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

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2answers
143 views

A strengthening of Raabe's test: $\sum a_n$ diverges if $\frac{a_{n+1}}{a_n} \geq 1 - \frac{1}{n} - \frac{A}{n^2}$ for $A>0$

The usual form of Raabe's test says that if $a_n>0$ and if for large $n$, $\frac{a_{n+1}}{a_n}\leq 1-\frac{p}{n}$ for $p>1$, then $\sum a_n < \infty$. A proof I've seen of this relies on the ...
0
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2answers
120 views

Partial Fractions - Calculus

Evaluate the integral: $$\int \dfrac{9x^2+13x-6}{(x-1)(x+1)^2} dx$$ For some reason I cannot get the right answer. I split up the equation into three partial fractions but I cannot seem to find A, B, ...
2
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3answers
92 views

Partial fraction $\int \frac{x^2 + 11x dx}{(x-1)(x+1)^2}$

I have been using the cover up method from this video lecture $$\int \frac{x^2 + 11x}{(x-1)(x+1)^2} dx$$ $$\frac{x^2 + 11x }{(x-1)(x+1)^2} = \frac{A}{x+1} + \frac{B}{x-1} + \frac{C}{x-1}$$ What do ...
5
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2answers
194 views

Using trig substitution to evaluate $\int \frac{dt}{( t^2 + 9)^2}$

$$\int \frac{\mathrm{d}t}{( t^2 + 9)^2} = \frac {1}{81} \int \frac{\mathrm{d}t}{\left( \frac{t^2}{9} + 1\right)^2}$$ $t = 3\tan\theta\;\implies \; dt = 3 \sec^2 \theta \, \mathrm{d}\theta$ $$\frac ...
0
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3answers
66 views

Consider the series $\sum_{n=0}^\infty \frac{(x-3)^n}{3^n \sqrt{n+1}}$

$$\sum_{n=0}^\infty \frac{(x-3)^n}{3^n \sqrt{n+1}}$$ Its interval of convergence is of one of the forms $(a,b)$, $(a,b]$, $[a,b)$ or $[a,b]$. What is $a$? What is $b$? I know the interval on ...
1
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2answers
251 views

The Integral Test enables us to bound the error approximation of the series

$$S = \sum\limits_{n=3}^\infty \frac{1}{ n(\ln{n})^5}$$ One partial sum is given by $$S_{40} = \sum\limits_{n=3}^{40} \frac{1}{n(\ln{n})^5}$$ What upper bound does it yield for the error ...
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1answer
369 views

What is the smallest value N for which we can guarantee that the error approximation of the alternating series?

$$S=\sum_{n=1}^\infty\frac{(-1)^n}{n^{7/2}}$$ $$\text{and by the partial sum }S=\sum_{n=1}^N\frac{(-1)^n}{n^{7/2}} \text{ is at most } 10^{-2}?$$
1
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1answer
205 views

Question over function twice differentiable if $D^2 f$ is constant

Let $E$ and $F$ be normed spaces. What can you say of a function $f:A\subseteq E\to F$ with $A$ open in $E$ twice differentiable, if $D^2 f$ is constant? This is a very open question that do not ...
2
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4answers
101 views

find minimum of given function

today my relative asked a problem,which had strange solution and i am curious, how this solution is get from such kind of equations. let say function has form $f(x)=a\sin(x)+b\cos(x)$ we should ...
1
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1answer
138 views

Proving the composition of two functions having partial derivatives has a partial derivative.

Let $N$ be open subset of $\Bbb R^n$, $x \in N$ The function $f : N \to \Bbb R$ has a partial derivative at point $x$ Let $I$ be open interval in $\Bbb R$ with $f(N) \subset I $ The function ...
4
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3answers
89 views

Integrating $\int^{e^3-1}_{0}\frac{dt}{1+t}.$

How can I integrate $$\int^{e^3-1}_{0}\frac{dt}{1+t}.$$ I tried to make $u=1+t$ which means that $du=dt$ but it's not giving me anything useful, but instead made things more complicated. Maybe I did ...
2
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2answers
120 views

Finding $\frac{dy}{dx}$ of $f(x)=(x^3-2x)^{inx}$

This is the function: $$f(x)=(x^3-2x)^{inx}$$ Is it possible for me to solve the problem with chain rule? Or there's another approach for this question? the answer is ...
1
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1answer
71 views

non differentiable, integrable function

Can a function be unable to be differentiated, but is integrable? By unable to be differentiated, I mean at any arbitrary x coordinate. Thank you
2
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3answers
146 views

Taking the derivative of $f(x)=x^{e^{e^x}}$

How can I take the derivative of $f(x)=x^{e^{e^x}}$? How do I apply the chain rule? Thanks for the help!
0
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1answer
45 views

Checking my question related to partial derivatives

I have a question. Is the soltion way true? If it is true, how do I show what I say formally mathematical way? Or if it is false, what is the solution? Please show me explanatorily. Thank you ...
0
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1answer
44 views

combination into an aggregate equation

Radiation flux ids defined by, $$\phi = - \frac{\dot E}{S} = \frac {1}{2} (\omega_{rad})^2\frac{ \omega_{rad}}{k_{rad}} = \frac {1}{2} \delta^2 \Omega(\omega_{rad})^2\frac{ \omega_{rad}^3}{k_{rad}} ...
2
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0answers
92 views

Checking my question: the function is continuously differentiable.

I solved a question related to first order partial derivatives. Please check my solution. Is it correct and ehough to get sufficient grade from an exam? I am not sure espacially part-b. Please check ...
0
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1answer
47 views

Trouble simplifying the pdf of minimum exponentially distributed r.v.

Given the following: $ X_i \sim EXP(1, \eta) $ Asked: show that $Q=X_{1;n}-\eta$ is a pivotal quantity. My approach: $\ \ \ f(x)=e^{-(x-\eta)} \Rightarrow F(x)=\int_{-\infty}^x ...
2
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1answer
46 views

How The Jacobian of the transformation can be shown to not depend on $X_i$ or $\bar X $ and is equal to the constant $n$

Transform the random variables, $X_i$, $i=1,2,\ldots,n$ to $$ \begin{align} Y_1 & =\bar X \\ Y_2 & =X_2-\bar X \\ Y_3 & = X_3-\bar X \\ & {}\ \vdots \\ Y_n & =X_n-\bar X ...
2
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4answers
304 views

Substitution for $\int \frac {dx} {ax^2 + bx + c}$

I'm looking for the substitution that makes easier to solve integral containing quadratic polynomial in denominator (!) when such polynomial cannot be broken into parts (if it can, then it's possible ...
1
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2answers
77 views

Taking the derivative of $x^{\sin(e^x)}$

How am I suppose to take the derivative of $f(x)=x^{\sin(e^x)}$? What should I make $u$ equals? I tried to make $u=\sin(e^x)$ and $u=e^x$ but they didn't work.
0
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1answer
65 views

A simple inequality in about integer part of numbers?

This question follows A simple inequality in calculus?. I have to solve this inequality in about $s$: $$\left(\left[\dfrac{r}{s}\right] + 1 \right) s \le 1,$$ ...
0
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1answer
82 views

Solution of $\tan x^3 = -\frac 32 x^3$?

How to solve $\tan x^3 = -\frac 32 x^3$? Could you give me advice?
1
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2answers
78 views

A simple inequality in calculus?

I have to solve this inequality: $$\left(\left[\dfrac{1}{s}\right] + 1 \right) s < 1,$$ where $ 0 < s < 1 $. I guess that $s$ must be in this range: $\left(0,\dfrac{1}{2}\right]$.But I ...
5
votes
0answers
280 views

Prove the equation: $\frac{2}{\pi} \int_0^\infty \frac{\cos kr - ak \sin kr}{k^2a^2 +1} \ldots $

Prove the following equation: \begin{equation} \frac{2}{\pi} \int_0^\infty \frac{\cos kr - ak \sin kr}{k^2a^2 +1} \left (\int_0^\infty \cos kr' \left [u(r')-au'(r') \right] dr' \right ) dk =u(r) . ...
1
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2answers
71 views

Pseudorandom numbers - Get a sequence of number not close each others

I have this function inside a software: a+(SeededRand(Round((SongTime-0.5+(Round(1000*c)))*1))-0.5)*b it creates a pseudo-random numbers sequence inside the ...
1
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1answer
204 views

How to properly evaluate the Riemann sum for the integral of $x^2$ for $x$ from $0$ to $3$

In my homework we start out with $$\int_{x=0}^3 x^2 \, dx=\lim_{P: \Delta x \to 0} \sum_{i = 1}^n f(x_i) (\Delta x)_i$$ Where I take $$P_i=[\frac{i-1}{n},\frac{i}{n}], x_i=\frac{i}{n}, (\Delta ...
0
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1answer
54 views

Name of integration techniques

Consider following integrations : $$\int \sec^3 x\ dx,\ \int \cos\ x\ e^x\ dx $$ These can be calculated by integration by parts. But here for instance to calculate the latter example, we meet ...
4
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3answers
185 views

calculate the derivative using fundamental theorem of calculus

This is a GRE prep question: What's the derivative of $f(x)=\int_x^0 \frac{\cos xt}{t}\mathrm{d}t$? The answer is $\frac{1}{x}[1-2\cos x^2]$. I guess this has something to do with the first ...
1
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1answer
77 views

convergent $f(a_n)$ for divergent $a_n$

Could you give an example of a divergent $a_n$ such that: 1) $\exp (a_n)$ is convergent 2) $a^2_n - a_n + 1$ is convergent $a_n$ need not to be the same for two cases.
2
votes
1answer
207 views

uniform convergence of $\sum\limits_{n=1}^{\infty}3^n\sin\left(\frac 1 {4^nx}\right)$ in $[1,\infty)$

I want to check the uniform convergence of $\sum\limits_{n=1}^{\infty}3^n\sin\left(\frac 1 {4^nx}\right)$ in $[1,\infty)$. The hint in the book was using Cauchy condition for uniform convergence. For ...
3
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2answers
250 views

What does δA mean in differentiation?

To be more specific, I met this when doing analytical mechanics involving the principle of least action:
2
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1answer
428 views

Input integral derivative in Wolfram Alpha

How to input $\frac{d}{dx}(\int_0^x \sqrt{t^2-t+1} \,dt)$ in Wolfram Alpha? If i change $dt$ by $dx$ it works, but the output is $\sqrt{t^2-t+1}$, there is no substitution for "$t$" there, if i am ...
1
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3answers
85 views

The boundedness of $x^k e^{-|x|}$

What is a simple way of showing that: for every $k$, there exists $M$ such that $$\sup_{x \in \mathbb{R}} x^ke^{-|x|}< M?$$
0
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3answers
139 views

Proving the function f , which has zero first order parital derivatives, is constant

Let the function $f: \Bbb R^{2} \to \Bbb R$ The first order derivatives of f are zero. i.e $f_x(x,y)$ = $f_y(x,y)$ = $0$ How can I prove that $f(x,y)$ is constant for all $(x,y)$
3
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1answer
145 views

Derivative chainrule on khanacadamy ignoring some terms

I watched the chain rule series on khanacademy.org and decided to do the "questions". One of the questions is: Let $y = \sin(6x^2−4x−1+3x^{−1}−5x^{−2})$ $dy/dx=?$ The answer is ...
1
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2answers
196 views

solving an equation of the type: $t \sin (2t)=2$ where $0<t< 3 \pi$

Need to solve: How many solutions are there to the equation, $t\sin (2t)=2$ where $0<t<3 \pi$ I am currently studying calc 3 and came across this and realized i dont have a clue as to how to ...
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3answers
150 views

Proving a complex sum equals factorial

I have just stumbled across the equality that: $$ \sum_{j=0}^{n}(-1) ^ {n + j} j ^ {n} \binom{n}{j} = n! $$ How would I go about proving this equality? Also, what is the left hand side equal to if ...
2
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1answer
294 views

Proof of Riemann-Lebesgue lemma: what does “integration by parts in each variable” mean?

I was reading the proof of the Riemann-Lebesgue lemma on Wikipedia, and something confused me. It says the following: What does the author mean by "integration by parts in each variable"? If we ...
4
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2answers
130 views

If $\sum a_k^2 /k$ converges, then $1/N \sum_1^{N}a_k \to 0$

I want to show that if $\sum a_k^2 / k$ converges, then $1/N \sum_{1}^Na_k \to 0$. Now, if $a_n \to 0$, then the result follows. But of course $a_n\to 0$ is not a necessary condition for $\sum ...
7
votes
3answers
220 views

Evaluating $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{2^{2n-1}}{(2n+1)\cdot 3^{2n-1}}}$

Calculate the summation $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{2^{2n-1}}{(2n+1)\cdot 3^{2n-1}}}$. So I said: Mark $x = \frac{2}{3}$. Therefore our summation is $\sum_{n=1}^{\infty} {(-1)^n ...
1
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1answer
65 views

How does $a² + b² = c²$ work with 'steps'? [duplicate]

We all know that $a²+b²=c²$ in a right-angled triangle, and therefore, that $c<a+b$, so that walking along the red line would be shorter than using the two black lines to get from top left to ...
1
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2answers
160 views

Understanding differentials

What is a good reference to learn about differentials and related topics. Some of my questions are: Why is it possible to split $dy/dx$ into individual terms $dx$ and $dy$? In a separated ...
5
votes
2answers
431 views

Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$

$$\int \frac{dx}{\sqrt{x^2 -9}}$$ $x = 3 \sec \theta \implies dx = 3 \sec\theta \tan\theta d\theta$ $$\begin{align} \int \frac{dx}{\sqrt{x^2 -9}} & = \frac{1}{3}\int \frac{3 \sec\theta ...
0
votes
1answer
74 views

Calculating body volume? (3-variables)

How do we calculate a body's volume in 3 coordinates? For example: Find the volume of the following body: $T = \{(x,y,z) \in \mathbb R^3 | 0 \leq x, x^2 \leq y \leq \sqrt{x}, 0 \leq z \leq xy \}$. ...
1
vote
1answer
60 views

solving arduous limit

How to simplify the following: $$\lim_{x\rightarrow 0} \frac {\sin (\pi / 2 - 10 \sqrt x) \ln ( \cos (2x))}{(2^x -1)((x+1)^5-(x-1)^5)}$$ Here is what I've done: $$ \sin (\pi / 2 - 10 \sqrt x) = ...
1
vote
1answer
42 views

evaluation of limit $\lim \frac 1n \cdot \frac {x^3}{1+x^2}$

$$\lim_{n \rightarrow \infty } \frac 1n \cdot \frac {x^3}{1+x^2}$$ I think that for any fixed x the limit is $0$ due to increasing n. But I don't feel confident. Could you help?
2
votes
4answers
178 views

How many times does the function $y=e^x $ meet $y=x^2$?

As you know $y=e^x$ and $y= x^2$ meet once on $x<0$. But I want to know whether or not they meet on $x>0$. Since $\lim_{x\rightarrow \infty } e^x/x^2=\infty$, if they meet once on $x>0$, ...
0
votes
3answers
121 views

Recurrence relation with periodic function

$$x_{n+1} = x_n + \sin x_n$$ $$x_{n+1} = \sin \left(\frac {\pi} {2} x_n\right)$$ How to solve these? Or, at least, what can be said about thier behavior and limits?
2
votes
2answers
70 views

Understanding Continuity of Functions

I know that graphically a function $f(x)$ is said to be continuous in $[a,b]$ if there are no breaks in the curve for $f(x)$ in the interval $[a,b]$ I also know that by definition, a function $f(x)$ ...