Analysis and Calculus

Hello, While learning about splines I've run into some confusion about the wikipedia article on cubic hermite splines: http://en.wikipedia.org/wiki/Cubic_Hermite_spline A few lines from the top of the article it reads: "The subinterval (xk,xk + 1) is normalized to (0,1) via t = (x − xk) / (xk + 1 − xk)." This is causing me some trouble. What does the "x" represent in the (x - xk) term? Also, if xk and xk+1 are vectors, then what does the division mean, since dividing two vectors is undefined? Is the t value a scalar or vector? In the article t seems to be used as a scalar term between (0, 1) which defines where the interpolated point on the splin…

i having problem of understanding log, the question is solve the equation [math] log_2 2p - log_2 (1-3p) = 1[/math] so, i calculate using this way [math] log_2 2p - log_2 (1-3p) = 1[/math] [math] log_2 2p - log_2 (1-3p) = log_2 2[/math] [math] log_2 \frac{2p}{(1-3p)} = log_2 2[/math] and I'm consider [math]\frac{2p}{(1-3p)} = 2[/math] so [math] p = \frac{1}{4}[/math] confirmed this?

Consider the function defined by the equation [math]\mathbf{ y^2-2y.e^{sin^{-1}x}+x^2-1+[x]+e^{2sin^{-1}x} = 0}[/math], Where [math] \mathbf{[x] = }[/math] Greatest Integer function. The find area of the curve bounded by the curve and the equation [math]\mathbf{x = -1}[/math]

Find the nth derivative of sqr(2x-1). That means we need to find a pattern. Here's what I've got so far, from 1st derivative to 5th: The pattern for the exponent is simple, 1/2 - n The pattern for the co-efficient is what I'm stuck on. Here's what it looks like. I'm not sure how to express it mathematically. 1 -> 1 = 1 2 -> -1 = 1(-1) 3 -> 3 = 1(-1)(-3) 4 -> -15 = 1(-1)(-3)(-5) 5 -> 105 = 1(-1)(-3)(-5)(-7) and so on... Anyone know how to make a function for the coefficient in terms of n?

I'm asking about the Zero Index which is a^0 = 1 can anyone explain 'mathematically' to me how this happen? P.S. how does people write Superscript, Subscript and fraction number in this forum?

In the serie [math] \sum_0^{\infty} a_n (x - c)^n [/math], the radius of convergency is: [math]R= \lim_{n \to \infty } |\frac{a_n}{a_{n+1}}|[/math]. My problem is : Find the radius of convergency when: [math] \sum_0^{\infty} \frac{(-1)^n}{(2n+1)!} \cdot x^{2n+1} [/math] i don't understand mainly who is [math]a_n[/math]. The answer is [math]R= \infty[/math]

Hello The problem is find the value of [math]\lambda[/math] for [math]\lim_{n \to \infty} \frac{a_{n+1}}{a_n} \nonumber\ [/math] < 1, where [math] a_n= \frac{(\lambda^nn!)^2}{(2n+1)!} \nonumber\ [/math] con [math]\lambda >0 [/math] I tried to do: [math]\frac{a_{n+1}}{a_n}=\frac{(\lambda^{n+1}(n+1)!)^2}{(2n+3)!}\cdot \frac{(2n+1)!}{(\lambda^nn!)^2}[/math] [math]=( \frac{\lambda^{n+1}}{\lambda^n}\frac{(n+1)!}{n!})^2\frac{(2n+1)!}{(2n+3)!} [/math] [math]\frac{(2n+1)!}{(2n+3)!}= \frac{3 \cdot 5 \cdot 7........(2n+1)}{5 \cdot 7 ........(2n+1) \cdot (2n+3)}[/math] [math]= ( \lambda (n+1))^2\frac{3}{(2n+3)} [/math] but the Answer is [math]\lambda…

Suppose for the real valued funtions f,g,h ,where f is strictly monotone and the other two monotone the following relation holds: g(f(x)) =h(f(x)) =x for all ,x, belonging to the real Nos. Can we rove that : g=h for all ,x, belonging to the real Nos?? The domain of all 3 being the set of real Nos

I might have already posted this in this forum quite some time ago, but the problem just came back to me, and now I want to solve it again, or at least know the solution. Here it is, more in depth: Arc length of a spiral Say you are givin a spiral with an initial radius r' (read r-prime), where r' is any real number greater than 0. As the radius "spins" it grows at the same rate as it "spins". So for every degree it rotates, the radius gains 1/360th of it's initial length (r'); exapmle- if r' = 1 cm, after 90 degrees the new radius is 1.25 cm, then after a full revolution it is 2 cm. And this goes on for however many revolutions. What I need it a solution fo…

How does one go about solving this algebraically? I tried creating a new variable (let a = log(base5)x) to get a^(1/2) + a^(1/3) = 2 but not sure what to do from there... If I can figure out how to get rid of the radicals, the rest should be simple. [EDIT ] A friend helped me with the answer. Let y = log_5(x)Let z = y^(1/6) Then the given equation becomes z^3 + z^2 = 2, that is (z-1)(z+1+i)(z+1-i) = 0. The solutions are r = 1, s = -1-i, and t=-1+i. Then x = 5^(r^6) =, 5^(s^6), 5^(t^6) are solutions of the original equation. The only real valued solution is x = 5. Alternatively, you can arrange the equation into cubic form, and get the roots using the c…

Could someone please setting an argument surrounding the following:- Does....... 48÷2(9+3) = 2 or does it equal 288 and why. Using BODMAS or BIDMAS one gets 2 because the brackets are sorted out first thus 48÷2(12) = 48÷24 = 2 However on my Texas TI-83 I get 288 viz. 48÷2(9+3) = 24*(9+3) = 24*12= 288 Thanks.

I am given a definition: One defines a generalized function [math]\chi(x)[/math] as a sequence [math]f_n(x)[/math] of functions (with certain not very restrictive properties) such that for any other function [math]g(x)[/math] with the same property, the limit [math]\lim_{n \to \infty}\int_{-\infty}^{\infty}f_n(x)g(x)dx = \int_{-\infty}^{\infty}\chi(x) g(x)dx[/math] exists. It seems to me this makes some sort of sense; since taking of limits and integration do not commute in general, it cannot be the case that [math]\chi(x) = \lim_{n \to \infty}f_n(x)[/math] But I am having a hard time seeing what the limit of integrals of a sequence might be (though you might …

There's something that's confusing me about what appears to be the standard form of stating the alternating series test in basic calculus. The four sources I looked up were James Stewart's CALCULUS, Howard Anton's CALCULUS, wolfram alpha's mathworld, and wikipedia. All four had essentially the same statement for the alternating series test: If the sum from i=0 to infinity of [(-1)^n][b_n], with b_n>0 for all n, satisfies a) b_(n+1)<=b_n for all n ({b_n} is a decreasing sequence), and b) the limit as n goes to infinity of b_n = 0, then the series is convergent. What I'm confused about is this: since all four sources made it clear that all of the b_n were stri…

Is 1 + 1 + 1 = 3 ? and are there alternative answers ? like for instance 2.99999999999999999999999999999999999999999999999999999999999999999999991

Hello fellow hobby-scientists, the normal differential equation-based definition of the trigonometric functions sin and cos are as solutions to the well-known second order DE y''(x) = -y(x). It occured to me that they can also be defined by a simple first order functional differential equation on the form y'(x) = y(-x), which for y(0) = 1 has the solution y(x) = sin(x) + cos(x). This obviously means it is possible to construct sin(x) as 0.5*(y(x) - y(-x)) and cos(x) as 0.5*(y(x) + y(-x)). As an attempted short section discussing this fact was swiftly and mercilessly rejected from the wikipedia article on trigonometric functions I'd like to ask you guys what …

If 0=2-(1+x), isn't the correct answer x=2? A simple question, before the author of the book is notified of the print error

I haven't done integrals for over a decade and I'm having trouble with them and my math skills are inadequate The function f(x) = 1/x^2 has a singularity at x=0. The definite integral of 1/x^2 is divergent, if it includes x=0. However, the integral from 1 to infinity, of 1/x^2, is 1. Are there examples of functions that have a singularity (where the function approaches infinity), with a convergent integral? For example of what I'm trying to get is... 1/x^2 remains non-zero for all finite values of x. Along the x axis, I imagine there's basically an infinitesimally tall rectangle that is infinitely wide, and yet it has 0 volume. Yet along the y axis …

Okay, Ironically, I don't even know if this is the right forum for it. Maybe Engineering would be better.. or computer science.. but it's mainly mathematical, so I was hoping the people who frequent this forum will be able to help me out. I want to model a system of objects, each with mass m, attached to each other by springs of constant k. But I have 20 such objects that move in time. My equation involves both an integer-counter x(s) where s is from 1 to 20 (counting the objects), but also x is a function of t (which is continuous). All I want to try and do is solve the relatively simple partial differential equation: d^2x/dt^2 - k/m x'' =0 …

I need a hint on how to solve the following DE. x'' = ex -1. The general answer to the equation will be the sum of inhom. + hom. solutions of the equation. Any help is appreciated.

If a quantity of 100 in year 1 changes to 11 in year 10, the change would be -89 %. But when this change is converted to an annual value (-89/10) and then applied to each year, year 2 would calculate as 91.1. This is repeated such that according to the annual value, year 10 quantity is 39, which is a big difference from 11! What is the correct way to apply an annual change, please?

I am not sure if this should be in Math section or in Computer section, but since it's about math, I'm hoping one of the experts here could help me out. I have an equation related to my research that I want to model in Mathematica. The point at the moment is to try and graph the angle on a circle of a group of particles (the 'n' index counts particles) per time (t index). I managed to produce a lineplot per time if I create individual functions for the particles (defining Phi1[t_], Phi2[t_], etc) but I'm looking for a way to model both indices at once so I can see the relationship between all particles. I looked things up in the Mathematica manual, but, alas, I ca…

Hi there, I hope I have placed this question in the right area. I am trying to workout the rule(s) to the below numbers. Sorry about the ---, but only way to line up numbers. I have managed to workout that if x=y then the result = x*1.414 I have also managed to workout that if x or y = 1 then the result (example if x=1) = y+(x/y)/2 I haven't managed to workout anything else. If you can give me a few tips or direct me in the right way, I would much appreciate it. Thanks for all your help in advance Vincent X 10--10.05--10.20--10.44--10.77--11.18--11.66--12.21--12.81--13.45--14.14 -9---9.06---9.22---9.49---9.85--10.30--10.82--11.40--12.04--12…

can someone pls give me a link or some proof of why the function [(a^n+b^n+c^n)/3]^1/n is monotonically increasing for n>1. i can see that am-gm, am-qm etc follow from this fact. also pf for mean of k nos will be preferable than for 3 stated above. also can i replace sigma ain / k by (definite integral from a to b xn dx) / b-a in the above inequality??

Hey everyone, My name is Anna and i live in Scotland, and am studying an introductory course with the Open University. This is my first time here! Got reccommended the site by a friend, as I have an assignment to do by tomorrow and am stuck on one particular maths question! If anyone has any pointers, or can possibly help explain this to me, it would be hugely appreciated! Question : © One cubic metre (1 m3) of atmosphere at sea level contains 3.80 × 102 ppm of CO2 and 5.00 × 103 ppm of water vapour. If there is a total of 2.6 × 1025 molecules in 1 m3 of air, calculate how many molecules of CO2 and water vapour there are in 1 m3 of air. Give your answers to the ap…

I have published a topic about how the odds of me existing is unlikely but here is two more mathematical probability reasons to doubt one life: Before I was born I was nothing, I was not I, I did not exist So relative to nothing there is no time so on that basis the odds of me coming into consciousness was not for just one particular time until it happened yet if I am supposedly alive just once then the odds of now being the time I am alive taken that time did not exist before I was born relative to me as I did not exist the odds of 2011 being the time I am alive versus all the other years on the universes time-line is in the billions/1 or another way of putting…