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

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75 views

Methods of Evaluating $\lim_{x\rightarrow 0} \frac{\sin x}{x}=1$ Multiple Choice Question

Which of the following techniques for evaluating limits cannot be used to show $\lim_{x\rightarrow 0} \frac{\sin x}{x}=1$ $a:$ The Squeeze Theorem $b:$ L'Hôpital's Rule $c:$ Using the graph $d:$ ...
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2answers
29 views

Using differential forms to define line integrals

I saw some similar questions and answers but they often included some information or mathematics I haven't learned/read so I'm hoping to get a somewhat simpler answer. Let $\beta:[a,b] \to ...
3
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3answers
65 views

Why is this claim about limits true

I didn't understand why this claim from wikipedia is true ...More specifically, when $f$ is applied to any input sufficiently close to $p$, the output value is forced arbitrarily close to $L$. ...
0
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1answer
90 views

Differentiation of Exponential Matrices

I have been struggling with the following question while solving an optimization problem $$\min_{y \in \mathbb{R}_{+}^G} F(y) = y^{T}\mathcal{K} + e^{-y^{T} \mathcal{A}} \mathcal{P},$$ where ...
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4answers
29 views

Differentiation of trigonometric function with $\theta/2$

Just wondering, what would be the method for differentiating the following: $$ \sin^{2}\left(\frac{\theta}{2}\right) $$
1
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1answer
49 views

Maximization problem on an ellipsoid [closed]

for three variables, $$\max f(x,y,z)= xyz \\ \text{s.t.} \ \ (\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{z}{c})^2=1$$ where $a,b,c$ are constant how to solve the maximization optimization problem? ...
0
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1answer
50 views

Inverse for $1-zb(z)$

I need to find the inverse of $1-zb(z)$ with $b(z)=\sum_{n=0}^{\infty}b_nz^n$. I have tried several approaches where I among other things have tried using the methods in my calculus book but nothing ...
2
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5answers
168 views

Evaluating $\lim\limits_{n \to \infty} \left(1+{3 \over n}\right)^n$

$$\lim_{n \to \infty} \left(1+{3 \over n}\right)^n$$ What are the general rules for limit of this kind, like $\lim_{n \to \infty} \left(1+{\alpha \over n}\right)^n$ or $\lim_{n \to \infty} ...
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0answers
26 views

Prove that $\chi$ is Riemann integrable.

Let $R=[-1,1] \times [-1,1]$, consider the set $D=${$(x,y); x^2+y^2 \leq1$}. $\chi_{D}$ is Riemann integrable in R. $\chi_{D}=$$ \left\lbrace \begin{array}{l} 1 \text{ if } (x,y)\in D \\ ...
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0answers
38 views

Reference on matrix differentiation?

Is there any good resource on matrix differentiation? I am seeing lots of questions like ...Consider $L(\vec{x},\lambda)=f(\vec{x})-\lambda^T (A\vec{x} - \vec{b}).$ Let's take derivative with ...
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2answers
42 views

If the first nonzero derivative at $a$ is of odd order $n\ge 3$, then $a$ is a point of inflection

Statement to Prove: Let $f$ be a real valued function such that for a fixed point $a$ , $$f^k(a)=0;1\le k\le n-1;\\and\ \ f^n(a)\neq 0.$$ Then if $n$ is odd then $a$ is a point of inflection. ...
4
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2answers
101 views

Volume of the solid intersecting 3 spheres

Let the next three spheres: \begin{array}{lcccl} S_1 : &(x-1)^2 &+ &y^2 &+ &z^2 &=1, \\ S_2 : &x^2 &+ &y^2 &+ &z^2 &=1, \\ S_3 : &(x+1)^2 &+ ...
3
votes
5answers
274 views

Limit with trigonometric function

Find $$\lim_{x \to \pi/4}\frac{1-\tan(x)}{\cos(2x)}$$ without L'Hôpital. $$\lim_{x \to \pi/4}\frac{1-\tan(x)}{\cos(2x)}=\lim_{x \to ...
1
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1answer
26 views

Test the convergence.

Test the convergence of $\left (a_{n} \right)$ if $$a_{n}=\sum_{k=1}^{n}\left( \sqrt[\left \lfloor \frac{m}{4} \right \rfloor]{k^2+1} -\sqrt[\left \lfloor \frac{m}{4} \right \rfloor]{k^2-1} \right), ...
0
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1answer
40 views

Functional inequality $f'(x)^3-f'(x)^2-9f'(x)+9\leq0$

f(x) meets the following conditions. $$ \left\{ \begin{align} &f'(x) \text{ is continuous for all }x\\ &f'(x)^3-f'(x)^2-9f'(x)+9\leq0\\ &f(0)=0\\ &f(2016)=4102 \end{align} \right. $$ ...
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2answers
52 views

Show that $\lim_{n\to \infty} \sum_{k=1}^n \frac n{n^2+k^2}=\frac \pi 4$ [duplicate]

My Work: $$\lim_{n\to \infty} \sum_{k=1}^n \frac n{n^2+k^2}$$ $$\lim_{n\to \infty} n\sum_{k=1}^n \frac 1{n^2+k^2}$$ $$\lim_{n\to \infty} ...
1
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0answers
17 views

Roots of an equation with normally distributed variable

Consider the following equation: $p\left(1-\int _{\mu}^{x} f(y)dy\right) \left[p\left(1-\int _{\mu}^{x} f(y)dy\right)+(1-p)q \right]-xf(x)p(1-p)q=0$, where $p,q \in [0,1]$, $f(\cdot)$ is the ...
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3answers
93 views

Calculate limit with floor function [closed]

How can I proceed to find the following limit: $$\lim_{n \rightarrow \infty }(n\sqrt 2-\lfloor n\sqrt2 \rfloor) $$ where $n$ is a natural number. Please if there is no limit, would you provide a ...
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4answers
2k views

Square of a second derivative is the fourth derivative

I have a simple question for you guys, if I have this: $$\left(\frac{d^2}{{dx}^2}\right)^2$$ Is it equal to this: $$\frac{d^4}{{dx}^4}$$ Such that if I have an arbitrary function $f(x)$ I can get: ...
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3answers
55 views

Alternate approaches to solve this Integral

Evaluate $$I=\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\:dx$$ I have used parts taking first function as Integrand and second function as $1$ we get $$I=x\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}-\int ...
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2answers
169 views

another product of log integral

Assuming one exists, and I think it does, find a closed form for: $$\displaystyle \int_{0}^{1}\log(1+x)\log(1-x^{3})dx$$ From it, I did manage to derive: $$\displaystyle ...
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4answers
40 views

Show that $f(x) = x^3 -3x^2 -1 = 0$ (unique root) on the open interval $]3.1, 3.2[$

Given the function: $$ f(x)=x^3 -3x^2 -1 $$ *Show that $f(x)=0$ admits a unique root on the interval $]3.1,3.2[$I first thought of using this rule $f(a).f(b) < 0$ since the function is monotonic on ...
0
votes
1answer
24 views

Multiple variable limit

I think $0$ is candidate for this limit, but I need help proving it: $$\lim_{(x,y)->(0,0)}(x^2+y^2)(\ln(x^2+y^2)-1)$$ Also, I'm not sure how to approach this limit ...
4
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3answers
84 views

$[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$

prove that $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$. Could someone help me to solve. I try many ways but I can't solve it ( I tried : Let $x=n^2 +k$ and don't know what to do afterwards). ...
1
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1answer
27 views

Optimal point selection to maximize length of closest points interval

Consider the following game: there are $n$ players, who pick $x_i\in I = [0,1]$ in turns, $1\leq i\leq n$. For each selection $x = (x_1,\dots,x_n)\in I^n$, the player's $i$ reward is the length of the ...
2
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1answer
80 views

Product of two hypergeometric functions

For $\Re a, \Re b, \Re c, \Re a', \Re b', \Re c'>0$, I would calculate the following product $$ {}_2 F_1(a, b; c; x^{-1}) \times \, {}_2 F_1(a', b'; c'; 1-\frac{x}{y}) $$ For all $y>x>1$. ...
2
votes
1answer
97 views

How to Evaluate this Integral $\int \frac{\cos 6x+\cos 8x}{1+2\cos 5x}dx$?

Evaluate the following integral $$I=\int \frac{\cos 6x+\cos 8x}{1+2\cos 5x}dx$$ My work I tried to change $\cos x$ into $\sin x$ and try to substitute. But I am not getting it ...
1
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1answer
59 views

How to Prove Chain Rule? [closed]

How the chain-rule formula for derivatives can be proved $$ {dy \over dx} = { \frac{\frac{dy}{dt}}{\frac{dx}{dt}} } $$ Further explanation will be appreciated. Thank you.
0
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1answer
30 views

Prove convergence of a series

Let $$M=\{2^k*3^l|k,l \in \mathbb{N}\}$$ and $(a_n)_{n \in \mathbb{N}}$ a sequence which takes the numbers from M and sorts them ascending i.e :$$1,2,3,4,6,8,9,12,16...$$ Let $$b_n=\frac{1}{a_n}$$ ...
2
votes
1answer
41 views

Prove area under curve is between two values

The question reads: Prove that $$0 \leq \int_\frac{\pi}{4}^\frac{\pi}{2} \frac{\sin(x)}{x} \, dx \leq \frac{\sqrt2}{2} \ .$$ I know that I should apply $$ m(b-a) \leq \int_a^b f(x) \, dx \leq ...
1
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2answers
66 views

Center of Mass of a Semi-Circle using Cartesian Coordinates

So I'm currently trying to figure this out but am not sure where to start. I know that you can figure the center of mass using polar coordinates, but I know that it's possible to do it using Cartesian ...
3
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1answer
85 views

Defining $e^{x}$ as an infinite number of integrals

I was thinking that using the Taylor expansion of $e^{x}$ we use a summation of derivatives is it possible to write it using integrals? Something along the lines of $\int\int\int...\int dxdxdx...dx$ ...
4
votes
2answers
97 views

Functional equation $x\space f(x^2) = f(x)$

How can I logically lead to the answer from the following conditions? $$ \left\{ \begin{align} & x \, f(x^2) = f(x) \text{ for all } x > 0, \\ & f(x) \text{ is continuous}, \\ &f(1) = ...
2
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0answers
55 views

finding the integral of: $1/((1+x^2)^n)$ [duplicate]

What is the integral of $1/((1+x^2)^n)$? I have tried adding and substracting $x^2$ from the numerator $$ \int \frac{1}{(1+x^2)^2}\:dx = \int \frac{1}{(1+x^2)^{n-1}}\,dx − \int ...
1
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0answers
23 views

f uniformly Lipschitz, integral from 0 to inf of |f(x)| < inf imply |f(x)| converges to 0 as x -> inf. [duplicate]

Suppose f: [0,inf) -> R is uniformly Lipschitz and the integral from 0 to inf of |f(x)| < inf. Then|f(x)| converges to 0 as x -> inf. Prove this by proving the contrapositive. ie. negate the limit ...
1
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1answer
47 views

Proof of the Darboux's Theorem with lemma.

Darboux's Theorem as specified in https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis) Let $g(x)=f(x)-dx$ on $[a.b]$ Then $g'(x)=f'(x)-d$ Then by hypothesis, we can say ...
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1answer
35 views

Limit of $(n+1)^a-n^a$

Im asked to prove the $\lim_{x\to\infty}(x+1)^a-x^a=0$ when $a<1$. I was going to do this in three steps, when $0\lt a \lt 1$, when $a=0$ and when $a<0$. Im unsure now that I think about it. It ...
0
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1answer
48 views

What is it called when $f_{xx}=f_{yy}$ and $f_{xy}=f_{yx}$?

The function $f(x,y) = x^2y + 2xy + xy^2$ has the properties such that $\frac{\partial^2 f}{\partial x^2}=\frac{\partial^2 f}{\partial y^2}$ and $\frac{\partial^2 f}{\partial x \partial ...
3
votes
4answers
720 views

How do I find the maximum perimeter of a rectangle inscribed in an ellipse?

The problem I've been stuck on is this: A rectangle is inscribed in the ellipse $$\frac{x^2}{20} + \frac{y^2}{12} = 1$$ What is the maximum perimeter of the rectangle? I don't even know if I'm ...
2
votes
2answers
31 views

Computing the hyperbolic cosine as a serie

So I want to prove that: $$\sum_{x=0}^\infty \frac{\lambda^{2x}}{(2x)!}=\cosh(\lambda)$$ But, proceeding backwards, the only thing I know is that ...
0
votes
1answer
29 views

Question about force on cable of varying density

My math teacher have us a question: A non-uniform cable of length 20 feet gets thicker toward the bottom.The density of the cable x feet above the bottom is 20-x pounds per foot.The cable is hanging ...
1
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1answer
67 views

How to Find $a$ and $b$ Such That the Function $f(x) = ax^3+bx^2$ has an Inflection Point at $ (2,64)$

I am trying to figure out how to find $a$ and $b$ of $f(x)$, such that the function $f(x) = ax^3+bx^2$. Also, $f(x)$ has an inflection point at $ (2,64)$. Now I know that $f(2)$ = 64, however, I am ...
1
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1answer
34 views

Determine if Riemann integrable and evaluae the integral

Let $f(x):=2$ if $0\leq x <1$, $f(1)=3$ and $f(x):=1$ if $1<x\leq 2$. Show that $f\in R[0,2]$ and evaluate the integral. I know what the graph looks like, I'm having a hard time figuring out ...
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2answers
43 views

Explain how the following expression was derived?

Can someone explain how the author gets to the expression after the words "This leads to:"
0
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1answer
248 views

Show that the curve x = 5 cos t, y = 4 sin t cos t has two tangents at (0, 0) and find their equations.

1) As x = 5cos(t); differentiating, dx/dt = - 5sin(t) 2) As y = 4sin(t)cos(t), y = 2sin(2t) [Since 2sin(t)cos(t) = sin(2t)] ==> dy/dt = 4cos(2t) 3) So, dy/dx = (dy/dt)/(dx/dt) = 4cos(2t)/-5sin(t) ...
0
votes
2answers
91 views

Find the exact length of the curve. $x = 1 + 12t^2,\ y = 4 + 8t^3,\ 0 ≤ t ≤ 1$

Find the exact length of the curve. $x = 1 + 12t^2,\ y = 4 + 8t^3,\ 0 ≤ t ≤ 1$ My answer was 245 units; however, it is wrong.
0
votes
2answers
46 views

Ratio test results argue Wolfram

$$\sum\limits_{n=1}^{\infty} \frac{1+2^n+3^n}{5^n}$$ Going through with the ratio test, I eneded up with L=1 which means the test is inconclusive. Wolfram, however, says that the series converges by ...
0
votes
0answers
26 views

Integrating an integral equation in time

I have derived the folowing identity: $$\frac{1}{2} \Vert (x + it \nabla) u \Vert_2^2 = \frac{1}{2} \Vert x u \Vert_2^2 - t \text{Im} \int_{\mathbb{R}^d} x \overline{u} \nabla u \ dx + t^2 \Vert ...
1
vote
0answers
42 views

Find the limit of $\frac{1^1+2^2+3^3+\dots +n^n}{ n^n}$ [duplicate]

What is the limit of the sequence: $$\frac{1^1+2^2+3^3+\dots +n^n}{ n^n}$$ I tried only to break it in $(1/n^n)+..+(n^n)/n^n$ and say that the first equals to zero and the last is $1$. I have no idea ...
-1
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2answers
38 views

Find the points on the curve where the tangent is horizontal or vertical.$x = t^3 − 3t, \ y = t^2 − 4$

Find the points on the curve where the tangent is horizontal or vertical. $x = t^3 − 3t, \ y = t^2 − 4$ (Enter your answers as a comma-separated list of ordered pairs.) horizontal tangent ...