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

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3
votes
1answer
41 views

Integral simplification

$$ \int_{-\infty}^{-1} e^{ikx} \left( \frac{-A}{-x-1+\sqrt{x^2-1}} \right)dx = \frac{A}{2}\int_1^\infty e^{-ikx} \left( 1 - \sqrt{\frac{x+1}{x-1}} \right)dx. $$ Hello, thank you very much for this ...
2
votes
1answer
65 views

Solving second order partial differential equation

I' trying to solve this differential equation: $$y^2 \frac{\partial ^2 u}{\partial y^2} - 2xy \frac{\partial ^2 u}{\partial x \partial y} + x^2 \frac{\partial ^2 u}{\partial x^2} + 2y \frac{\partial ...
0
votes
1answer
83 views

Existence of a unique maximizer of a strict quasi-concave function defined over a convex set

Set $S \subset \mathbb R^2$ is compact and convex. A typical element of $S$ is $s=(s_1,s_2) \in S$. Also, $d \in \mathbb R^2$ is a fixed element such that there exists $s \in S$ such that $s \gt d$. ...
1
vote
2answers
49 views

Integration Using Trigonometric Substitutions with $x^2$ and $\sqrt x$ in denominator

How can I integrate a function in this form: $$\int\frac{1}{x^2\sqrt{a+x^2}}dx$$ I know I need to use trigonometric substitution but don't see how to apply it for this form of function. What ...
2
votes
2answers
35 views

Finding integral of P(x)

When $0\le x\le 1, P(x) = 2$. When $x>1, P(x) = -\frac{2}{x}$ Then what is $$\int P(x)\,dx$$ I could not find the second part when $x > 1$.
3
votes
1answer
81 views

Multivariate integration by parts

Let $f,g:\mathbb R^n \rightarrow \mathbb R^n$, $(x_1,\dots, x_n)\mapsto f,g(x_1,\dots, x_n)$ be smooth functions with compact support. Use the following notation: for an index $\alpha = ...
1
vote
1answer
82 views

Geometric vs Arithmetic returns differences

Been reading some notes that say when calculating returns, using the geometric methodology (1+returns, performing a division between the two returns and - 1 from the result) for computing returns is ...
0
votes
1answer
20 views

Methods to create a function that represent directly definite integral?

Exist some method to create directly from integral some function $g$ that $$\int_{a}^{b}f(x)dx=F(b)-F(a)=g(b-a)$$ Maybe that only some kind of $f(x)$ can be transformed in this way and many other ...
1
vote
1answer
38 views

Second derivative test failed

I have $$H(\psi_1(t),\psi_2(t),x_1(t),x_2(t),u(t))= -1 + \psi_1(3x_1+x_2) + \psi_2(4x_1+3x_2 +u)$$ Note: $\psi_1 = -2Ae^t + 2Be^{5t}$, $\psi_2 = Ae^t + Be^{5t}$ Hence we have $$\frac{\partial ...
2
votes
1answer
69 views

Show that if f'' is positive on an interval [a,b], then f' has at most one zero in [a,b]

Show that: $f''>0$ on [a,b], then $f'$ has at most one zero in [a,b]. How do I prove this? $f''>0$ implies that it is a concave up function right?
0
votes
1answer
48 views

Is there free software that can be used to generate a chain rule tree graph?

I'm in multivariate calculus and we just finished up the chain rule. One of the methods for solving this is to produce a tree graph and traverse it. An example would be this tree graph taken from my ...
3
votes
3answers
108 views

How to solve the functional equation $f(2x) = (e^x+1)f(x)$?

I need to solve $f(2x)=(e^x+1)f(x)$. I am thinking about Frobenius type method: $$\sum_{k=0}^{\infty}2^ka_kx^k=\left(1+\sum_{m=0}^{\infty}\frac{x^m}{m!}\right)\sum_{n=0}^{\infty}a_nx^n\\ ...
2
votes
2answers
107 views

Elementary proof of an inequality with $e^x$ when $|x|<1$.

Assume $|x| <1 $ and we already know $0 \leq e^x - 1 - x$. Note that last inequality was god given, we know it is true, but we do not know how it was proved. Can we deduce from here that $e^x - 1 - ...
1
vote
1answer
28 views

Find the positive values of x for which the series x^ln(n) n=2 to infinity converges

I know its a geometric series so I figured it would be x from 0 to 1, but I got the wrong answer.
0
votes
1answer
47 views

$4\int_{-4}^{0} \sqrt{-y} \,dy$

Isn't $$4\int_{-4}^{0} \sqrt{-y} \,dy = 4(0-16/3)=-64/3 ?$$ How come the answer I see is $64/3$, without the negative sign?
3
votes
1answer
39 views

find the moment generating function of a pdf

Let $X$ be a random variable with pdf $$f_x(x)=\frac{1}{2\sigma}e^\dfrac{-|x-\mu|}{\sigma}$$, $-\infty< x<\infty$, $-\infty< \mu<\infty$, and $\sigma>0$. I have to find the mgf of $X$?. ...
4
votes
1answer
55 views

Triple integral over an ellipsoid

Let $E$ be the solid ellipsoid $E = ${$(x,y,z)$ | $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1$} where $a > 0,\: b > 0,\: c > 0$ Evaluate $\int\int \int xyz\: dxdydz$ over: a. ...
2
votes
3answers
62 views

Show that $\int_{-1}^1f(t)dt = \sum_{i=1}^nw_if(a_i)$

If $a_1,...,a_n$ are distinct reals, show that there are scalars $w_1,...,w_n$ such that $$\int_{-1}^1f(t)dt = \sum_{i=1}^nw_if(a_i)$$ for all polynomials $f(t)$ in $P_{n-1}$. I ...
2
votes
2answers
58 views

Changing to polar coordinates

I am stuck with this problem: $$\int_0^1\int_{x^2}^x\sqrt{x^2+y^2}\,dy\,dx$$ I would like to change to polar coordinates, but I am having a hard time parametrizing the domain. Do you guys have any ...
0
votes
1answer
14 views

finding A using with restriction $1 \leq a \leq 20$ in GCD

For what $1 \leq a \leq 20$ you are finding $a$ is it true that $a^m+a^n=x^2$ for positive integers $a,m,n,x.$ I did $a^m+a^n=x^2.$ $=a^m(a^{n-m}+1)=x^2$ We know that since $(a,b)=1$ since the ...
2
votes
1answer
38 views

Convergence of $x_{n+1} = x_n + \dfrac{(\vert x_n \vert)^{1/2}}{n^2}$

Let $(x_n)$ be the sequence defined by $x_{n+1} = x_n + \dfrac{(\vert x_n \vert)^{1/2}}{n^2}$ for $n \geq 2$ and $x_1$ be any real number. Then I want to prove that $x_n$ is convergent. It is ...
1
vote
1answer
46 views

Find the sum of the series $\sum^{\infty}_{n=2}\ln(1- \frac{1}{n^2})$

I got $0$ which is the wrong answer. I got $0$ because $1/n^2$ just goes to $0$ so you're left with $\ln(1)$ which is $0$.
0
votes
2answers
41 views

Given a list of N integers, how to find out if the second derivative is positive or negative?

Let's say I have a list where N=10, such as [45,34,56,22,33,44,34,34,43,35]. I would like to know if the second derivative is positive or negative, in other words, if the rate of change of these ...
2
votes
2answers
114 views

How to prove the lower bound of $\frac{x^2}{\sin^2x}$?

How to prove $$1+\frac{x^2}{3}\leq \frac{x^2}{\sin^2x}, x\in (0,\pi/2)?$$ I do want to show it by intermediate value theorem as $$\frac{1}{\sin^2x}-\frac{1}{x^2} =\frac{2}{\xi^3}(x-\sin ...
1
vote
2answers
41 views

Understanding the process behind finding a sum to an infinite series

I'm struggling to understand why the sum of $$\sum\limits_{n=1}^\infty (e^{1/n}-e^{1/{n+1}})$$ is equivalent to/converges to $$e-1$$ I'm also unsure of the process to cancel terms to yield ...
8
votes
2answers
175 views

Can help me to find $\sum_{n=1}^{\infty }\frac{1}{(4n-1)^3}$?

Can help me to find $\sum_{n=1}^{\infty }\frac{1}{(4n-1)^3}$?
12
votes
3answers
200 views

How to prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$? [duplicate]

How can I prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$, where $a<1$? Thanks.
6
votes
4answers
804 views

Why do oscillating sequences diverge?

The sequence I'm presented with is defined by $$a_1=3,$$ $$ a_{n+1}=\frac{a_n}{a_n-1}$$ I see that the values of this sequence are $$3, \frac{3}{2},3, \frac{3}{2},3,...$$ So they oscillate between ...
1
vote
5answers
67 views

Partial Fractions Decomposition

I am failing to understand partial fraction decomposition in cases like the following: Provide the partial fraction decomposition of the following: $$\frac{x+2}{(x-4)^3(x^2 + 4x + 16)}$$ I see this ...
2
votes
1answer
360 views

Find the second derivative of a double integral

Problem: Find $F''(\pi)$ if $$ F(x) = \int_x^{\sin x} \left(\int_0^{\sin t} (1+u^4)^{0.5} \,du\right)\,dt $$ Context I don't know how to integrate the inner part. What to do to the integral inside ...
6
votes
1answer
155 views

Prove this closed-form of sum of ${_4F_3}$ hypergeometric functions

I think the following identity is true. How could we prove it? $${_4F_3}\left(\begin{array}c 1,1,1,1 \\\tfrac54,2,2\end{array}\middle|\,1\right) + ...
8
votes
1answer
60 views

Intuition behind a certain limit.

We want to find $\displaystyle\lim_{\theta\to\frac{\pi}{2}} b_1-a_1$, we are given $c=1$ and that $\cdot=90^{\circ}$ This is my solution; $$\begin{equation}\sin \theta=\frac{b_1}{a_1} \iff b_1=a_1 ...
0
votes
2answers
20 views

Limits-related question

If I know that: •$f(x)$ is an increasing function •$x<x+a$ •$\lim_{x\to +\infty}f(x)=\lim_{x\to +\infty}f(x+a)$ Can I conclude that $\lim_{x\to +\infty}f(x)=\infty$ ? Sorry if the question is ...
1
vote
2answers
48 views

Calculus - $\sum_{k=1}^\infty (\frac{e}{\pi})^{k-1}$

I have this problem: $$\sum_{k=1}^\infty (\frac{e}{\pi})^{k-1}$$ I need to find the sum. $$S_n=(\frac{e}{\pi})^0+(\frac{e}{\pi})^1+(\frac{e}{\pi})^2+(\frac{e}{\pi})^3+....+(\frac{e}{\pi})^{k-1}$$ ...
0
votes
1answer
27 views

Is it possible to write all of the functions in terms of polar form?

Is it possible to write all functions in terms of polar form? For example, the equation of the circle with radius one can be written like $r=1$ I'm wondering whether reform the equations of all curves ...
7
votes
0answers
149 views

Closed form of $\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$

Find the closed form of $$\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$$ I know the closed form for smaller powers like $2, 3$ exists, but I'm not sure if there is a closed form for this ...
1
vote
1answer
32 views

$\int_{0}^{1}(f'(x))^2-(f'(x))^3f(x)\; dx \geq 0$?

Is it true, that for all functions $f$ that $f(0)=f(1)=0$ : $$\int_{0}^{1}(f'(x))^2-(f'(x))^3f(x)\; dx \geq 0$$ I've tried to find counterexample, but I've not found.
0
votes
1answer
17 views

If $s_1 +s_2 \gt 1$ and $(t_1,t_2)$ be a convex combination of this with $(0.5,0.5)$ then show that $t_1t_2 \gt 0.25$

Let $(s_1,s_2)$ be such that $s_1 + s_2 \gt 1$. Let $(t_1,t_2)=((1-\epsilon )(0.5) + \epsilon s_1 , (1- \epsilon)(0.5) +\epsilon s_2)$, where $0< \epsilon \lt1$. I need to show that for $\epsilon ...
3
votes
1answer
48 views

Calculating derivative by keeping all but one $x$ constant and adding the results together for different $x$

In this answer there's a comment which says That's not wrong; that's a perfectly valid method. You get the derivative of any expression with respect to $x$ as the sums of all the derivatives with ...
0
votes
1answer
41 views

Transformation to elliptical coordinates

I'm currently struggling to make any progress with this question. I'm a little bit thrown by the inclusion of cosh and sinh. I am aware of all of the definitions, just need guidance with approach.
5
votes
1answer
213 views

Find center of mass of triangle with varying density

My objective: Find the center of mass of a thin triangular plate bounded by the y-axis and the lines $y= 7x+3$ and $y= 36-4x$. Assume that the density is given by $\delta(x,y) = 7x+2y+2$. In ...
1
vote
0answers
33 views

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \to \sqrt{x}$

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \rightarrow \sqrt{x}$ for $x > 0$. Let $\epsilon > 0$ be arbitrary, want to find $N \in \mathbb{N}$ such that $n \geq N \Rightarrow ...
0
votes
1answer
29 views

Does integral of f(x) = the integral of conjugate of f(x)?

$$\int f(x) = \int \overline f(x)$$ Is the above statement true?
1
vote
4answers
57 views

Evaluate $\int_1^\infty \frac{\ln x}{x^4}$

I evaluated the integral down to $[-\frac{\ln x}{3x^3} - \frac1{9x^3}]$. So I plug in infinity and get $-1$. because you get $-\frac\infty\infty$ for $-\frac{\ln x}{3x^3}$. $\frac1{9x^3}$ just goes ...
0
votes
2answers
32 views

Derivatives using chain rule. $f(x) = (7x^3 + 2)^3(6x^2 - 1)^4$

I understand how to use the chain rule with simple functions. But in this there are two core's too choose from. Can't wrap my head around how to even start. Please help, been stuck on this for too ...
-1
votes
2answers
51 views

How to solve for $e^{-x}$?

I'm sure I learned this at some point, I think it has something to do with $\log$-ing the function. Could someone please walk me through it? $$0=4x^2e^{-x^2}-2e^{-x^2}$$ By the way, this is the ...
0
votes
2answers
26 views

Convert a triple integral to cylindrical coordinates?

Find the volume determined by $$z \le 6-x^2-y^2$$ and $$z \ge \sqrt{x^2+y^2}$$ I used cylindrical coordinates to change the bound for $z$ to $r \le z \le 6-r^2$. However, I am not sure how to find ...
0
votes
2answers
42 views

How to prove conjugate symmetry on an integral?

Seems strange that proving complex conjugate symmetry of integrals isn't discussed on here yet. In general, how would you prove the conjugate symmetry aspect of an integral?
1
vote
3answers
59 views

How to find the sum of the series $\sum_{k=2}^\infty \frac{1}{k^2-1}$?

I have this problem : $$S_n=\sum_{k=2}^\infty \frac{1}{k^2-1}$$ My solution $$S_n=\sum_{k=2}^\infty \frac{1}{k^2-1} = -\frac{1}{2}\sum_{k=1}^\infty \frac{1}{k+1} -\frac{1}{k-1} = ...
1
vote
0answers
31 views

Find the $cot^{-1}(x)$ series without using the Taylor series????

Find the $cot^{-1}(x)$ series without using the Taylor series?