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

learn more… | top users | synonyms

3
votes
1answer
18 views

integral of logarithm and rational function

i'm wondering how can i evaluate this integral using real methods: \begin{equation*} \int_{0}^{\infty}\frac{\log x}{1+x^{2}}dx. \end{equation*} I tried using mclaurin series of $\log x$ but really ...
1
vote
1answer
14 views

Find directional derivative - simple

The directional derivative of $f(x,y)$ at $(1,2)$ in the direction of $\vec a =\vec i + \vec j$ is $2\sqrt{2}$. We also know that the directional derivative of $f(x,y)$ at $(1,2)$ in the direction of ...
0
votes
1answer
27 views

Area between two curves in terms of x

I am given two equations and a graph. The equations are $$x=-y^3+4y+9$$ $$x=y^2-5y$$ The problem shows a graph with a shaded region, and I am only to find the area above $y=-1$. I want to set up ...
-3
votes
0answers
12 views

thumbnails list index out of range? [on hold]

Hello I need help when i put picture to my project in django 17.4 with aplication thumbnails i see this error my models is image = ImageWithThumbsField(upload_to='static', sizes=((100,150),)) and my ...
1
vote
6answers
54 views

How do I find the limit of this function as $x\to1^-$?

I need to find $$\lim_{x\to1^-}\frac{\sqrt{1-x^2}}{\sqrt{1-x^3}}.$$ I tried l'Hôpitals, however it seems like no matter how many times you differentiate, it will still be in the indeterminate form. Is ...
0
votes
2answers
76 views

To prove that no such function can be continuous.

Suppose $f: [a,b] \to R$ is two to one. that is, for each $y$ in $R$, $f^{-1}({y})$ is empty or contains exactly two points. How to prove that no such function can be continuous.
3
votes
4answers
50 views

How often is $x\to\infty$ used to denote ($x\to +\infty$ or $x\to -\infty$)?

How often is $x\to\infty$ used to denote ($x\to +\infty$ or $x\to -\infty$)? Both my textbook and my teacher use $x\to\infty$ as above, so e.g. it's false for us that ...
3
votes
5answers
68 views

Why $\int_0^af(x)dx=\int_0^af(a-x)dx$?

Is it true that: \begin{equation*} \int_0^af(x)dx=\int_0^af(a-x)dx? \end{equation*} Because I know that: \begin{equation*} ...
1
vote
4answers
62 views

Solve $\lim_{x\to\infty}\frac{\sqrt{x-1} - \sqrt{x-2}}{\sqrt{x-2} - \sqrt{x-3}}$

I'm trying to solve this limit $$\lim_{x\to\infty}\frac{\sqrt{x-1} - \sqrt{x-2}}{\sqrt{x-2} - \sqrt{x-3}}.$$ I've tried to rationalize the denominator but this is what I've got ...
2
votes
1answer
28 views

Fourier transform of Gaussian?

For the Fourier transform defined as $$\frac {1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\alpha x}\,dx$$ I know there is simple formula for the Fourier transformation and inverse transformation ...
2
votes
0answers
24 views

detemine a limit involving log function

I just came up with the following limit $$ \lim_{n\to \infty} \frac{\displaystyle\log_a\left(\sum_{\substack{k\in \mathbb{N}\\k\leq n~(1-\frac{1}{a})}}\binom{n}{k} (a-1)^k\right)}{n} \qquad \text{ ...
-1
votes
1answer
56 views

How to find $\int_{- \infty}^{x} e^x dx$ step by step ??

this is a improper integral that i need in probability but i can't seem to find a way to find it, any help ? $$\int_{- \infty}^{x} \lambda e^{-\lambda x} dx\ \ ; x >0; \lambda > 0;$$ Is there ...
1
vote
2answers
50 views

Proving $\frac{x}{x-\lfloor \sin x \rfloor}$ has no limit as $x\to 0$ using the definition of limit

I need to show that $$\lim_{x\to 0} \left(\frac{x}{x-\lfloor \sin x \rfloor} \right )$$ doesn't exist using the definition of limit (its negation). I was asked to formulate it in terms of ...
1
vote
1answer
60 views

Calculate the Taylor series of $f(x) =\ln( 1 -x +x^2) $ and the domain of convergence

I just stuck at the following exercise: Show that the function f has a Taylor series and calculate it, with $x_0 = 0$. $$ f(x) = \ln{(1-x+x^2)}$$ Because I already know the Taylor series from ...
1
vote
0answers
47 views

$f(x) = x^3$ is a smooth bijective map of manifolds. [on hold]

Show that $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$ is a smooth bijective map of manifolds.
4
votes
0answers
52 views

Calculating $\int_0^{\infty } \frac{\log (v+1)}{\sqrt{(v+1)^2+1} \sqrt{(v+1)^2+4 \sqrt{(v+1)^2+1} (v+1)+4}} \, dv$

What tools would you recommend me for this one? $$\int_0^{\infty } \frac{\log (v+1)}{\sqrt{(v+1)^2+1} \sqrt{(v+1)^2+4 \sqrt{(v+1)^2+1} (v+1)+4}} \, dv$$ It's related to Calculate in closed form ...
4
votes
2answers
45 views

compute improper integrals using integration by parts

Compute \begin{equation*} \int_0^\infty \frac{\sin^4(x)}{x^2}~dx\text{ and }\int_0^\infty \frac{\sin (ax) \cos (bx)}{x}~dx. \end{equation*} For the first integral I tried letting $u = \sin ^4 x$ ...
1
vote
5answers
46 views

How can I find the indefinite integral of $\int\sin^3x \cos^3x dx $?

$$\int\sin^3x \cos^3x dx $$ I'm not sure if I started this right but I broke the terms up like this: $$\int\sin^2x sinx \cos^2x cosx $$ Edit: I got $${1 \over 4}\cos^4x + {1 \over 6}\cos^6x + C $$ ...
-3
votes
1answer
24 views

How much should Carl put in a savings account to have 3000 after 2 years [on hold]

What is Carl's initial deposite to the account that pays 3.2% interest compounded quarterly to have $3000 in 2 years?
0
votes
0answers
18 views

Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
0
votes
0answers
20 views

derivative of a scalar wrt matrix

Let $y = \|A^T\mathbf{x} + \mathbf{b}\|_2^2$ where A is a matrix of size $d \times D$, $\mathbf{x}$ and $\mathbf{b}$ are $d\times 1$ vectors. What is the derivative of y wrt A? Is it ...
2
votes
2answers
38 views

Solve differential equation $y' = |1.1 - y| + 1$

How can the following differential equation be solved analytically? \begin{equation*} y' = |1.1 - y| + 1, \\ y(0) = 1. \end{equation*} I guess one must rewrite the differential equation piecewise ...
0
votes
1answer
17 views

A question about Darboux sums.

Let $f$ be a bounded real function defined on a bounded, closed set $[a,b]$. Let $P=\{x_0,x_1,\ldots,x_n\}$ be a partition of $[a,b]$. For $1\leqslant i\leqslant n$, we define $$m_i=\inf\{f(x)\mid ...
2
votes
1answer
49 views

Question regarding Fourier Series

Things I understand (scroll down to see question in bold): Let $T$ be the function's period Let $w_0 = \frac{2π}{T}$ A function $x(t)$ can be written as the sum of its even and odd parts, that is ...
2
votes
1answer
37 views

What is connection here between $x$ and interval: $\sin^{-1}(2x(\sqrt{1-x^2})=2\sin^{-1}x$ .

For the expression \begin{equation*} \sin^{-1}(2x(\sqrt{1-x^2})=2 \sin^{-1}x,~x\in[\frac{-1}{\sqrt2}, \frac{1}{\sqrt2}], \end{equation*} I know there is a connection between the interval and the ...
-1
votes
2answers
32 views

Solve the integral $\int \frac{3\cos x+7\sin x}{5\sin x+2\cos x}dx$ [duplicate]

I think it is necessary to do the replacement. That's just what? $$\int \frac{3\cos x+7\sin x}{5\sin x+2\cos x} dx$$
1
vote
2answers
35 views

Optimization of $e^{x^2 + y}$ on $x+y \leq 2$

Let $f(x,y) = e^{x^2 + y}$ and $M = {(x,y): x+y \leq 2}$. A. $f(x,y)$ on M is bounded above and not bounded below B. $f(x,y)$ on M achieves global minimum(a). C. (0,0) is point of ...
0
votes
2answers
46 views

Solve the integral $\int \frac{xdx}{\sqrt{1+\sqrt[3]{x^2}}}$ [on hold]

Help me, please, again $$\int \frac{xdx}{\sqrt{1+\sqrt[3]{x^2}}}$$
1
vote
4answers
67 views

Evaluate $ \lim_{x\to 0} \frac{\tan(4x)}{\sin(7x)}$

Evaluate $$ \lim_{x\to 0} \ \frac{\tan(4x)}{\sin(7x)}$$ I am stuck after I convert tan(4x) into sin (4x) / cosine(4x)
0
votes
2answers
44 views

Limit (Calculus)

Can you help me to solve step by step this limit, please? (I've found out that its result is equal to 12) $$ \lim_{n\to\infty}\frac{n\cdot \sin\left(\frac{2}{n}\right)\cdot ...
3
votes
2answers
54 views

Evalutating $\lim_{x\to +\infty} \sqrt{x^2+4x+1} -x$

I'm looking to evaluate $$\lim_{x\to +\infty} \sqrt{x^2+4x+1} -x$$ The answer in the book is $2$. How do I simply evaluate this problem? I usually solve limits such as this with the short cut ...
4
votes
1answer
57 views

Finding this weird limit involving periodic functions with periods 5 and 10.

If $f(x)$ and $g(x)$ are two periodic functions with periods 5 and 10 respectively, such that: $$\lim_{x\to0}\frac{f(x)}x=\lim_{x\to0}\frac{g(x)}x=k;\quad k>0$$ then for $n\in\mathbb N$, the value ...
1
vote
2answers
27 views

How to deduce the derivative of a function from the formal definition of the derivative?

Define $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ by $$ f{x \choose y} = \left\{ \begin{align} \frac{xy^2}{\sqrt{x^2+y^2}} ,\,& {x \choose y} \ne \mathbf{0} \\ 0 ,\, & {x \choose y} = ...
2
votes
2answers
62 views

Solve the integral $\int \frac{dx}{\:\sqrt[4]{\left(x+2\right)^5\cdot \left(x-1\right)^3}}$ [on hold]

Here is a indefinite integral must be solved. Help, who knows. Although it would be like casual. $$\int \frac{dx}{\:\sqrt[4]{\left(x+2\right)^5\cdot \left(x-1\right)^3}}$$
0
votes
1answer
18 views

Find the coordinates of the point(s) at which the tangent to the curve is parallel to the x-axis.

Find the coordinates of the point(s) at which the tangent to the curve \begin{equation*} y=3x^2+5x-7 \end{equation*} is parallel to the $x-$axis. I don't know how to solve this.. help!
0
votes
0answers
27 views

Orthogonal polynomials

I was put on hold 2 times already for this question, I don't know how to solve it (If i knew how to solve it I wouldn't be bothering you ) and I don't know why it doesn't fit the rules of this site or ...
2
votes
1answer
47 views

Proving $\lim_{x\to\infty} \frac{x}{x+\sin{x}}=1$ using limit definition

So I need to prove that $\lim_{x\to\infty} \frac{x}{x+\sin{x}}=1$ using the limit definition. Given $\varepsilon>0$ I'm trying to find an $M$ such that for all $x>M$, $$\left | ...
0
votes
2answers
39 views

Image under Möbius transformation

I would like to find the image of $$ {z \in C: |z|<1, Im{z}>0 } $$ under the complex map $$ w(z) = \frac{2z-i}{iz+2} $$. Well, since $w(2i)=\infty$ the interval $[-1,1]$ and $ {z \in C: ...
0
votes
3answers
28 views

convergence, finding limit

I just came across an exercise, however I don't know how to find the limit of $$\lim_{n \to \infty} \frac{2^n}{n!}$$ can any body help? Of course this is not homework, I'm only trying out example ...
2
votes
1answer
49 views

Compute $\lim\limits_{n\to \infty}{\prod\limits_{k=1}^{n}a_k}$ Where $a_k=\sum_{m=1}^{k} \frac{1}{\sqrt{k^2+m}}$

I showed that: $$\frac{1}{\sqrt{1+{1 \over {k}}}} \leq a_k \leq \frac{1}{\sqrt{1+{1 \over {k^2}}}}$$ And then $$\lim\limits_{n\to \infty}{\prod\limits_{k=1}^{n}a_k} \leq \lim\limits_{n\to ...
0
votes
3answers
55 views

$\underset{x\rightarrow\infty}{\lim}x-x^{2}\ln\left(1+\frac{1}{x}\right)$

Give me please a hint, how to find $$\underset{x\rightarrow\infty}{\lim}x-x^{2}\ln\left(1+\frac{1}{x}\right)$$ I tried to use substitution $h = \frac{1}{x}$ and, then, apply L'Hopital's rule, but ...
0
votes
2answers
56 views

Integration of $\int\frac{x^{4}}{x+1}dx$

Could you help me with the following integral: $$\int\frac{x^{4}}{x+1}dx$$
1
vote
1answer
31 views

Statement is true for any $a,b,c$?

Let us have $a,b,c$ arbitrary positive numbers. Prove: $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \ge 3$ I have tried many things, but none of them seemed to work. Any ideas?(If duplicate, I am sorry, ...
2
votes
1answer
35 views

Divergence of $\int_{0}^{\pi/2}\frac{3\sin x\cos x}{x^{2}-3x+2}dx$

How can I make sure that $$\int_{0}^{\pi/2}\frac{3\sin x\cos x}{x^{2}-3x+2}dx$$ diverges? I see that the integrand has a discontinuity at $x=1$ (a root of polynomial in the denominator). But may be ...
2
votes
1answer
23 views

Prove or disprove: $p(x)$ diverges to infinity for $a_{n}>0$ [on hold]

Prove or disprove that for any $n$ degree polynomial, $p(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{1}x+a_{0}$, if $a_{n}>0$, then $p(x)$ diverges to infinity as x tends to infinity. This is not homework.
0
votes
2answers
51 views

Prove if f is continuous at x=a, f'(a) does not necessarily exist

I can come up with an example that the right-hand limit does not equal to the left-hand limit, for example: $f(x)=\begin{cases} 2 &\text{ for } x\le 0 \\ 4 & \text{ for } x\ge 0\end{cases}$. ...
0
votes
1answer
26 views

“$\frac{dy}{dx}= -0.11$. It is easy to see that except at one particular point, dy will be of a different size from dx”.What point?

From "Calculus made easy" by Thompson. A ladder of fixed length is against a horizontal wall.Height of ladder is y,distance of base of ladder from wall is x.For positive increment to x, there will be ...
-4
votes
1answer
25 views

calculus II with physics application [on hold]

can someone help me with question 32? It has a physics application and I am completely stuck The question is screenshoted below http://imgur.com/mXldL7b
1
vote
0answers
24 views

Can't find solution for equation

$100z=a(z+i)(z-i)^2(z-2)+b(z-i)^2(z-2)+c(z+i)^2(z-i)(z-2)+d(z+i)^2(z-2)+e(z+i)^2(z-i)^2$ I already got $b=5+10i, d=-5+10i $ and $e =8$ by eliminating the factors using $z=i, z=-i, z=-2$ but I cant ...
2
votes
1answer
21 views

PDE Manipulation - Calculus

I need help for this question, its a lot of calculus but I'm confuse. let $$ u= \dfrac{(x-b)^{2}+y^{2}-q^{2}}{(x-b-1)^{2}+y^{2}-q^{2}-1} $$ I need show that $$ u_{x}^{2}+u_y^{2}= ...