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

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3answers
21 views

Consider $F(x,y)=f(x+3y,2x-y)$…

If $f: \mathbb{R}^2\rightarrow\mathbb{R}$ where $F(x,y)=f(x+3y,2x-y)$ with $f$ is defferentiable and $\nabla f(0,0)=(4,-3)$ compute the derivate at the origin in the direction of unit vector ...
1
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0answers
13 views

Functional equations with involutions

Having seen that the topic of functional equations is loved by StackExchange, I have constructed this problem hoping that it will please readers. Solve the functional equation $$ ...
0
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1answer
19 views

Limit in electric field far from a charged disk

Let $\sigma$ be the areal charge density $\frac{Q}{\pi R^2}$ of a disk of radius $R$; then the electric field on the line pependicular to the disk and passing through is centre, if we use its ...
0
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1answer
15 views

Calculate the rate of change of the slope of a tangent line of a graph, given the equation, value of x, and rate of change of x.

Heree is the full question from Supp. problem 20.26, Beginning Calculus, Schaum's Outlines, 3rd. ed. An object is moving along the graph of y = 3x - x^2, and its x-coordinate is changing at the ...
2
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1answer
37 views

Derivative of $\frac{x}{\|x\|}$ w.r.t. x where $x\in \mathbb{R}$ ($x \neq \theta_n$)

I want to find the Hessian of a function. I have already computed the gradient of the function. So, I have to again differentiate it w.r.t. $x \in \mathbb{R}^n$ to get the hessian, but I am facing a ...
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0answers
18 views

The highest direction of the trace operator

Let $W$ be a real and symmetric matrix ${m \times m}$ from the set $\widetilde{W_m}$, and $T:\widetilde{W_m} \rightarrow \mathbb{R}$ a function defined by $T(W) = trace(W^3)$. We are interested to ...
0
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1answer
37 views

what's the greatest volume of a cylinder using calculus?

I have a rectangle that has the perimeter of 38cm. I need to make this rectangle into a baseless cylinder and find the greatest volume of it, by deriving. so far i came with this: for the rectangle ...
2
votes
1answer
25 views

How to determine the function from the following?

The graph of a certain function contains the point $ (0,2)$ and has the property that for each number 'p' the line tangent to $y = f(x)$ at $(p, f(p))$ intersect the x-axis at p + 2. Find $f(x)$ The ...
2
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1answer
32 views

$\lim_{x\to 0} \frac{f(x)}{x}=-1 \implies \lim_{x\to 2}\frac{f(x^2-4)}{x-2}=-4$.

I'm trying to prove if $\lim_{x\to 0} \frac{f(x)}{x}=-1$, then $\lim_{x\to 2}\frac{f(x^2-4)}{x-2}=-4$. I've tried everything, substitution, limit composition, etc. Anyone could help me to solve this ...
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0answers
10 views

For the classical diffusion equation ut = r (5ru) (in 3 space dimensions)

fi nd TWO changes of variables which changes the di ffusion constant from 5 to D = 1 for the new coordinate system?
4
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3answers
78 views

Why does WolframAlpha's expression for $\int\frac{dx}{x\sqrt{x^4-4}}$ disagree with my own?

$$\int\frac{1}{x\sqrt{x^4-4}}$$ My teacher gave us these notes and I'm unsure if they're correct. Wolfram gives a different answer, and when I derive I might have messed up. Thanks.
1
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1answer
18 views

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$.

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$. I am having difficulty in the problem. Please help.
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0answers
9 views

Inquiry about Second order condition for Lagrange Equations

Consider the programming problem: $ min\hspace 3mm x_1^2 + x_2^2$ subject to $ q = x_1+x_2$. The choice variables are $x_1$ and $x_2$. Establish that the second order necessary condition holds ...
1
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2answers
14 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
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1answer
42 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 $xy^3z^7$ ...
4
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1answer
39 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
28 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 ...
4
votes
0answers
57 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 ...
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1answer
35 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 $f$ ? ...
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1answer
25 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
31 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
15 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
179 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
49 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
88 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: ...
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0answers
35 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
57 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 $f(x) = |x|$ ...
0
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0answers
21 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
29 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
37 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
30 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
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 ...
0
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0answers
13 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
31 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
votes
1answer
53 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
25 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
41 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
20 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
45 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
45 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
32 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
15 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 ...
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votes
2answers
20 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
79 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
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0answers
14 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
91 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 ...