc-Vb)/R3 + (Vc-0)/C3 = 0 and since Vc = Vout…

Area of Circumscribed (around a circle) n-gon = n/tan[(n-2)(180)/(2n)] square: Find: As, pi Given: r=1, n=4 As=n/tan[(n-2)(180)/(2n)]=4/tan[(4-2)(180)/(2x4)]=4/tan[2(180)/8]=4/tan45 = 4 hexagon: Find: Ah, pi Given: r=1, n=6 Ah=6/tan[(6-2)(180)/(2X6)]=6/tan[(4)(180)/12]=6/tan60 = 3.464 decagon: Find: Ad, pi Given: r=1, n=10 Ad=10/tan[(10-2)(180)/(2x10)]=10/tan[8(180)/20]=10/tan72 = 3.249 hectagon: Find: Ah Given: r=1, n=100 Ah=100/tan[(100-2)(180)/(2x100)]=100/tan[98(180)/200]=100/tan88.2 = 3.14263 chiliagon: Find: Ac Given: n=1000 Ac=1000/tan[(1000-2)(180)/(2x1000)]=1000/tan[998(180)/2000]=1000/tan89.82 = 3.…

I have been given the following problem (see attached). I believe I have almost solved it but for the final part. Please any suggestions would be fantastic

Hello everyone, Great forum, wondering if you can help me with a problem. Ive attached the question along with my calculations so far. Any help would be fantastic

In an online videolecture as well as in some texts regarding Fourier Series, i have come across a statement "Periodicity results from symmetry" and they quote the example of heat distribution in a ring. My doubt is that won't be the heat distribution still be periodic even after the symmetry of the ring is broken- say there is a kink in only one half of the ring.

I'm currently taking a class on computer algorithms where we are trying to determine efficiency classes. I realized in this process that I am unclear on how to compute certain summations and was wondering if someone could point me in the right direction. We have been given the following example: [latex]\sum\limits_{i=0}^{n-2} \sum\limits_{j=i+1}^{n-1} 1 = \frac{n(n-1)}{2}[/latex] I am not clear on how the RHS can be obtained from the LHS algebraically? I'm also a little uncertain on how to approach the following when asked to compute the summation: [latex]\sum\limits_{j=1}^n 3^{j +1} [/latex] I know this may be an easy question for some of you to…

Hey I just want to find out if anyone knows if it is possible to enter complex numbers into a matrix of a scientific calculator. I have a CASIO fx-991ES PLUS scientific calculator. For example: If 20cos(200t -45)= 70jI1 + 90jI2 .....1 and 0=2jI1 +8jI2 ....... 2 Is there any way to solve these equations in a 2x2 matrix on a calculator? (The j's are imaginary, used so that there is no confusion with the variable vectors of I1 and I2). It is easy to solve these equations by hand but when it gets up to 4 equations to solve it gets far more tricky and time consuming. I am getting fed up with all the math that is involved to solve simple circuits, so I t…

Hi there, I have been doing some calculus exercises, and was wondering if I had this right. Prove that [math]f[/math] is continuous at [math]a[/math] if and only if [math]\lim_{h \to 0}f(a+h) = f(a)[/math] I'm not sure if I got this correct, but I assumed that as we take [math]h[/math] to [math]0[/math] then [math]f(a+h) = f(a+0) = f(a)[/math] and so we are done? Thanks