Analysis and Calculus

Hey there, mmm, you will see, I'm a calculus student, today as I was coming back home I remembered the logarythm properties, the thing is, I found out I just accepted them as they told me, and never questioned them, so as soon as I got home I tried to analyze the properties from a deeper perspective, I was wondering, there's someone that can explain me and demonstrate me the reason of the properties in logarythms? I would really appreciate it Also, as I was experimenting I came with a log 9, but I (feeling guilty of not having questioned the properties told to me before) decided to solve them using differential calculus instead of a calculator so I made like this: …

Hi all! how can I approximate the series: [math] S_{n} = \sum_{k=0}^{n}\binom{3n}{k} [/math] ? The approximation is necessary, since no closed form can be found. Solutions are: [math] S_{n} \sim 2\binom{3n}{n} [/math] or more precise [math] S_{n} = \binom{3n}{n}(2-\frac{4}{n}+\mathcal{O}(\frac{1}{n^2})) [/math] but how did I get there? Thanks! safak

Имеем бесконечное количество яблок. Распространение их на бесконечное множество ofidentical число groups.In каждой группы 5 apples.Now от eachgroup вывезти 2 яблока. Осталось 3. Theremainder множества разбит на бесконечное множество groups.In eachgroup и 7 яблок. Теперь возьмите за 2 яблока. Остается 5apples. Theremainder множества разбит на бесконечное множество groups.In eachgroup, и 11 яблок. Теперь возьмите за 2 яблока. Остается 9apples. Andso на неопределенный срок. Захват 2/n. Сохранить N-2/N) . N-простых чисел в order.Question: как manyapples останется? Конечного или бесконечного?

Write equations for the two straight lines that pass through the poing (2,5) and are tangent to the parabola y=4x-x^2. I drew the figure and the lines. I fought the slope for the tangent lines (2 and -2) and eventually the equations y=-2x+9 and y=2x+1 When I plugged the three equations into my calcuator to see if I was right it seems that I was off on y=-2x+9, that it doesn't reach 5 when x is two but at 4. The second one seems right. Am I right and the calcultar be wrong? Thanks.

A math professor of mine was recommending that I go over integration techniques before doing courses involving differential equations. Since I plan on doing at least 3 such courses in the fall, (comp. mechanics, classical mechanics, and diff. eq. itself) I need to catch up on this stuff. So, I was doing some integration by parts practice problems, and I've run into trouble already. One such problem was to find the indefinite integral of LN(2x+1)dx with respect to x. So I set up u = LN(2x+1) and dv = dx, implying v = x and du = 2/(2x+1) Using the formula INT(u)dv = uv-INT(v)du, this yielded LN(2x+1)-INT(2/(2x+1))dx So for the integral within an integral, I …

((x+1)/(x-2))-((x-1)/(x+2))

Ok, so supposedly in calculus it's a rule that the limit of a ratio of two functions should be logically equivalent to the ratio of the limits of those two functions. Supposedly even for discrete sequences the same applies to limits at infinity. So what if two functions (or sequences) tend to infinity on their own? Let's say we have f(x) and g(x), both of which tend to infinity as x tends to infinity... however, the limit at infinity of f(x)/g(x) is a finite number, let's say c. If c were greater than 1, would it be logically equivalent to say that lim f(x) > lim g(x) even though both tend to infinity? Similarily, if c were less than 1, would it be logicall…

Hi there! Does someone know an analitic expression of fourier transform of error function, or at least, if it exist? Thank you in advance.

Hi Guys, I have a fundamental doubt on differential of y = f(x) = ln(ax) where, ln - natual logorithm to base e, a - constant x - independent variable. I need to find the value of dy/dx from first principles. Can anybody help me on this? Thanks, Srini

What is the limit of this thing [math] \lim_{n\rightarrow 0}\frac{a^n-1}{n} [/math] It has come up in an attempt of mine to find the derivative of [math]a^x[/math]. According to my derivatives table, the derivative of [math]a^x[/math] should be [math]a^x \ln(a)[/math], suggesting that the limit should converge to [math]\ln(a)[/math]. The question is, how do I arrive at this?

Hi! How are you? I'm doing a project on chemistry and i need some help on analysis of data, if possible. I think i can get more help from a mathematician then a chemist. I'm studying a property (CMC, critical micellar concentration) of a solution, and to treat data i trace a plot of concentration (m) Vs specific conductivity (k), like this. Then with a linear fit i get two straight lines, and the intersection is the CMC: The problem begins when the variation isn't relevant, like this: Now, i've found this paper where get a complicated data treatment and get a derivate-like result: Paper: http://www.scienceforums.net/forum/attachme…

What is the graphical interpretation of the mixed partials? The partial derivative is pretty straightforward graphically, but I don't know about the mixed. Is it just a measure of the magnitude of the slope of the plane tangent to the surface at a given point?

1. The problem statement, all variables and given/known data [math]\mbox{Check whether } \sum_{n=0}^\infty \frac {1}{e^{|x-n|}} \mbox{ is uniform convergent where its normaly convergent}[/math] 3. The attempt at a solution [math]\mbox{I choose } \epsilon = 1/2[/math] [math]a_n=\frac {1}{e^{|x-n|}}\ ,\ b_n= \frac {1}{n^2}[/math] [math]\lim_{n\rightarrow\infty} \frac {a_n}{b_n}=0\ \Rightarrow\ \sum_{n=0}^\infty a_n \mbox{ is convergent for all x }\in\ R. [/math] [math]f_k(x)=\sum_{n=0}^{k} a_n\ ,\ f(x)=\lim_{k\rightarrow\infty} \sum_{n=0}^{k} a_n[/math] [math] x_n=n\ \Rightarrow\ \sup_{x_n \in R}|f_k(x_n)-f(x_n)|\geq \sum_{n=k+1}^\infty \frac {1} {e…

1. The problem statement, all variables and given/known data Suppose: [math]f(x)\ and\ f'(x)\ are\ continuous\ for\ all\ x \in R [/math] [math]For\ all\ x \in R\ and\ for\ all\ n \in N[/math] [math]f_n(x)=n[f(x+\frac{1}{n})-f(x)][/math] [math]Prove\ that\ when\ a,b\ are\ arbitrary,\ f_n(x)\ is\ uniform\ convergent\ in\ [a,b][/math] 3. The attempt at a solution [math]\lim_{n\rightarrow \infty} f_n(x)=\lim_{n\rightarrow \infty} n[f(x+\frac{1}{n})-f(x)] = \lim_{t\rightarrow0} \frac {f(x+t)-f(x)}{t}=f'(x)[/math] [math]\max_{[a,b]}|n[ f(x+\frac{1}{n}) -f(x)]-f'(x)|=|n[ f(x_0+\frac{1}{n}) -f(x_0)]-f'(x_0)|=[/math][math]|\frac {f(x_0+t)-f(x…

So, I basically blew off the last couple weeks of calc 2 because they dropped the lowest test grade so I could skip the final test. Thus, I missed the last half of infinite series. Will I need to teach it to myself for Calc 3 over the summer? "Geometry and vectors of n-dimensional space; Green's theorem, Gauss theorem, Stokes theorem; multidimensional differentiation and integration; application to 2- and 3-D space." There is the course description.

f(x + x[math]_{0}[/math])=[math]\sum[/math]1/n! x[math]_{0}[/math]^n(d/dx)^n f(x) i saw this at my book but didnt understand .

Hi everyone There two problems that bother me for quite some time. 1)If [math]\lim_{x\to\infty}\frac{f(x+1)}{f(x)}=2[/math] then find [math]\lim_{x\to\infty}\frac{f(x+2)}{f(x-2)}[/math]. 2) I do not know if this the right place to ask for this problem but here it goes. Can we prove the De Morgan law? Thank you

Concerning differentials, when is 1/dy/dx not equal to dx/dy? When can the derivative not be thought of as a ratio?

Prove using contradiction the following: [p<=>(q <=>r)] =>[(p<=>q)=> r]

In proving : [math] x+y\leq 2[/math] given that : [math]0\leq x\leq 1[/math] and y=1,the following proof is pursued: Since [math]0\leq x\leq 1[/math] ,then (x>0 or x=0)and (x<1 or x=1) ,which according to logic is equivalent to: (x>0 and x<1) or (x>0 and x=1) or (x=0 and x<1) or ( x=0 and x=1). And in examining the three cases ( except the 4th one : x=0 and x=1) we end up with : [math] x+y\leq 2[/math]. The question now is : how do we examine the 4th case so to end up with [math] x+y\leq 2[/math]??

In defining implication we say that: p => q is false, if p is true and q false, and is true for all other cases. The question is if the definition could be other wised. For example instead of F =>T defined as T ,why not defined as F

In a certain book i read and i quote : "As in mechanics this work is defined by the integral. W = [math]\int F.dl[/math]. where F is the component of the force acting in the direction of the displacement dl.In a differential form this equation is written. δW = F.dl. where δW represents a differential quantity of work" Is that correct??

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

The product rule: [math] y=u(x)\cdot v(x) [/math] [math] y = u\cdot v [/math] [math] y+dy=(u+du)\cdot(v+dv) [/math] [math] y+dy=uv+udv+vdu+dudv [/math] now [math]dudv[/math] is discarded on the grounds of being "too small". If I were to include it, later on (by subtracting y and dividing through by dx), it would become [math]\frac{dudv}{dx}[/math]. What does that mean?

If we are presented with the following arguments : 1) If 1<0 ,then 1-1< -1 => 0<-1. But since [math]\neg(1<0)[/math] ,then [math]\neg(0<-1)[/math]. 2) if 1>0 then 1+1>0+1 => 2>1 . And since 2>1 ,then 1>0 What true facts can we use so that we can decide whether the above arguments are valid or non valid??