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

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1answer
9 views

Average Value of a Surface

For a project I am working on I'm attempting to get the average height of a surface. That is, for a function $z = f(r, \theta)$ I would like to obtain the average z value within a specified radius and ...
0
votes
1answer
33 views

Maximum of $xy^3z^7$ in the plane $x+y+z=1$

A friend gave to me this problem and on having seen that I could not solve it in the first instance helped me with the hint of using the AM-GM inequality. PROBLEM.- To maximize the product $xy^3z^7$ ...
3
votes
1answer
33 views

Splitting up a double integral

I need to compute the following integral: $$ 2\pi\nu^2\int^a_be^{x^2}\int_{-\infty}^xerfcx(-y)dydx, $$ where $erfcx(x)=e^{x^2}erfc(x)$, $erfc(x)=1 - erf(x)$, and $erf(x)$ is the error function. The ...
3
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3answers
27 views

A question about the formulation of the definition of a limit for sequences

So I know the definition of a limit of a the sequence is: $a$ is a limit of a sequence $\{x_n\}$ if given $\epsilon>0$ there exists a positive integer $N$ such that $|x_n-a|<\epsilon$ for all ...
3
votes
0answers
42 views

Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$

(This is a slight variation of another question, already answered) Can we find a closed form of the following series? $$S=\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}\tag1$$ Using some non-rigorous ...
1
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1answer
32 views

how to show that $f_n \uparrow f$

How to show that $f_n \uparrow f$ where $$f_n(x)=\min\left(\frac{\lfloor 2^nf(x)\rfloor}{2^n},n\right)$$ It is clear to me that $f_n(x) \leq f(x)$ But how do I show that the limit is indeed $f$ ? ...
-3
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1answer
22 views

Find parametric equations? [on hold]

Find parametric equations A.) Part of line that goes through points $(2,5)$ and $(3,2)$ and $y∈[1,2]$. $\mathbf{r}(t)=\mathbf{a}+t(\mathbf{b}-\mathbf{a}),\;\; t\in\mathbb{R}$ B.) Intersection ...
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0answers
24 views

Find the maximum value of $\log_{10}(\frac {c_2}{x})\log_{10}(x+1-c_1)$, where $c_1 ,c_2$ are real constants and x is a real number,$x\in [c_1,c_2]$

What is the maximum value of: $$\log_{10}\left(\frac {c_2}{x}\right)\log_{10}(x+1-c_1)$$ where $c_1$ & $c_2$ are real constants and $x$ is a real number, $x\in [c_1,c_2]$. For which $x$ is this ...
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0answers
12 views

a problem involving binary entropy function

let $\alpha<1/2$ such that $2^{H(\alpha)}\le 2^{1-\epsilon}$,when $H$ is binary entropy function. how can i prove that then we have: $2^{n(1-\epsilon)}\ge \sum\limits_{i\le \alpha n } {n \choose ...
9
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3answers
154 views

If $u_{n+1}\le u_n+u_n^2$ and $\sum u_n$ converges, prove that $\lim\limits_{n\to +\infty}(n\cdot u_n)=0$

Given the positive sequence $\{u_n\},n\in \mathbb{N}$ that meets the conditions: $\boxed{1}$. $u_{n+1}\le u_n+u_n^2$ $\boxed{2}$. Exist the constant $\text{M} >0$ so that ...
-8
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1answer
45 views

Can somebody integrate this function for me? [on hold]

This is the function. $\frac{1}{6.08 \cdot \sqrt{2\pi}}\exp\left(-\frac{(x-10.75)^2}{2 \cdot 6.08^2}\right)$ Thanks in advance!
3
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4answers
83 views

How I can evaluate $\lim_{(x,y) \rightarrow (0,0)} xy(\frac{1+xy}{x^3+y^3})^{1/3}$

I don't have idea how I can evaluate this double limit $$\lim_{(x,y) \rightarrow (0,0)} xy \left(\frac{1+xy}{x^3+y^3} \right) ^{1/3}$$ could you help me please! I try prove that $f$ is continuous: ...
-4
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0answers
32 views

How to solve for x in x^x-c*x+c=0, where c is a constant [on hold]

How does one solve for $x$ in $x^x-c*x+c=0$, where c is a constant?
0
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2answers
54 views

What is the anti derivative of $ \frac{f(x)}{g(x)}$

I'm working on a formula just for fun and I need to know what is the antidervative of one function divided by another like $\displaystyle \frac{f(x)}{g(x)}$ And then specifically where $f(x) = |x|$ ...
9
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0answers
128 views

A very tough integral $\int_0^{\pi} \arctan^3\left(\frac{\sin (x)}{2 \sqrt{2}}\right)\csc ( x) \, dx$

My research shows that $$\int_0^{\pi} \arctan^3\left(\frac{\sin (x)}{2 \sqrt{2}}\right)\csc ( x) \, dx$$ $$=\frac{3}{16} \pi \sinh ^{-1}(1) \log ^2(2)-\frac{1}{96} 85 \pi \log ^3(2)+\frac{61}{16} ...
0
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0answers
17 views

Upper bound for incomlete Gamma function

It is well-known, that for real arguments $a \geq 0$ and $x \geq 0$ the upper incomplete Gamma function $$\Gamma(a,x) = \int_x^\infty e^{-t} t^{a-1} \, \mathrm{d} t$$ behaves for sufficiently large ...
0
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1answer
13 views

Find the distance such that the angle will be the gratest

Rectangle shaped screen in a cinema is 8m high. It is place on a wall in such a manner that the upper edge of the screen is 12m above the floor. Find the distance between the viewer and the wall where ...
0
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1answer
28 views

Finding the differential equation, given a solution

I am unable to understand how to find the differential equation when a general solution has been given. Here are a few example solutions, which require their differential equations to be found: (a) ...
0
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3answers
35 views

fundamental theorem of calculus 2 [on hold]

Differentiate the following equation with respect to $x$: $$8 + \int_a^x \frac{f(t)}{t^2}\, dt = 2 x^{1/2}$$ Hence, find a function $f(x)$ and real number $a$ such that the above equation is true ...
0
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1answer
29 views

Volume bounded by $y^2+z^2=x$ and $x=y$

I need general help in solving for the area bounded by $y^2+z^2=x$ and $x=y,\ z=0$. I'm trying to get the limits of integration for $\int \int \int dzdxdy $. Here's my attempt so far: $0\leq z\leq ...
0
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1answer
27 views

Second derivative with implicit differentiation

Question: Determine whether the given relation is an implicit solution to the give differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit ...
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0answers
12 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
0
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0answers
29 views

Kirschenhofer Ramanujan functional equations part I(alternative form) [duplicate]

Ramanujan analyzed $$\sum _{k=1}^{\infty } \frac{e^{-k x}}{e^{-2 k x}+1}=\sum _{k=1}^{\infty } \frac{\pi \operatorname{sech}\left(\frac{\pi ^2 k}{x}\right)}{2 x}+\frac{\pi }{4 x}-\frac{1}{4}$$ it ...
1
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1answer
12 views

Real analysis: Characteristic property for unconditional divergence

A convergent series $\sum_{k=1}^\infty a_k$ is called unconditional convergent, when it's value is invariant under any permutation $\sigma:\mathbb N\to\mathbb N$ of it's summands, i.e. ...
3
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1answer
52 views

Kirschenhofer Ramanujan functional equations

Ramanujan analyzed $$\sum _{k=1}^{\infty } \frac{e^{-k x}}{e^{-2 k x}+1}=\sum _{k=1}^{\infty } \frac{\pi \operatorname{sech}\left(\frac{\pi ^2 k}{x}\right)}{2 x}+\frac{\pi }{4 x}-\frac{1}{4}$$ it ...
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0answers
19 views

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? The function is defined as $$f(x) \colon= \sum_{x_n < ...
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0answers
36 views

Functional derivative or chain rule?

Just a quick question... I have two functions – $V(a,b,c)$ and $F(a,b,c)$ – and I wish to calculate the derivative of one with respect to another ($\frac{\partial V}{\partial F}$). Am I right in ...
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0answers
19 views

Find minimum distance between the plane and the beginning of Cartesian plane.

Find minimum distance between the plane: $S=\{\left(x,y,z\right) \in \mathbb{R}^3: x+yz=2012 \}$ and the beginning of Cartesian plane $(0,0,0)$. I want to minimize this with use of lagrange's ...
1
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2answers
43 views

How to prove $2\arccos(x)+\arccos(1-2x^2)=π$ on $x\in[0,1]$ from MVT

First what I did was use the cosine addition formula: $$2\arccos(x)+\arccos(1-2x^2)=π$$ $$\cos(2\arccos(x))=\cos(π-\arccos(1-2x^2))$$ $$2x^2-1=-(1-2x^2)$$ $$0=0$$ However, this is inconsistent with ...
0
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1answer
40 views

How do we calculate the upper sum and lower sum of an Integral?

How do we calculate the Upper and Lower Sum of an Integral? I am trying to calculate it to for : $$\int_1^2 (3-4x) dx$$ Is there a Formula?
2
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0answers
19 views

The remainder estimate for the integral test

The remainder estimate for the integral test states that if $a_k=f(k)$ where $f$ is a continuous, positive, and decreasing function on $[n,\infty)$ and $R_n=s-s_n$ (where $s_n$ is the $n$th partial ...
1
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1answer
31 views

Simple Harmonic Motion under Periodic disturbing force

A particle of mass $m$ is executing a SHM in a straight line under an acceleration $n^2 \times (distance)$. If a periodic force $mk \cos{pt}$ be introduced and the time period of forced vibration ...
1
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1answer
14 views

Paramaterization of paraboloid and plane.

Consider the paraboloid $z=x^2+y^2$. The plane $2x-4y+z-6=0$ cuts the paraboloid, its intersection being a curve. Find "the natural" parameterization of this curve. I have set each equation equal ...
0
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2answers
19 views

Find the point at which the line intersects the plane. Is the intersection perpendicular?

Find the point at which the line $$x = 1 - t \\ y = 3 + t \\ z = 7 + 2t \\$$ intersects the plane $$x + 2y + z = 20$$ Is the intersection perpendicular? I have found the point of intersection to be ...
5
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1answer
76 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
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0answers
11 views

Determine the number of saddle points under specified conditions

Suppose a function with two variables $f(x, y)$ is smooth enough everywhere. If it has a local minimum and a local maximum, can we say that there are at least two saddle points as well? If so, how can ...
2
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3answers
87 views

Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
1
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1answer
24 views

L'Hopital's rule and limiting variables

I'm working some problems from a calculus text and came across this question: If $f(x)$ is a function that's differentiable everywhere, what is the value of the limit $$\lim\limits_{h \to ...
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0answers
40 views

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. [on hold]

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. I'm stuck on how to do this problem. Any solutions ...
0
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1answer
27 views

Can't figure out this basic algebra

Been a while since I did math but I'm trying to understand how they got the final equation in this step: http://i.imgur.com/Y09bqwT.png When I solve for P I get this: $$ P(t) = ...
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0answers
30 views

What are the name of these signals

It might be funny but there are two signals which confuse me about how to call them. Signal1: http://s3.postimg.org/ffefhwqyr/Capture1.png Signal2: http://s3.postimg.org/samf4o683/Capture2.png I am ...
0
votes
2answers
31 views

How to solve an integral with the use of arcsine

The specific question is the following, $$\int_{-a}^x \sqrt{a^2-x^2}\,dx$$ We are also given that $0\le x\le a$ Thank you very much for helping.
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0answers
6 views

Find the value of X in this point slope equation based on r (no values given)

I've been working on a shrinking circle problem for my calculus class where two circles cross at a point. I haven't been given any values other than the center of the larger circle is at (0,0). The ...
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1answer
41 views

Show that a polar equation describes a circle

I want to prove that this polar equation: $$r^2 + 2r(\cos(\theta) - 3\sin(\theta)) = 4$$ describes a circle. I tried converting the equation into a cartesian equation and got $$r^2 + 2x - 6y = 4$$ ...
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2answers
37 views

Solving an equation with $\sin(x)$ in the exponent: $2^{\sin(x)} \cdot \cos(x) + 1 = 1$

Hi I need help with a trig problem: I have $2^{\sin(x)} \cdot \cos(x) + 1$, and I need this to equal $1$ between $x = -3$ and $3$. I keep going in circles with substitution, etc. Any help would be ...
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0answers
11 views

For single variable, is quasi convex function also a quasi concave function?

To be a quasi concave or quasi convex, function must be monotone. Meaning, function must either be increasing or constant. It seems like for single variable function, both of quasi concave and ...
0
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0answers
13 views

upper-band of the Integral expression

Consider below integral expression $$\int_{0}^{\infty}g(y)[\int_{a}^{\infty}(1-e^{-(k+y)x})f(x)dx ]dy \ \ \ \ (1)$$ Where, we know: $$f(x)>0\ ,\ \ a\leq x \leq \infty$$ $$\ k>0$$ $$g(y)>0\ ...
1
vote
1answer
28 views

Using L'Hospital's Rule on Parametrics [on hold]

Stuck on this problem... Let C be the curve given by the parametric equations $x = f(t)$, $y = g(t)$ and let $$\left(f\left(t_0\right), g\left(t_0\right)\right)$$ be a point on the curve. Let $m(t)$ ...
0
votes
3answers
27 views

Help with a derivative of integral please.

I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. ...
0
votes
1answer
46 views

Does the following limit always hold? [on hold]

Let $y$ be a function that is differentiable everywhere. Is it true that the limit as $k$ tends to $n$ of the $k$th derivative of $y$ is the $n$th derivative of $y$?