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

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3 views

Prove of limit related to $|f(x)|$

Question: Prove that if $\displaystyle \lim_{x \to a} f(x) = L$ then there is a number $\delta > 0$ and a number $M$ such that $|f(x)|<M$ if $0 < |x - a|< \delta$. This means: For every ...
0
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0answers
14 views

How to compute the integral $\int^{\pi/2}_0\ln(1+tan\theta)d\theta$?

How to compute the integral $\int^{\pi/2}_0\ln(1+\tan\theta)d\theta$. If we let $t=\tan\theta$, then the integral becomes to ...
1
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0answers
23 views

Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$

Nowadays I encounter an integral which is difficult for me to evaluate it. Please help me to evaluate it. Thank you. $$\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} ...
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0answers
16 views

Partial Fraction Decomposition of Exponential Generating Functions

I want to see if it is possible to write $$ \left(\frac{x}{e^x-1}\right) \left(\frac{x^2/2! }{e^x-1-x}\right) \left(\frac{x^3/3!}{e^x-1-x-x^2/2}\right)$$ as a linear combination of the factors ...
0
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1answer
10 views

Show that $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$ for any n by n matrix

Prove that for any n by n real matrix $v\in {\mathbb R}^{n\times n}$, $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$, where $t\in\mathbb R$, $I$ is the identity matirx, and $trv$ denotes the trace of ...
1
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1answer
15 views

Spline approximation for $g(t) = \frac{t e^{-t}}{(x+t^2)^2}$

Is there any nice way to do a spline approximation for $$ g(t) = \frac{t e^{-t}}{(x+t^2)^2}\,, $$ where $x$ is some constant? I tried finding nice interpolation points, however this proved very ...
1
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1answer
22 views

convergence of the series to inf or not

let $a_n = \dfrac{e^{-(1/2) \times a^2 \times\log(n) }}{a\sqrt{2\pi \log(n)}} $, $a$ is a constant, and the question is if $S_n = \sum a_n$ converge to a finite number. I wonder if I should ...
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2answers
42 views

Does alternating test show divergence?

My book states the alternating tests' convergence requirements. However, my book doesnt point out, if $a_n$ fails one of the convergence requirements, is it true that is diverges? Such as the limit ...
0
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0answers
26 views

how to add supremums

I need to prove that $$\sup(S)+\sup(T)=\sup(S+T)$$ I don't understand what $\sup(S+T)$ means, can you show me examples for groups $S$ and $T$ so this equation works. Thanks
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0answers
10 views

Are these two option valuation formulas equivalent? Why?

I have been reading a finance paper that claims that the following function, which is a value for a financial derivative (1): $$V(s,t)=E_{Q} \left[\zeta\big(S(T)\big)e^{-\int_t^T r_F(\nu) ...
2
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2answers
66 views

Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$

Evaluate: $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$ Using real methods only. I am not sure what to do. I tried finding a power series, which was too ugly. I just need some ...
1
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0answers
17 views

derivative of t distribution cdf wrt degrees of freedom

Given the cdf of a t distribution as follows: $T_\nu(x)=\frac{1}{2} + x\Gamma(\frac{\nu+1}{2}) + \frac{_2F_1 ...
1
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2answers
59 views

Does $\int_0^\infty |f'(x)| dx < \infty$ conclude $\lim_{x\to \infty} f(x)<\infty $

$f:[0,\infty) \to \mathbb R $ is $C^1$ and $$\int_0^\infty |f'(x)| dx < \infty$$ then can we prove that $\lim_{x\to \infty} f(x)$ exists and $$\lim_{x\to \infty} f(x)<\infty $$ My attempt: ...
0
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1answer
19 views

Area of the region inside $r = 1 - \cos(\theta)$ and also inside $r = \cos(\theta)$

Pretty simple polar integration question that I've been having trouble with... The question says it all. I identified the limits of integration by setting $1 - \cos(\theta) = \cos(\theta)$ so that ...
0
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1answer
28 views

Integrate $dx/(4x^2-1)^{3/2}$

I have trouble using trig sub. After I get that x = 2x+1, should I substitute back into the original problem's $4x^2$ with $(4(2x+1)^2)$?
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3answers
37 views

How to differentiate the function $f(x) = [ \frac{a+x}{b+x}]^{a+b+2x}$?

It has been given that, $$f(x) = \Big[ \frac{a+x}{b+x}\Big]^{a+b+2x}$$ How to prove , $$f'(0) = 2\ln \frac{a}{b}+ \frac{b^2-a^2}{ab}\Big[\frac{a}{b}\Big]^{a+b}$$ Do I have to take the logarithm of ...
1
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0answers
47 views

Calculating an integral with sine, cosine

I've recently calculated the Fourier transform of $\dfrac{\sin \pi ax}{\pi x}$. Now I'm trying to calculate $$\int _{\mathbb{R}} \frac{\sin ^2 \pi ax}{\pi ^2 x^3} \cos \pi bx\;\mathrm dx$$ The ...
1
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1answer
46 views

How to evaluate $\int \cot^2(x) \;\mathrm dx$?

How do you find the antiderivative of $\cot^2x$? My steps to find it First $$ \csc^2 x = \cot^2 x+ 1 $$ because of Pythagorean Identities, so $$ \cot^2 x= \csc^2 x-1$$ so $$ \int \cot^2 x\, ...
2
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2answers
45 views

Integrate $\int \csc^6(2x)\, dx$

I know to use the identity $1+\cot^2(2x)$. I'm not sure how to use $u$-substitution to substitute the $2x$ from the problem. I would have to use a $u$-substitution and then another $w$-substitution. ...
0
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1answer
30 views

If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$

Let $f$ be a twice differentiable function on the open interval $(-1,1) $such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $and $f''(x) \le f(x)$, for all $ x\ge 0$. Show that ...
0
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2answers
53 views

Why doesn't $\ln (x)$ have an asymptote since its derivative is $1/x$?

My understanding is that the derivative gives the gradient of the function at that point. So for the function $x^2$, its gradient at point $x=10$ is equal to $20$. Extrapolating this to $\ln (x)$, ...
9
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2answers
345 views

Why doesn't it work when I calculate the second order derivative?

Let $y=y(x)$ be determined by the equation \begin{align*}\begin{cases} x=t-\sin{t}\\ y=1-\cos{t}.\end{cases} \end{align*} I understand the solution: ...
2
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1answer
40 views

How to prove $\lim\limits_{t \to 1^-} \frac{\sqrt{1-t^2}}{2\pi}\int_{S^1}\frac{f(x,y)}{1-tx}ds=f(1,0)$?

$f(x,y)$ is a continuous function defined on unit circle $\ S^1 :$ $x^2+y^2=1$, prove $$\lim\limits_{t \to 1^-} \frac{\sqrt{1-t^2}}{2\pi}\int_{S^1}\frac{f(x,y)}{1-tx}ds=f(1,0)$$ I have tried to ...
-1
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2answers
23 views

Maximal value, several variables [on hold]

Let $x_i>0, \quad n=1,...,n, \quad \sum_{i=1}^nx_i=1$. Show that the function $\sum_{i=1}^nx_i\log_2\frac{1}{x_i}$ attains a maximal value at $x_i=\frac{1}{n}, \quad i=1,...,n$. Give me a hint, ...
2
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2answers
49 views

Proof : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$

I need to proof this : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$ then there is $d \in (a,b)$ so $f'(d)=0$. I'm not sure ...
3
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1answer
28 views

Volume when rotated about the line $y=-1$

Find the volume when the region enclosed by $y=x^2$, $y=4$ is revolved around the line $y=-1$ My teacher has given the following answer: I assume she has done this through the method of shells, ...
1
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0answers
64 views

Integrals and f(x)dx

Suppose $$\int_0^2 f(x)\,dx=3$$ $$\int_0^5 f(x)\,dx=8$$ Compute $$\int_2^5 f(x)\, dx$$ $$\int_0^2 f(2x)\,dx$$ For the first one, I know that by subtraction $$\int_2^5 f(x)\,dx = \int_0^5 ...
0
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0answers
9 views

Logistic model. Did I set up the differential equation $(1)$ correctly?

Update: I fixed it. The major mistake I made was that originally put $I(t) = \beta\cdot(P-y(t))$ while it of course is supposed to be $I(t) = \beta\cdot y(t)$. NB: I came up with this problem ...
0
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1answer
18 views

How to find the integration bounds when calculating area

To calculate an area between curves, I need to integrate with respect to x between the curve $y=\sqrt{2x}$, the x-axis and the line $y=\frac{4x-12}{5}$ My understanding, using google to display plot ...
3
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1answer
55 views

Factorial identity $\left(\tfrac{1}{2}\right)!$ to get Waallis

I asked the wrong question here, my fault :( How does one see, using $n! = \prod_{k=1}^\infty \left(\frac{k+1}{k}\right)^n \frac{k}{k+n}$, that $$\left(\frac{1}{2}\right)! = \frac{}{} ...
0
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0answers
23 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
1
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1answer
39 views

When integrating, can only one term of an equation be integrated or must entire equation be integrated to maintain equality?

Is integration considered a basic operation in the sense you have to do it to all parts of the equation? $y dy - x dx = 0$ Is it valid to do $\int y dy - \int x dx = \int 0$ but invalid to leave out ...
2
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1answer
42 views

Surface of revolution of an ellipse

I have been working on this question, but I end up getting the wrong answer overtime: The ellipse $$\frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1$$ where $a>b$ is rotated about the $x$-axis to form a ...
3
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1answer
36 views

Rectangle circumscribed to an ellipse of max area/perimeter

I could solve the classical problem of maximizing the area (fixing the perimeter) or maximizing the perimeter (fixing the area) of an inscribed rectangle, but I don't know how to solve ...
1
vote
2answers
62 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
0
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2answers
23 views

Chain rule and implicit dfferentiation

We are given that $y(x)=e^{z(x)}$. I want to show $$z''(x) = \frac{y''(x)}{y(x)}-\left(\frac{y'(x)}{y(x)}\right)^{2}$$ But I can't seem to get to this result. Since $y=e^{z}$ then $z=\ln{(y)}$, so ...
2
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1answer
56 views

power series of function

I am studying on summation theory on power series of functions. My question is to find the sum of power series $$e^{-n} \sum_{k=0}^{\infty} \frac{n^k\frac{k}{k+1}}{k!}.$$ I tried apply ...
3
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1answer
37 views

Prove with Cauchy's limit definition ($\epsilon, \delta$) that $\lim_{x \rightarrow 0} \frac{x^2-8}{x-8}=1$

Prove with Cauchy's limit definition ($\epsilon, \delta$) that $$\lim_{x \rightarrow 0} \frac{x^2-8}{x-8}=1$$ Got really troubled with the proper technique of solving this. Any assistance will be ...
2
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2answers
50 views

Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $

How does $$\Pi_{k=1}^\infty \sqrt{\frac{k+1}{k}}\frac{k}{k+\tfrac{1}{2}} = \frac{\sqrt{\pi}}{2} = \frac{\sqrt{2(\tfrac{\pi}{2})}}{2} = \frac{1}{2}\sqrt{2 \Pi_{k=1}^\infty ...
0
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2answers
56 views

How to find the derivative of $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$?

For a real number $t>0$, let $\sqrt t$ denote the positive square root of t. For a real number $x>0$, let $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$. If $F'$ is the derivative of $F$, then ...
5
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3answers
85 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
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0answers
63 views

Determine $1^{x}+2^{x}+3^{x}+4^{x}+5^{x}+6^{x}+7^{x}+8^{x}+9^{x}=10^{x}$ [on hold]

there is only one real solution : $$1^{x}+2^{x}+3^{x}+4^{x}+5^{x}+6^{x}+7^{x}+8^{x}+9^{x}=10^{x}$$
1
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1answer
26 views

Solution for a complexed equation

Find $z$ for the equation $e^z + e^{-z} = 0$. So $$e^z + e^{-z} = 0 \iff e^z = -e^{-z} \iff e^z = e^{\pi i - z} \iff z = \pi i -z + 2\pi ik$$ I understand all expect the $2\pi ik$. Can you ...
1
vote
2answers
33 views

Matrix representation of the derivative of a smooth function

Let $V:\mathbb R^n\to\mathbb R$ be a smooth function and define the Hamiltonian function $H:\mathbb R^n\times\mathbb R^n\to\mathbb R$ (kinetic plus potential energy) by $$H(x,y):=\frac ...
1
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1answer
25 views

find roots in the complexes

Find the roots of: $$ z^2 -3z +4iz = 1-5i $$ Rearranging the terms: $z^2 + z(4i-3) + 5i - 1 $ Solving by using the quadratic formula: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
6
votes
1answer
99 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
-1
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1answer
33 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
2
votes
1answer
20 views

$ \int_{ABC} f = \int_{CDA} f $

Problem from this year's MIT-PRIMES application: Let $f$ be a continuous function on the plane. In any rectangle $ABCD$ so that $AB$ is parallel to the $x$-axis and $B$ has a greater ...
0
votes
0answers
13 views

Arc length and curvature for logistic curves [on hold]

How can arc length and curvature for logistic or sigmoid curves can be calculated? Consider the logistic curve given by $y = \frac{y_i-y_f}{1 + \left(\frac{x}{C}\right)^{1/B}} + y_f$ where, ...
0
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1answer
34 views

Interchange of derivatives

Given Euler-Lagrangian equation $$\frac{d}{dt}\frac{\partial L}{\partial \dot q}-\frac{\partial L}{\partial q}=0$$ Can I equivalently write as $$\frac{\partial \dot L}{\partial \dot q}-\frac{\partial ...