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

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0
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2 views

Why does max. increase have to be along the x,y,z axis in gradient?

$$\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})$$ These components are the rate of increase along the x,y and z directions respectively and ...
0
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0answers
20 views

HJM Model vs Leibniz integral rule

I state that I'm an electronic engineer (undergraduate), then the my knowledges about advanced mathematics are almost null. A colleague asked to me an help about one point of the proof of the theorem ...
0
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0answers
16 views

Solving a trigonometric equation

Let $n=3m$ and $k=3t$ be positive integers. Does the following equation have any solutions for $0 \leq j \leq n-1$ $$\cos \left ( \frac{2\pi j (k+1)}{n}\right )+\cos\left (\frac{2\pi j (k-1)}{n} ...
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0answers
18 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\ ...
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1answer
16 views

Which are the good books,resources,extensive question banks to learn real analysis,calculus

Which are the good books,resources,extensive question banks to learn real analysis,calculus(indefinite,definite,area under curves),differential equations for IIT plus plus level.Foreign authors are ...
0
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1answer
16 views

Can single variable function be represented by field?

Is field concept in mathematics directly related to multi variable functions? Can single variable function be represented by field?
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0answers
29 views

The highest direction of the trace operator

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

Does semi positive definite function imply quasi convexity? [on hold]

Alternatively does semi negative definite function imply quasi concavity?
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2answers
17 views

Parametric Representation for a Square with Side $1$ Centered at the Origin as a Function of the Angle Measured from the Positive $x$-Axis

While playing with some graphics progamming in OpenGL, I've encounterd this problem: Find the Parametric representation for a square with side $1$ centered at the origin as a function of the angle ...
2
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0answers
20 views

Coming up with a function or a single graph, given its characteristics (pre-calculus)

Give an example of a function or a single graph which has the following characteristics: Hole at $(3,-1)$ Domain is all real numbers except $3$ Local minimum at $(-1,-2)$ Local ...
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0answers
6k views

How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} ...
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2answers
48 views

Solve L'Hopitals problem

$$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x}{{\sec^2 3x}} $$ I used LH: $$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x \tan x}{6\sec 3x \sec 3x \tan 3x}$$ then: $$\lim_{x\rightarrow ...
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2answers
33 views

Finite series identity [duplicate]

How would I prove this statement? I know that it's a finite series. I don't know how to approach this at all. $$\sum_{i=1}^N i^3 = \left(\sum_{i=1}^N i \right)^2$$
3
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2answers
51 views

Derivation of Series Expansions

I have the series expansion for $$I=\frac{1}{\sqrt{1-x^2}}$$ $$|x|\lt1$$ which is $$1+\sum_{k=1}^\infty\frac{1.3.5.7......(2k-1)x^{2k}}{2^kk!}$$ or so I am assured. Given the above what ...
3
votes
3answers
44 views

Sum of infinite geometric series

How do I evaluate this (find the sum)? It's been a while since I did this kind of calculus. $$\sum_{i=0}^\infty \frac{i}{4^i}$$
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1answer
22 views

Area under a parabolic trajectory

I have this problem: "prove that the area under the trajectory described by a parabolic shot that has: $f(x)=\tan(\theta)x - (\frac{g}{2v^2\cos^2(\theta)})x^2$ and $x=v\cos(\theta)t$ is defined ...
0
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0answers
16 views

Trig substitution triangle restrictions

I apologize if this is a dumb question, or if I am a little slow, but I've been thinking about this for all of yesterday and today and I just can't figure it out, despite googling it. I am confused ...
8
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1answer
71 views

What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?

This was inspired by similar posts like this one. Define the function, $$F(p) = \lim_{n\to\infty}2^n\sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}_{n \textrm{ square roots}}}$$ We know that, ...
1
vote
1answer
20 views

Identifying the formula for a quartic graphic

I am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't ...
0
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0answers
15 views

Fiding the most general antiderivative of a function bounded by two x's.

At first I thought this problem would simply become a definite integral since it appears two be bounded by two x's. However, I feel as though I may be wrong and I'm curious as to how I would approach ...
1
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1answer
36 views

Use L'Hopital's with this problem?

The problem is: $$\lim_{x\rightarrow 0^+} \left(\frac{1}{x}\right)^{\sin x}$$ I know the answer is $1$ because I checked with my graphing calculator, but how exactly do I get there? I got this far: ...
11
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3answers
169 views

Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing

I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing. $\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for ...
-3
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1answer
9 views

Calculus :Work of an inverted right circular cone

A tank in the shape of an inverted right circular cone has height 6 meters and radius 4 meters. It is filled with 5 meters of hot chocolate. Find the work required to empty the tank by pumping the hot ...
2
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0answers
57 views

solving definite integral problems without complex line integral

It is well known that some definite integrals such as $$\int_{0}^{\infty} \frac{dx}{a+\cos{x}}$$ $$\int_{0}^{\infty} \frac{\sin{x}}{x}dx$$ are solved by using complex analysis techniques. (It uses ...
3
votes
2answers
39 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...
1
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1answer
20 views

Divergence theorem and applying cylindrical coordinates

This time my question is based on this example Divergence theorem I wanted to change the solution proposed by Omnomnomnom to cylindrical coordinates. $$ \iiint_R \nabla \cdot F(x,y,z)\,dz\,dy\,dx = ...
6
votes
2answers
105 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
1
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1answer
85 views
+300

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 < ...
0
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2answers
76 views

Why is convergence required for a series to be differentiable? [on hold]

Since moderators marked this question as "unclear" I will repeat the title maybe this won't be marked. Why is convergence required for series to be differentiable? I want intuitive explanation - not ...
3
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2answers
63 views

Since $\lim\limits_{x\to0}\frac{\sin kx}{kx}=1$ for constants $k$, is it also true for general arguments?

To be more specific, is it true that $$\lim_{x\to0}\frac{\sin f(x)}{f(x)}=1~~?$$ I'm tempted to say yes at first glance, so long as $f(x)\to0$ as $x\to0$. The reason I ask is to verify this limit ...
1
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2answers
94 views

Prove the limit is $\sqrt{e}$.

How do you show $$\lim\limits_{k \rightarrow \infty} \frac{\left(2+\frac{1}{k}\right)^k}{2^k}=\sqrt{e}$$ I know that $$\lim\limits_{k \to \infty} \left(1+\frac{1}{k}\right)^k=e$$ but I don't ...
5
votes
1answer
101 views

How to integrate $\int_0^{\infty}\frac{e^{-(t+\frac1t)}}{\sqrt t} dt$?

This is a problem given in my homework . I have to find the integral$$\int \limits_{0}^{\infty} \frac{e^{-(t+\frac{1}{t})}}{\sqrt t}dt$$ I am trying to use integral representation of the gamma ...
0
votes
2answers
23 views

interpreting $(1-t)f(a)+tf(b)$ from $f((1-t)a+tb)\leq (1-t)f(a)+tf(b)$

For a convex function $f((1-t)a+tb)\leq (1-t)f(a)+tf(b)$ holds. I understand how the graph looks like but why is the equation of the secant line $(1-t)f(a)+tf(b)$? Can anyone pleasae give me a ...
5
votes
2answers
91 views

$ \lim_{n\rightarrow \infty}n^{-\frac{1}{2}\left(1+\frac{1}{n}\right)}\cdot \left(1^1\cdot 2^2…n^n\right)^{\frac{1}{n^2}}$

Evaluate the limit $$ y=\lim_{n\rightarrow \infty}n^{-\left(1+1/n\right)/2}\times \left(1^1\times 2^2\times 3^3\times\cdots\times n^n\right)^{1/{n^2}} $$ My Attempt: When $n\rightarrow ...
1
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0answers
20 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
5
votes
6answers
386 views

What is the limit of $x/(x+\sin x)$ as $x$ approaches infinity?

I am trying to determine $$\lim_{x \to \infty} \frac{x}{x+ \sin x} $$ I can't use here the remarkable limit (I don't know if I translated that correctly) $ \lim (\sin x)/x=1$ because $x$ approaches ...
0
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1answer
24 views

Consider a function $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all real roots of $f(x)$ and $t=|s|$. Then…

the real number $s$ lies in the interval (A)$(-0.75,-0.5)$ (B)$(-0.5,0)$ (C)$(0,1)$ (D)$(-0.25,0)$ and the area of region bounded by $f(x),y=0,x=0$ lies in the interval (A)$(0.75,3)$ ...
0
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3answers
63 views

How to prove that a sequence is unbounded

I want to ask, how to start a proof that shows a sequence to be unbounded Define a sequence $\{X_n\}$ by $$X_1 = 1 ,\quad X_{n+1} = X_n + \sqrt{X_n} \quad \text{for}\ n \geq 1$$ Prove that ...
-2
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0answers
58 views

Rolle's theorem question

Let $f(x)=\sin2x/e^{2x}$. Note that $f$ is continuous on $[0,\pi/2]$, and differentiable on $(0,\pi/2)$, with $f(0)=f(\pi/2)=0$. So by Rolle's theorem, there exists a $c\in(0,\pi/2)$ with $f'(c)=0$. ...
-1
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1answer
27 views

simple percentage problem [on hold]

a man sold a watch of rs 2400 at a loss of 25%.at what rate should he sold the watch to earn a profit of 25%
6
votes
2answers
221 views

Is it possible to develop Analysis solely from Peano's axioms

...and a few definitions on the way? When I studied Calculus using Spivak's book It was clearly shown that, in order to prove some fundamental theorems (intermediate value theorem being one of them), ...
0
votes
1answer
14 views

Calculating the length of a helix

I have a pipe and I want to put a wire through it in a helix form. I need to calculate how long the wire (wl) has to be. I know the internal diameter (id), and therefore the circumference (c). I know ...
1
vote
0answers
30 views

Continuity at a point II

For a function $f$ to be continuous at a point $x_0$, is it necessary that there exists a ball centered at $x_0$ such that $f$ is continuous at all points within this ball? Put another way, does $f$ ...
25
votes
3answers
650 views

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
-2
votes
1answer
32 views

Area under the given curve [on hold]

The area under the curve $\displaystyle y = \frac{|x-3| + |x+1|}{|x+3| + |x-1|}$ , $x$-axis and the ordinates at $x = -3$ and $x = 1$
0
votes
1answer
25 views

Finding the gradient of a function.

A function $f=f(x,y)$ has continuous partial derivatives , and assume that maximal directional derivative of $f$ at $(0,0)$ is equal to $100$ and is attained in the direction towards $(3,-4)$ , we ...
0
votes
0answers
9 views

Solving for the poisson rate

Say I have an equation of the form $$ 0 = -a + \sum_{k=0}^\infty f(k, \lambda)R(k)\\ 0= -a + \exp(-\lambda)\sum_{k=0}^\infty \frac{\lambda^{k}}{k!}R(k) $$ where $f()$ is the Poisson mpf, $a$ is a ...
0
votes
1answer
89 views

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $?

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .
1
vote
2answers
31 views

Sketching functions $f(x) = \frac{e^x}{x^2} \quad \text{ and } \quad g(x) = \frac{1}{x}$ - First Derivative test and domain restriction

when working on a "Sketching a function" problem, some textbooks have a step-by-step procedure. The first one is usually stating the Domain of a function. When working with functions like $$ f(x) = ...
1
vote
1answer
19 views

Class $C^1$ function on a compact set

The problem is: Let g be of class $C^1$ on $\Delta$⊂$ℝ^n$ and K be a compact subset of Δ. Show that there is a number C such that |g(s)-g(t)|≤C|s-t| for every s,t∈K. I have proved that it is true ...