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

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0answers
24 views

Geometric Interpretation of Fractional Derivatives

I was looking for a geometrical interpretations of fractional derivatives and fractional integrals. I would be glad to see any kind of intuitive and preferably visual interpretation of the objects ...
-2
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0answers
25 views

Help with a Differential Equations problem. [on hold]

Having problems with this one, i just got into Diferentials Equations.
0
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1answer
74 views

Spivak's calculus: Chapter 7 problem 18 d)

In cases (a) and (c) [where it was proven that such a number exists for a continous $f$ on $\textbf{R}$], let $g(x)$ be the minimum distance from $(x,0)$ to a point on the graph $f$. Prove that ...
2
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3answers
51 views

How to differentiate $F(x,y)=\int_x^y \sqrt{e^{tx}+3y}dt$

I want to compute $D_1f$ and $D_2f$, two partial derivatives. The only tool I have now is the fundamental theorem of calculus and chain rule. Maybe I can write $F(x,y)$ as some composition functions ...
1
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1answer
36 views

domain of $\sqrt {\cos^{-1}(\cos x)-\lfloor x\rfloor} $

Here is my question where I got stucked. The domain of $\sqrt{\cos^{-1}(\cos x)-\lfloor x\rfloor} $ where $\lfloor \cdot\rfloor$ denotes the greatest integer function (floor function).
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5answers
49 views

How to prove $\displaystyle\lim_{x \to 0} \dfrac{\sin^{-1} x}{x} = 1$?

How to prove this? Is there any geometrical proof? I have proved , btw, $\displaystyle\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ by Sandwich Theorem and little geometry.
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0answers
10 views

Lagrange Multiplier, Boundary

In many cases when we have to optimize a function under a constraint, i.e $f(x,y)=e^{-xy}$ with constraint $x^2+4y^2 \le1$, Lagrange multipliers only help with finding the extreme values at the ...
3
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4answers
108 views

Find the limit $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$

Find the limit if it exists $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$$
1
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1answer
292 views

Volume of ellipsoid bounded by two planes.

I need to find the volume of ellipsoid: $$5x^2 + {y^2\over25} + {3z^2\over4} = 1$$ if the ellipsoid is bounded by $z={-1\over2}$ and $z=1$ planes. I was able to find the total volume of the ellipsoid ...
0
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0answers
64 views

Find the upper and lower sum of an integral with a floor

I'm having some trouble and looking for some help with a problem i'm trying to solve. Without the floor function it would be easy but the floor has made it a bit trickier: Find the upper and lower ...
0
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1answer
7 views

error in approximation a monotonic function in L^1

I am trying to solve this problem, but I am getting an incorrect solution. Here is the problem and my approach: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize ...
2
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1answer
26 views

Changing order of summation including a min in the summation

Lets say I have the following expression: $$ h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x) $$ Now my goal is to have the $v$ ...
16
votes
3answers
132 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
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2answers
33 views

Is $L=\sin^2(t) \frac{d}{dt}$ a linear differential operator?

Consider the differential operator $$L=\sin^2(t) \frac{d}{dt}$$ If it acts on the sum of two functions, $y_1(t)$ and $y_2(t)$, you get $$\begin{align*} L(y_1(t)+y_2(t))&=\sin^2(t) ...
0
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0answers
15 views

How this result in archived in Fourier series

I was reading some notes about functions of symmetry in Fourier series and came across the following result for a function with symmetry of an odd quarter wave $$\begin{align} ...
0
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1answer
13 views

On the finite limit of a product of real functions, one of then being unbounded

Given functions $f,g:\mathbb{R}\to \mathbb{R}$ such that $\lim_{x\to\infty} f(x).g(x) = 1$ and $\lim_{x\to\infty} g(x) = \infty$, is it true that $\lim_{x\to\infty} f(x) = 0$ by force? If so, how to ...
3
votes
3answers
961 views

General term of Taylor Series of $\sin x$ centered at $\pi/4$

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
2
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2answers
851 views

Concentration of a drug after repeated injections

After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ ...
2
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2answers
57 views

Prove with $\epsilon$ and $\delta$ that $\lim_ {x\to 0} |x|=0$

Prove with $\epsilon$ and $\delta$ that $\lim_{x\to 0} |x|=0$. I just have to find the $\delta$ that works with $\left||x| -0\right|< \epsilon$. Following the absolute value definition I found ...
0
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0answers
33 views

What is the exact solution to this PDE?

I'm in a numerical methods class for my senior year of college, and it's been about 3 years since I took diff eq. We have a problem in which we are using numerical methods to approximate the solution ...
1
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4answers
70 views

For what $a$ is $\lim_{x\to\infty}(\frac{x+a}{x-a})^x=e$?

I am trying to figure out for which $a$ the following equation is true: $$\lim_{x\to\infty}\left(\frac{x+a}{x-a}\right)^x=e$$ It seems like an application of L'Hospital could work perhaps, but I'm ...
1
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2answers
7k views

Derivative of cross-product of two vectors

In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?
1
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3answers
39 views

Find ${\rm d}y/{\rm d}x$ and simplify as much as possible, $y=x/(2x+5)^3$

My answer is $-6x(2x+5)^2+(2x+5)^{-3}$. I am wondering if my answer is right and whether I am able to simplify more?
0
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0answers
8 views

Explanation for differential element relation

Given $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, my textbook makes the substitutions to write $\int \exp(-\xi^2 /2) \dfrac{\partial^2}{\partial x^2}\exp(-\xi^2 /2)dx= \sqrt{\frac{m\omega}{\hbar}} \int ...
2
votes
2answers
55 views

Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$

Evaluation of Integral $\displaystyle \int\frac{1}{x^2\left(x^4+1\right)^{\frac{3}{4}}}dx$ $\bf{My\; Try::}$ Let $\displaystyle x = \frac{1}{t}\;,$ Then $\displaystyle dx = -\frac{1}{t^2}dt\;,$ ...
0
votes
1answer
1k views

Proving that the limit of 1/x as x approaches negative infinity equals 0

I am trying to prove that the limit of $1/x$ as $x \to -\infty$ equals $ 0$. I get stuck in trying to find a proper epsilon. I know that it is supposed to be $-1/\epsilon$ but I don't understand how ...
-2
votes
1answer
100 views

Calculate the upper sums Un and lower sums Ln, on a regular partition of the intervals, for this integral:

sorry new to this site. Can someone please help me with this? I have tried for such a long time and have yielded no correct answers. $$\int_1^7 (3−5x)dx$$ We have $n$ rectangles, so what I did first ...
0
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2answers
30 views

How can I find the radius and interval of convergece of $\sum_{n=0}^\infty {(x+5)^n} $, and for what value of x does the series converge?

$$\sum_{n=0}^\infty {(x+5)^n} $$ We talked about this briefly but I'm still pretty confused about how to start this problem.
1
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2answers
35 views

Prove that $F=\left(-x^2+2\right)\cdot \cos\left(x\right)+2x\sin\left(x\right)$ don't have limit

We have $f\colon\mathbb{R}\rightarrow \mathbb{R}$, $f\left(x\right)=x^2\cdot \sin\left(x\right)$ and $F$ its primitive. We have to prove that $F$ doesn't have a limit at $\infty $. What I can say ...
1
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2answers
29 views

How can I make a series expansion of $F(x) = \int_0^x \exp -{(t^2)}\ dt$?

$$F(x) = \int_0^x \exp -{(t^2)}\ dt$$ We need to find the series expansion for $F(x)$. I tried differentiating $F(x)$ but couldn't establish certain pattern so that Taylor series formation may help.. ...
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0answers
34 views

How can I evaluate: $\lim_{n\to\infty}\frac{n}{n!^{1\over{n}}}$? [on hold]

What is the value of this limit. $$\lim_{n\to\infty}\frac{n}{n!^{1\over{n}}}$$
-1
votes
1answer
72 views

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ [on hold]

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ and negative for $|x|<1$ ? I don't understand .. I can't see.. damn it!
0
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1answer
22 views

Why is just $0$ extreme point? v22

We have $f:R\rightarrow R,\:f\left(x\right)=x^3-3x+2$ and we need to find extreme points for $g:R\rightarrow R\:,\:g\left(x\right)=\int _0^{x^2}\:f\left(t\right)e^tdt$. Here is all my steps: ...
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0answers
32 views

What radius circle to remove from unit circle to make golden earring?

A circular lamina of radius $x$ is removed from a circular lamina of radius $1$. If the center of gravity is at the edge of the smaller circle (along the line connecting the two centers) what is $x$? ...
-3
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3answers
33 views

A problem of Schwarz derivative [on hold]

I need help with the following problem analysis: Suppose $f$ is defined on an interval around $x$. The limit $$\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2},$$ if it exists, is called the Schwarz ...
2
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1answer
26 views

How can I find monotonicity intervals? v18

We have $F:\mathbb{R}\rightarrow \mathbb{R}$, $F(x)=x\int _0^x (1+\cos(t)) \, dt$ and we neeed to find monotonicity intervals and I don't know how... Here is what I try to do: $$F'(x)=\int _0^x ...
0
votes
1answer
17 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
0
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0answers
15 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
2
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2answers
28 views

A function that integrates to zero against a sequence of weights

Fix any $a\in(0,1)$. Is there a nontrivial continuous function $f:[a,1]\to\mathbb R$ so that $$ \int_a^1t^{-2n}f(t)dt=0 $$ for all integers $n\geq0$ and $f(a)=f(1)=0$? I would prefer explicit ...
1
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1answer
30 views

Proving a sequence is convergent and calculating its limit

In my assignment I have to solve the following question. I think I have an idea how to solve it, but I suspect there is a little thing in my solution which is wrong. If you can tell if my solution is ...
2
votes
2answers
50 views

Solve the complex equations

I have a question from complex calculus. How to solve this two equations: a) $$ \sin(z)=2015 $$ I know that $\sin(z)$ equals to $$ \frac{e^{iz}-e^{-iz}}{2i} $$ And i don't know whats next. b) $$ ...
2
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1answer
29 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
3
votes
2answers
146 views

Integration $\int \left(x-\frac{1}{2x} \right)^2\,dx $

$$\int\!\left(x-\frac{1}{2x} \right)^2\,dx $$ From U-substitution, I got $u=x-\frac{1}{2x},\quad \dfrac{du}{dx} =1+ \frac{1}{2x^2}$ , and $dx= 1+2x^2 du$ and in the end I come up with the answer to ...
2
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2answers
32 views

Convergence: infinite series

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}, n\geq\text{some integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
0
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0answers
15 views

Calculus 1- Derivatives: What Overall Dimmensions Will minimize the Amount of Paper Used? [on hold]

You are designing a poster with 50 sq.in. of printing, a 4in. margin at the top and bottom, and a 2 in. margin on each side. What overall dimensions will minimize the amount of paper used? Explain ...
2
votes
3answers
70 views

How to solve $\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$?

$$\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$$ I tried to do it by substitution with no sucess. Anyone can solve it?
0
votes
2answers
61 views

Derivative of a Matrix w.r.t. a Matrix

I have a matrix product with $\mathbf{X} \in \mathbb{R}^{m\times n}$ as $\mathbf{F(X)} = \mathbf{XAA}^T$ where $\mathbf{A}$ is a constant matrix w.r.t. $\mathbf{X}$. I see that I can write the ...
0
votes
2answers
43 views

Why Riemann sum is convergent? [on hold]

Why $\frac{1}{n}\sum _{k=1}^nf\left(\frac{k}{n}\right)$ is convergent? I don't understand how we can prove that is bounded and monotone... For instance: $f:R\rightarrow R,\:\:f=\frac{1+x}{1+x^2}$, ...
0
votes
0answers
14 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula [on hold]

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
0
votes
4answers
47 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...