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

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3
votes
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
40 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
vote
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
votes
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 ...
0
votes
0answers
23 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 ...
0
votes
0answers
11 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
votes
3answers
134 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
votes
1answer
44 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
votes
4answers
80 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
votes
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
votes
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
votes
0answers
126 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
votes
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
votes
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
votes
1answer
27 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
votes
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
votes
1answer
28 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
votes
1answer
25 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 ...
0
votes
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
votes
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
vote
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
votes
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 ...
0
votes
0answers
12 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? I'm having trouble trying to visualise what such a ...
1
vote
0answers
34 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 ...
-1
votes
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
vote
2answers
42 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
votes
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
votes
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
vote
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
vote
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
votes
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
votes
1answer
75 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 ...
1
vote
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
votes
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
vote
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 ...
0
votes
0answers
39 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
votes
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) = ...
0
votes
0answers
29 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
30 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.
0
votes
0answers
5 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 ...
1
vote
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 $$r^2 + 2x - 6y = 4$$ ...
0
votes
2answers
35 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 ...
0
votes
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
votes
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 $y$?
0
votes
0answers
24 views

Why don't we check the exactness of differential equation with Inspection cases?

When solving the differential equations which are reducible to exact differential equations using Inspection cases for example: Solve: $2xy^2 + ye^xdx = e^xdy$ The integrating factor would $1/y^2$ ...
-3
votes
1answer
23 views

How do you solve these for Intersection Points [on hold]

How do you get the intersection points between these algebraically? $$\sqrt{4-y} = (y-2)^2+2$$
-1
votes
0answers
15 views

Is the following derivative also differentiable with respect to $n$?

Let $f(x)$ be the $n-th$ derivative with respect to $x$ of $x \exp (n-1) log (x-1)$ evaluated at $x=1$. Is $f(x)$ differentiable with respect to $n$ ?