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

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3answers
19 views

Evaluate the Indefinite Integral Given

$$\int {x+3\over x^2+6x+10}dx$$ Could anyone help me with this substitution problem?
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2answers
16 views

Intervals and Signs

In the first and second derivative tests, I find whether the derivative is positive or negative by picking a random number within that open interval. The number I pick is arbitrary; however, what ...
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1answer
33 views

Minimizing a function in Mathematica

Edit: I simplified the function using $\textbf{Simplify[...]}$ How can I minimize this function of $x$, where $l$ is a positive constant? $$\frac{1}{2} \sqrt{\frac{x}{l}+\frac{l}{x}+4 x^2-2}$$ ...
2
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2answers
46 views

Proof of Max (x,y)

The problem states that $ \max(x,y) = \dfrac { x+y+|y-x|} {2} $ where $x,y \in \mathbb{R}$ Part 1) Prove that this is true. Part 2) Derive a formula for $\max (x,y,z)$. 1) Intuitively i see this as ...
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2answers
25 views

Given a curve, such as $1/x$, how to find which tangent is closest to its OWN interception with the y-axis

As title mention, if I have a function such as $\dfrac{1}{x}, x>0$, how can I find which tangent of the curve is closest to its y-axis interception. Using pythagorean theorem, one sees that the ...
2
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0answers
20 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
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3answers
82 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
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3answers
62 views

Want to ensure my proof is rigourous enough.

Question. Prove: $ 0 \leq x < y $ then $ x^n < y^n$ $ \forall n \in \mathbb{N} $ I'm particularly bad at proving obvious things but here it goes. ( please be super strict on analyzing my proof ...
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1answer
53 views

How to approach, substitution - definite integral

So I have this problem $${\int^{\pi/2}_0} {{\cos\theta \sin\theta}\over \sqrt{\cos^{2}\theta +8}}d\theta $$ and I'm not sure if this is the right direction to begin. If I have $u = \cos\theta$ ...
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2answers
24 views

Solution of given differential equation using Laplace Transforms.

I need solution of DE $$y'' + 2y' + 5y = 0$$with initial conditions $$y(0)= 1 \text{ and } y'(0)=0$$ I tried this but problem came when i started taking laplace inverse of F(s), so i need a complete ...
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2answers
38 views

How to PROVE there are only finite number of sub limit in this sequence that not converge

for example, let $A,B,C\:\in \mathbb{R}\:$ be some constants, and $$ a_n=\begin{cases} A, & n=3k-2,\ k\in \mathbb{N} \\ B, & n=3k-1,\ k\in \mathbb{N} \\ C, & n=3k,\ k\in \mathbb{N} ...
0
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4answers
69 views

Differentiating $ \left( 1 - \frac {1}{x} \right)^x $

I have a calculus question. How does one differentiate $\left(1-\frac{1}{x}\right)^x$, for x>1? It should be positive right?
2
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1answer
62 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
1
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1answer
43 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
-1
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2answers
28 views

Finding a differentiable inverse of $f(x)=x/\cos x$

Let $$ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \rightarrow \mathbb{R} $$ be defined by $$ f(x) = \frac{x}{\cos x}. $$ We're supposed to show that $f$ has a differentiable inverse $$f^{(-1)}$$ ...
0
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2answers
39 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
2
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1answer
44 views

Writing a proof of the convergence of a series defined recursively

Define the sequence $a_n$ recursively by $a_1=1$ and $$a_{n+1}=\frac13\left(a_n^2+\frac1n\right)$$ (a) Prove, by induction or otherwise, that $(a_n)$ is decreasing. (b) Prove that the series ...
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1answer
31 views

Finding the surface area of a sphere

Today I was going over some calculus that I had long forgotten, and I made the following mistake when trying to find the area of a sphere: I though it would be this: ...
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0answers
18 views

CDF of RVs taking infinite values

How can we define the CDF of a RV that takes positive infinite values with a tagged probability? Thanks in advance
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0answers
26 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 ...
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3answers
96 views

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

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 ...
8
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3answers
118 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)} ...
1
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1answer
37 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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2answers
23 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
21 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
25 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
47 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
31 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
15 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
78 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 ...
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0answers
18 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
61 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: ...
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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
33 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
39 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 ...
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0answers
56 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
51 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
48 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
36 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
61 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
364 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 ...
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2answers
24 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
52 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
29 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
74 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
11 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
votes
1answer
20 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
votes
1answer
58 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
24 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 ...