For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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2
votes
0answers
31 views

Composing a smooth even function and square root

Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $f(-x)=f(x)$ for all x. Define $g:[0,\infty)\to\mathbb{R}$ by $g(x)=f(\sqrt{x})$. Is $g$ necessarily smooth at $0$? I guess the answer is ...
1
vote
5answers
112 views

Limit of factorial function: $\lim\limits_{n\to\infty}\frac{n^n}{n!}.$ [duplicate]

I am studying for a test and I am given this problem: $$\lim_{n\to\infty}\frac{n^n}{n!}.$$ How do I go about solving this limit? Intuitively I see how the numerator is growing much faster, but how ...
0
votes
1answer
68 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
0
votes
1answer
44 views

Chain rule application in fundamental Theorem of Calculus

I have attached a question that I came across in trying to understand the fundamental theorem of calculus. The solution (highlighted with an arrow). I have difficulty understanding why the chain rule ...
0
votes
1answer
24 views

How to get polarised electromagnetic TE wave differential equation from Maxwell's Equations?

I wish to understand how the following equation: $\frac{\partial^2 E_x}{\partial y^2} + \frac{\partial^2 E_x}{\partial z^2} + n^2 k_0 E_x = \frac{\text{d} (\ln \mu)}{\text{d}z}\frac{\partial ...
-1
votes
1answer
39 views

Find the volume generated! [closed]

Find the volume generated by revolving the curve bound by $y=4-x^2$, the $y$-axis and $x$-axis about the $x$-axis using the disk method.
2
votes
2answers
66 views

How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell

Solving a diffusion-type ODE across a spherical shell, the equation is: $$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$ with boundary conditions $f(r_1)=f_1$ and $f(r_2)=f_2$. The solution is: ...
3
votes
1answer
45 views

Double integral proof, where is my mistake?

The bounds are 0 < x < b , 0 < y < b. $$ \int_0^b \int_0^b e^{-(x^{2}+y^{2})} dxdy $$ Since it is a square, x=y so we can write: = $$ (\int_0^b e^{-(x^{2}+x^{2})} )^{2} dxdy $$ = $$ ...
0
votes
1answer
33 views

Total derivative proof [closed]

The wikipedia article does not prove it http://en.wikipedia.org/wiki/Total_derivative Neither the top articles in google search. Could somebody help me proving it? I've found this: ...
3
votes
3answers
283 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
3
votes
0answers
98 views

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx$

Please help me to evaluate this integral: $$I={\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx.\tag1$$ Mathematica could not evaluate it in a closed form. A numerical ...
3
votes
2answers
74 views

Evaluate $\lim_{x \to 1} \frac{\sqrt[3]{x} -1}{\sqrt{x} -1}$

Evaluate $\lim_{x \to 1} \frac{\sqrt[3]{x} -1}{\sqrt{x} -1}$ I want to solve this limit by employing the strategy of introducing a new variable $t$ in such a way as to make the problem simpler. I've ...
2
votes
2answers
132 views

Asymptotic Behaviour Of A Bizarre Function

It is relatively easy to show that the asymptotic behaviour of $f(x)$, where $$ f(x)= \left[\frac{x}{2}\right] + \left[\frac{x}{4}\right] + \left[\frac{x}{8}\right] + \left[\frac{x}{16}\right] + ...
2
votes
1answer
39 views

Uniform convergence of telescoping series

find if $$\sum_{n=1}^\infty \left(nxe^{-nx^2} - (n-1)xe^{-(n-1)x^2} \right)$$ uniformly converges in [0,1]. Let $$a_n = nxe^{-nx^2}$$ and let $S_k(x)$ be the partial sum such that $$S_k(x) = ...
0
votes
2answers
31 views

Valid way of evaluating limits?

Calculate the following limits $$\lim_{x \to 0} \frac{e^{\sin x} - \sin^2x -1}{x},\,\,\,\,\,\,\,\, \lim_{x\to0} \frac{\sin x \cos x - x}{x^2 e^x}.$$ I've evaluated these using the asymptotic ...
0
votes
1answer
31 views

Simpler way of proving series convergence?

Determine whether the following series converges $$\sum_{n=1}^{\infty} \left (\frac{n^4}{n^4 + 2}\right)^{n^5-3}.$$ I've found convergence using the root criterion in the following way. $\sqrt[n]{ ...
1
vote
0answers
48 views

How prove this $\sum_{cyc}\frac{x+y-2z}{(x+y)^2+z^2}=0$

let $x,y,z\in R$, show that $$\dfrac{x+y-2z}{(x+y)^2+z^2}+\dfrac{y+z-2x}{(y+z)^2+x^2}+\dfrac{z+x-2y}{(z+x)^2+y^2}=0$$ My idea: let $$x-y=a,y-z=b,z-x=c,a+b+c=0$$ maybe this is very easy,and I think ...
0
votes
1answer
56 views

Just calculus, the integral = 0 and the argument inside integral = 0?

It is really hard for me to make a title to describe my question. Below is my question: Suppose $f(y-x)$ is a known Gaussian function defined as $$ f(y-x) = \frac{1}{\sqrt{2\pi}} \exp ...
7
votes
1answer
520 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
5
votes
2answers
107 views

A series $\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\frac{(i-1)! (j-1)!}{(i+j)!}H_{i+j}$ and $\zeta(3)$

We have $$ \sum_{j=1}^{\infty}\sum_{i=1}^{\infty} \displaystyle \frac{(i-1)! (j-1)!}{(i+j)!} H_{i+j} =\displaystyle 3 \: \zeta(3) $$ where $\displaystyle H_{n}:=\sum_{1}^{n} \frac{1}{k}$ are ...
-1
votes
2answers
93 views

Evaluate $\lim_{x\to 2}\frac{x^3 - 8}{x-2}$.

$$ \lim_{x \to 2}\frac{x^3 - 8}{x-2}$$ I solved it my answer is 8. But the solution says it is 12. How?
2
votes
3answers
93 views

Is not Wiktionary’s definition of “step function” incorrect?

Wiktionary says that a step function is, “A function from the real line to a finite subset of the real line”. I realize that there is some variation on how things are defined, but this seems too much, ...
2
votes
2answers
62 views

Differentiating the Taylor expansion of $e^x$

It is well known that a) $\frac{d}{dx}\exp x = \exp x$ and b) $\exp x = \sum\limits_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + ...$. Therefore, it should be possible to ...
9
votes
1answer
157 views

Evaluate $\int_{0}^{\large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\frac{\ln{(\sin x)}}{\cot x}+\frac{\ln {(\cos x)}}{\tan x}\right)dx$

How do I find the value of this integral? $$I=\int_{0}^{\Large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\dfrac{\ln{(\sin x)}}{\cot x}+\dfrac{\ln {(\cos x)}}{\tan x}\right)dx$$ I tried ...
1
vote
1answer
42 views

Evaluating $\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$

$$\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$$ My Attempt: For $\lim_{h\rightarrow 0}\frac{\sin^{8}(\pi/6+h)-\sin^{8}(\pi/6)}{h}=f'(x)=8\cdot ...
-1
votes
1answer
12 views

Equation of a line with a positive gradient [closed]

Two straight lines passing through the point (0,2) are tangent to the graph of the function y=1-x^2. Find the equation of the line with a positive gradient.
1
vote
0answers
24 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
6
votes
1answer
489 views

Is calculus not rigorous?

While studying single and multivariable calculus during my first year some people complained that calculus wasn't rigorous enough, when I asked about this no one seemed to be able to really specify ...
-4
votes
0answers
47 views

how do calculus this derivate? [closed]

how do calculus this derivative? $$f(x) = (2x-x^2)^{1/2}$$ And how do I calculus this derivative? $$F(x) = (\sin{x}/(1+\cos{x}))^2 $$
7
votes
1answer
196 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
1
vote
4answers
96 views

Evaluate the limit: $\lim_{x\to \infty}$

Evaluate the limit: $$\lim_{x\to\infty} \frac{(2x^2 +1)^2}{(x-1)^2(x^2+x)}$$ The answer is 4 and I don't understand why, but why can't I just do something like:$$\frac{(\infty)}{(\infty)(\infty)} = ...
-1
votes
2answers
46 views

What and how do derivate? [closed]

How do I derive this function? $f(x) = x(e^{-x^2})$ I need the first and second derivative.
0
votes
1answer
29 views

Can someone help me, I cannot find the partial derivatives of this

Need help find the partial derivatives...I keep keeping cos(4x-3y+z)
1
vote
1answer
72 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
0
votes
1answer
20 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
0
votes
2answers
40 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
2
votes
1answer
67 views

Evaluate the area of the region bounded by the ellipse, where is my mistake?

$ (10x^2+6xy+y^2=2)$ => $ ((x/\sqrt2)^{2} + ((3x+y)/\sqrt2))^{2} = 1 $ so if I change the variables to $u$ and $v$, $u = x/\sqrt2$ $v= (3x+y)/\sqrt2) $ Then my bounds of integration become $-1 ...
0
votes
0answers
16 views

How do you obtain the version of Simpson's rule required as well as deduce the composite integration rule? [closed]

Consider the function $$g(x)=f(a+(x−1)h)$$ and obtain a version of Simpson’s rule applicable to an integral $$\int_{a+h}^{a−h}f(x)dx.$$ Then deduce the composite integration rule ...
7
votes
4answers
157 views

Meaning behind differentials

So I think I understand what differentials are, but let me know if I'm wrong. So let's take $y=f(x)$ such that $f: [a,b] \subset \Bbb R \to \Bbb R$. Instead of defining the derivative of $f$ in ...
-3
votes
1answer
69 views

How to calculate these limits using L.Hopital Rule [closed]

How to evaluate the limits, using L'Hospital rule? $$\lim_{x \rightarrow 0} \frac{x-\sin x}{x^2}$$ $$\lim_{x \to 0^+}\frac{\ln(x^2 + 2x)}{\ln x} $$
1
vote
1answer
49 views

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin t}{t}J_m(t)\,dt$

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin(t)}{t}J_m(t)\,dt$ $$\int_R \frac{\sin(t)}{t}J_m(t)\,dt=\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt =\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt ...
4
votes
2answers
119 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
2
votes
0answers
46 views

Calculating work [closed]

How much work is required to lift a $1000\mathrm{kg}$ satellite from the surface of the earth to an altitude of $2\times10^6$ meters? The gravitational force is $F=\frac{GMm}{r^2}$, where $M$ is the ...
1
vote
1answer
41 views

Finding radius of convergence using root test

Find the radius of convergence of the following power series $$\sum_{n=1}^{\infty} \frac{2^n + 1}{n} x^n.$$ Using the ratio test, I have found that the radius of convergence is $R = \frac{1}{2}$. I ...
0
votes
1answer
49 views

Help with math steps, chain rule.

I'm trying to to understand the math steps to go from Eqn. (1) to Eqn. (2). $$\tag{1} q(x,t)=\frac{-V_t(1+\delta f(c,g))}{P(x,t)}\cdot \left(\frac{dP_o}{dt}\right)$$ $$\tag{2} \frac{-V_t ...
0
votes
1answer
37 views

How do you solve the second part of the question where i am required to derive Simpson’s integration rule?

When $v(x) = A + Bx + Cx(x − 1)$ show that $$\int_0^2v(x)dx= 2A + 2B + \frac23.$$ By choosing A,B and C so that $y = v(x)$ fits a given curve $y = g(x)$ at $x = 0$, $x = 1$ and $x = 2$ derive ...
1
vote
3answers
65 views

For What Values Of $x$ Is $f$ Continuous

For what values of $x\in\mathbb{R}$ is $f$ continuous? $f(x) = \left\{ \begin{array}{lr} 0 & \text{if}\, x \in \Bbb Q\\ 1 & \text{if}\, x \notin \Bbb Q \end{array} ...
1
vote
1answer
53 views

What function to use to get geometric mean in trapezoidal rule?

When deriving a trapezoidal rule an integral of $f(x)$ is switched to integral of new function $g(x)$ approximating the first one $$\int_a^b {f(x)dx}\approx \int_a^b {g(x)dx}$$ where $g(x)$ is a ...
4
votes
1answer
117 views

How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$

How do I find $\displaystyle\int\dfrac{dx}{1+x^8}$? My friend asked me to find $\displaystyle\int\dfrac{dx}{1+x^{2n}}$ for a positive integer $n$. But looking up I am getting pretty noisy answer for ...
4
votes
1answer
46 views

Existence of improper integral

Prove that $$\int_{0}^{\infty} \frac{(\arctan x)^2}{x^2} dx$$ converges. This is my attempt: The above integral is equal to $$\int_{1}^{\infty} \frac{(\arctan x)^2}{x^2} dx + \int_{0}^{1} ...