Analysis and Calculus

Could anyone help me find this integral? I've been going back to it occasionally on and off for a year and a half. I once had access to mathematica, which gave me a strange string for the integral. At the time I couldn't understand it, but I may be able to understand it now. Here is the integral I'm trying to find: [math]\int^{1}_{-1}\left[(-x)^{k}(1-x^{k})^{1-k} + 1\right]^{\frac{1}{k}}dx[/math] If anyone could help I would be very much obliged, it is an interesting integral.

Hi, my name is Daniel. I am in first semester calculus, and the teacher is very disorganized and skips sections whenever he cares to. Earlier in the semester he skipped the section on epsilon/delta limit proofs. By reading the book, I was able to understand the idea of finding d as a function of e such that for any e > 0, there is a corresponding e that satisfies |f(x)-L| < e if 0 |x-a| < d and that this refers to the distances on the f and x axes... The problem is not with my understanding of epsilon delta proofs, so much as it is the techniques that the book uses to proove them. It only has a few examples, and the techniques seem very diverse and rando…

lim x-->0 [(1/x)+x] Can I split it into limx-->(1/x) + limx--->0 (x)? A principle said it can if both of the functions exist. But does the former exist?

Hello, over the summer I was checking out this infinite series, the alternating harmonic series, and was able to reach two contradictory answers. Let the sum of the series be S: [math]S = \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + - +...[/math] Group the terms in to the odd and even fractions: [math]S = \left(\frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} +...\right) - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} +...\right)[/math] Now by the distributive property, pull out a half: [math]S = \left(\frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} +...\right) - \frac{1}…

hey all, my textbook has this sample question... "Find the shape of a reectangular box with no top having given volume V and the least possible total surface area of its five faces." now the book has the soluition, i.e. you just do S = xy + 2yz + 2xz and V = xyz and you end up finding just one critical point. my question is why should this point be a minimum? -sarah

the question is f(x)=(2x+sin(x))/(2x) as x approaches 0 split the equation and I get f(x)=(2x)/(2x) + sin(x)/(2x) (2x)/(2x)=1? sin(x)/(2x)=(1/2)? (since sin(x)/x=1) add the 2 together and I get the answer 3/2 now this is where the trouble starts, when I graph this function, as x approaches 0 from possible and negative side. the y value is 1.00873 so which answer is correct? the 3/2 from algebra or 1.00873 from graphing? I probally did something wrong somewhere, help me

Define [math]\Lambda(n):=\log(p)[/math] if n is a power of a prime p and 0 if n = 1 or n is a composite number Prove that [math]\Lambda(n)=\sum_{d|n}\mu(\tfrac{n}{d})\log(d)[/math] The hint says to look at [math]\sum_{d|n}\Lambda(d)[/math] and apply the Mobius inversion formula. So far I have got [math]\sum_{d|n}\Lambda(d)= \sum_{i=1}^r \log(p_i)= \log(\prod_{i=1}^r p_i)[/math] assuming that n has r distinct primes in its expansion. So help Don't mind the above, I have figured it out. I will post more questions if any in this thread instead.

Hello all. I just had a calculus question. Were learning all about derivatives and everything right now. I was just wondering how do you get the anti-derivative of something? Like working backwards from the derivative. I know that if [math] f'(x) = 2x [/math] then working backwards it'd be [math] f(x) = x^2 + c [/math] where c could be anything, but I dont know why. If anyone could show me an equation of some sort that gives you an anti-derivative I'd be happy.

Extra Credit on the test (already taken)... Here's the question: You have a 5 meter long trough. A cross section of the trough is an isololes trapazoid with upper base 3 meters, lower base 2 meters and an altitude of 2 meters. You are pouring water into the trough at a rate of 1 cubic meter per second. What is the rate of change of the height of the water at a height of .5 meters... here is what I got out for givens and the variable I need to solve for... V'=1 l=5 b1=2 h=.5 h'=? [math]V=\tfrac{b_{1}+b_{2}}{2} \cdot h \cdot l[/math] b1 is 2, l=5, but I can't figure out b2! Then it's a simple matter of related rates, and diferentiate with respect…

OK... I was sick and missed this class (Damn my luck!). And the book is no help figuring it out. Related Rates. here's the example problem... Volume of a cone... [math]V=\frac{\pi r^2 h}{3}[/math] it's talking about how if a conical tank of water is being drained... it's all a funtion of t,r,h and V... and it says to implicitly diferentiate... [math]\frac{d}{dt}V=\frac{d}{dt}(\frac{\pi r^2 h}{3})[/math] [math]\frac{dV}{dt}=\frac{\pi}{3} (r^2\frac{dh}{dt}+2rh\frac{dr}{dt})[/math] and then it says that you can see that t is related to r and h! I am REALLY confused.... first off, to that last form... and where is the t? Since the rest…

I just started my lesson on my elementary limit recently, and I did many exercises to consolidate my knowledge. I figured out something interesting but I'm not sure whether it is correct. That's why I need your superb help to prove or to reject it. There are several ideas of mine. 1.[Math]\lim_{f(x)\to 0} \frac{1}{f(x)} = 0[/Math] \lim_{f(x)\to\0} \frac{1}{f(x)} = 0 My idea: The limit does not exist. I don't know how to prove it, because if the function is a fraction, I've heard of inverse function. But I have no idea to prove my thought. 2.[Math]\lim_{x\to -2} \frac{1}{[\sqrt(x+2)]} = 0[/Math] The limit does not exist. 3.[Math]\lim_{x\to 1} \sqrt{x-1} = 0[/Math…

http://www.howtodogirls.com Girls in bikinis teaching calculus, I found it on iFilm. But I must say, I want to order it, and I'm only doing pre-calculus maths

Can you guys help me out. I am in calc2 and having trouble. I was given the test took it (wasnt good). Well we got to take the test home handed in work on seprate sheet of paper. I wanted to try and do them at home to see if its possble for me to even do the problem. I took it to the calc2 student tutor and it took him 2 hours to figure out 2 problems !!! How am i suppose to do 10 problems in 1 1/2 hrs and it takes this tutor (who is in calc3) longer to do 2 probelms ?!?! anyway long story short i am having problems. CAn you guys check these sample problems out and walk through the steps. If you give me the answer can you explain how you got it, which methods used (sub, i…

Can you please check if my answer is correct. [math] y=sin(sin(sinx)) [/math] [math] y'=cos(sin(sinx))cos(sinx)cosx [/math] (My calculus professor doesn't want our answers simplified.)

^ as title states anyone?

tan(θ°) = (viv) / (vih) = (1.28 ± 0.0676m/s) / (0.514 ± 0.0676m/s) = 2.55 ± 0.467 m/s θ° = 68.0 ± 3.65° This answer may not be correct to the decimal place but it shud b relatively close. Could someone plz explain to me how 3.65° becomes the uncertainty for the angle? Plz. explain very clearly and simply. Im in a hurry. THanks in advance

Can you guys tell me if my answer is correct? Determine if the following functions satisfy local or uniform Lipschitz condition. 1). te^y I found d/dy (te^y) = te^y, and this is not bounded above for any value of y, so this made me conclude that it has locally Lipschitz condition since the Lipschitz constant here changes as the reagion changes? Am I right? I used the equation | f(t,y_1) - f(t, y_2) | = d/dy(f(t,y)) + | y_1 - y_2| 2). y t^2/ (1 + y^2) I used the same approach here and d/dy (f ) = t^2 - 2y + y^2/ ( 1 + y^2)^2, which is clearly could be bounded above by a constant but this constant changes as the reagion changes so it is l…

This is the problem in my book. If g(x)=x/e^x, find g^(n)(x). I don't really understand what the problem is asking me to find. It is in the differentiation section of the book, if that helps at all. I think it may be asking for a formula... By the way, the n in the formula represents how many times to take the derivative of g(x).

Ok... Our class (as our homework assignment) needs to find the following derivatives using implicit.. [math]sin^{-1}(x), cos^{-1}(x), tan^{-1}(x), cot^{-1}(x), csc^{-1}(x), sec^{-1}(x)[/math] and, since our book doesn't give these derivatives, I don't know if I got them right (because we have to then use them in the assignment...) [math]sin^{-1}(x)=y[/math] [math]sin(sin^{-1}(x))=sin(y)[/math] [math]x=sin(y)[/math] [math]1=cos(y)y'[/math] [math]y'=\frac{1}{cos(y)}=sec(y)[/math] using triangles and trig identities, since arcsin(x)=y, y is the angle with x the side opposite... so x/1 would be opposite over hypotinuse... so the last side is…

I am reading ODE ( Ordinary DE) notes, and there is a statement that says " A function that is not uniform, even a continuous function that is not uniform, cannot have a lipschitz constant. As an example is the function 1/x on the open interval (0,1). I want to see if I understood this correctly. I will assume that we can find a number k >= 0 such that | f(x) - f(y) | =< K*|x - y| we have x and y in (0,1) so 0 < x < 1 0 < y < 1 .....(1) Now, | 1/x - 1/y | =< k*|x - y| Find common denominator | y - x|/|xy| =< k*|x-y| | y - x | = |x - y| then we cancel the term from both sides of inequality, so we get 1/…

Hi, i'm just wondering if anyone could teach me calculus online. I'm only a sophomore in Geometry and Geometry is just too easy so I just really need to learn more... Or Advanced Algebra even?? I don't care just anything harder than Geometry... please...

Here is the question: [math]\lim_{x\to\infty}\left(1+\frac{2}{x}\right)^x[/math] I thought that (1 + 2/x) as x approaches infinity is just 1... and 1 to the power of anything (including x) is still 1. But it turns out that 1 is the wrong answer. What's up with that?

Can someone tell me the limit of (x/(2x-2))-(1/((x^2)-1)) as x approaches 1.

Can someone help me with the following proof: Suppose f ,g: X -> [- infinity, + infinity] are measurable. Prove that the sets { x : f(x) < g(x) } , {x: f(x) = g(x) } are measurable.

http://www.math.gatech.edu/~cain/notes/calculus.html seems to be a good online textbook for multibvariable, just wondering if its missing coverage on anything.