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

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0
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0answers
6 views

Calculus scenario involving instantaneous and speed (sequences)

The scenario is nearly always the same as Wilie is standing at the end of a road that is 1 kilometer long, and there at the other end is that Roadrunner, he’s just standing there, sticking his tongue ...
1
vote
1answer
24 views

How to express sum as triple summation

I am trying to express the following sequences as summations: $$ 1+2^2+3^2+4^4+5^4+6^4+7^4 $$ and $$ 1+(2+3)^2 + (4+5+6+7)^4 $$ as summations. I think they will likely be triple summations, so ...
0
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1answer
31 views

How do we prove $\int \frac{\ln(1+x)}{x}dx = -\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$?

After working on the integral $\int_{0}^{1} \frac{\ln(1+x)}{x}dx$ for a couple of hours, I became convinced its antiderivative was not elementary. So I looked it up on Wolfram Alpha, and it found that ...
3
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2answers
51 views

Hint on how to find $\int \frac{x^2}{1+x^2}dx$

I am almost sure that this would have been asked before, but how can one find $$ \int \frac{x^2}{1+x^2} dx? $$ If I had a $x^2 - 1$ in the denominator, then I could factor into $(x-1)(x+1)$ and use ...
2
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0answers
14 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
0
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3answers
51 views

Evaluate $\lim _{n\to \infty }\int_1^2\:\frac{x^n}{x^n+1}dx$

We have $$I_n=\int _1^2\:\frac{x^n}{x^n+1}dx$$ and we need to find $\lim _{n\to \infty }I_n$. Have any ideea how we can evaluate this limit?
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1answer
31 views

sum of a geometric series

check if the series converges and if so what is the sum $$\sum_{n=1}^\infty \ln(6^{\frac{1}{4^{n}}})$$ By the ratio test: $$\frac{\frac{1}{4^{n+1}}\cdot \ln{6}}{\frac{1}{4^{n}}\cdot ...
0
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0answers
7 views

$\lim_{T \to \infty} \frac{\sin (T(x+i(c-1)i))}{T(x+i(c-1)i}$?

How can I calculate $\displaystyle\lim_{T \to \infty} \frac{\sin (T(x+i(c-1)i))}{T(x+i(c-1)i}$? I´ve seen that $\displaystyle\lim_{T \to \infty} \frac{\sin (T(x-a))}{T(x-a}=1_a x$ Is this correct?
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5answers
19 views

How do I find the radius and interval of convergence of $\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $

$$\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $$ I used the ratio test to test for absolute convergence, but I'm sort of stuck on: $$n(x+2) \over n+1$$
4
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0answers
16 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
2
votes
4answers
53 views

sum of a telescoping series

calculate the following $$\sum_{n=1}^{\infty} \ln \left ( \frac{\left (\frac{n+3}{n+1}\right ) ^{n+1}}{\left (\frac{n+2}{n}\right )^n} \right )$$ I have manage to written it as ...
0
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1answer
52 views

Find the integral: $\int \frac{( x-6)^2}{x^4}\mathrm{d} x$

Find the integral: $\int \frac{(x-6)^2}{x^4}\mathrm{d} x$ I have so far $\int (u)^2(u-6x)^{-4}\mathrm{d} x$ $u= x-6$ and $du=dx$ and $u-6=x$ Am I on the right track?
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4answers
39 views

Finding the derivative of an inverse function.

Let $f(x) = (-x^2)/(x^2+1)$. If g(x) is the inverse function of f(x) and f(1)=-1/2, what us g'(-1/2)? Can someone explain how to do the above problem as I am not even sure where to start. Would I ...
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1answer
22 views

Multiplying logarithms of different bases

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
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2answers
29 views

fifth order Differential EQuation

find the general solution of higher order linear differential Equation? find the general solution of Differential equation using auxiliary equation? $2y^{(5)}-7y^{(4)}+12y"'+8y"$=0
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2answers
43 views

A Sequence That has No Upper Bound But Does Not Tend To Infinity

Let there be a sequence $a_n$ which has no upper bound. Give an example to a sequnace that disprove $a_n\rightarrow \infty$ Any hint?
1
vote
2answers
44 views

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$ Can I multiply and distribute the $ \ x^{7/2}\ $ and $ \ sec^2 \ $ together. What is the strategy to solve this problem.
1
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1answer
23 views

Evaluating the sum of a partial geometric sequence using Sigma notation

I have a worksheet from my instructor with this problem on it, but the solution he has given is different from what I got, and I don't know why. I'm not sure how to input the Greek letter sigma, but ...
0
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1answer
10 views

Darboux sums inequality with relation to Sup|f'(x)|

Assuming f is continuous on [a,b] and differential on (a,b) and assuming f ' is bounded on (a,b) ; denote k = sup(|f '|) prove that, for all P a partition of [a,b]: 0 ≤ U(f,P) - L(f,P) ≤ k(b-a)Δ(P) ...
6
votes
5answers
40 views

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly ...
1
vote
1answer
32 views

Are there only a few 'universally convergent' Taylor Series?

The taylor series for $sin(x)$, centered at any point, converges for all x. The taylor series for $e^{x}$ and $cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
4
votes
2answers
28 views

Roots of unity, where $\omega^3 = 1, \omega \neq 1$.

Say that $\omega^3 = 1$ and $\omega \neq 1$. Find the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$. I'm not very good at the roots of unity. May I have a couple of hints to get started? ...
1
vote
5answers
26 views

Proving convergence/divergence via the ratio test

Consider the series $$\sum\limits_{k=1}^\infty \frac{-3^k\cdot k!}{k^k}$$ Using the ratio test, the expression $\frac{|a_{k+1}|}{|a_k|}$ is calculated as: $$\frac{3^{k+1}\cdot ...
0
votes
4answers
21 views

Difference between $\nabla T$ and $\nabla \cdot E$

Why is $\nabla T = (\frac{\delta T}{\delta x},\frac{\delta T}{\delta y},\frac{\delta T}{\delta z})$, but $\nabla \cdot E \neq (\frac{\delta E}{\delta x},\frac{\delta E}{\delta y},\frac{\delta ...
0
votes
1answer
27 views

Partial ordering of functions

Let $X$ be the set of all real-valued functions $x$ on the interval $[0,1]$ and let $x \leq y$ mean that $x(t) \leq y(t)$ for all $t \in [0,1]$. Does it define a partial ordering/ total ordering? Does ...
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2answers
38 views

For a strictly increasing function $f$, prove that $f(x)=cx, x\in\mathbb Q\implies f(x)=cx,x\in\mathbb R$.

For a strictly increasing function $f:\mathbb R\to\mathbb R$, prove: If $f(x)=cx$ for $x\in\mathbb Q$, then $f(x)=cx$ for $x\in\mathbb R$. I found this statement on M.SE. How can I prove ...
0
votes
3answers
41 views

Integral of $\cosh^3(x)$

What is the integral of $\cosh^3(x)$? And how exactly can I calculate it? I've tried setting $\cosh^3(x)=(\frac{e^x+e^{-x}}{2})^3$ but all I get in the end is one long fraction.
2
votes
4answers
45 views

Find the integral: $\int_0^{1/2} x \sin(\pi x^2)\,dx$

Evaluate the integral $$\int_0^{1/2} x \sin(\pi x^2)\,dx$$ I have: $-\cos (1/4) -1$.
1
vote
1answer
25 views

How calculate: $\lim \limits_{(x,y) \to (0,0)} \frac{ln(1+2x^2+4y^2)}{arctan(x^2+2y^2)} $?

How do I calculate: $\lim \limits_{(x,y) \to (0,0)} \frac{\ln(1+2x^2+4y^2)}{\arctan(x^2+2y^2)} $? Is there any certain path that is recommended to go through? I tried to use the inequality: $\ln(1+x) ...
2
votes
1answer
40 views

Integrate $\int_{0}^1 (1 + 4y^2)^{1/2} dy$ [duplicate]

$$\int_{0}^1 (1 + 4y^2)^{1/2} dy$$ So, how do I integrate this without the use of trigonometrical substitution? Can anybody give me a hint? Thank you!
2
votes
1answer
58 views

Trouble solving an integral

So I have been trying to solve this equation, The given answer is, I began by using substitution to change the integral. Substituting t back in where t is taken from 0 to infinity. ...
2
votes
2answers
32 views

Computing limits using Monotone Convergence theorem

I am trying to compute the limits of $\lim_{n \rightarrow \infty} \int\limits_0^{\infty} \dfrac{1}{(1+\dfrac{x}{n})^n \sqrt[n]x}dx $ by using Monotone convergence theorem of integrals and switching ...
0
votes
2answers
28 views

How would I find power series $\sum_{n=0}^\infty {(3^nx^n) \over n!} $ radius and interval of convergence

$$\sum_{n=0}^\infty {(3^nx^n) \over n!} $$ I have no idea how to start this problem, the only thing that looks familiar is $x^n/n!$ which I know as a sequence goes to 0 when you take the limit, but I ...
2
votes
1answer
24 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?

We call two metrices equivalent if for all sequences $x_n,y_n\in\mathbb R$ it holds $\lim_{n\to\infty}d_1(x_n,y_n)=0 \iff\lim_{n\to\infty}d_2(x_n,y_n)=0$ . I have given $d_1(x,y)=|x-y|$ and ...
1
vote
2answers
43 views

Why the continuity of $f$ is not a necessary condition?

I am quite new to functions and continuity, and now I am reading the slides regarding the intermediate value theorem, which is related to continuity of functions. While reading, I found the ...
0
votes
4answers
31 views

How can I find the radius and interval of convergence of $\sum_{n=1}^\infty {(3x-2)^n \over n} $, and for what value x would it converge to?

$$\sum_{n=1}^\infty {(3x-2)^n \over n} $$ Not sure where to start with this problem. I'm thinking the ratio test because the numerator is raised to n, but n is also in the denominator.
2
votes
1answer
48 views

Is it true that $\int_a^b f(y)y' dx = \int_{y(a)}^{y(b)} f(y)dy$?

For a couple of years now I've been using the formula $$\int_a^b f(y)\frac {dy}{dx} dx = \int_{y(a)}^{y(b)} f(y)dy$$ and if asked about it I'd wave my hand saying it comes from the FTC. But the FTC ...
1
vote
1answer
36 views

Understanding a particular transformation of an integral given in a proof

Using the theorem of mean values find the sign of the integral... $$\int_{0}^{2 \pi}{\sin x \over x}dx= \int_{0}^{\pi}{\sin x \over x}dx+\int_{\pi}^{2 \pi}{\sin x \over x}dx$$ Then: $[x-\pi=t ; ...
1
vote
3answers
41 views

Evaluate the integral $\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x$

Evaluate the integral $$\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x.$$ If I transfer $(x^2 + 4)^5$ to the numerator, how do I integrate?
1
vote
0answers
10 views

Existence of real valued function continuous at $\mathbb Q$ discontinuous at $\mathbb R\backslash \mathbb Q$ [duplicate]

Does there exist a real-valued function of a real variable which is continuous at every rational point and discontinuous at every irrational point?
2
votes
3answers
48 views

Find the integral: $\int ( 4x -1 +3 \sqrt{x})\mathrm{d}x$

I have to find the following integral: $$\int ( 4x -1 +3 \sqrt{x})\mathrm{d}x$$ My answer is $2x^2 -\ 1x + \frac{2\sqrt{27}}{3}$. Am I right?
3
votes
1answer
47 views

$\int_a^b f(x) g'(x) dx = 0$ implies $f$ is constant

Given $f$ is continuous on $[a,b]$, $\forall g$ which is a continuously differentiable function on $[a,b]$, with $g(a)=g(b)=0$, the following equation is satisfied: $\int_a^b f(x) g'(x) dx = 0$. I ...
1
vote
0answers
17 views

If $\lim_{x \to a}(f(x)+g(x))=\lim_{x \to a}(f(x)g(x))=0$ then $\lim_{x \to a}f(x)=\lim_{x \to a} g(x)$?

Is the following true? If $\lim_{x \to a}(f(x)+g(x))=\lim_{x \to a}(f(x)g(x))=0$ then $\lim_{x \to a}f(x)=\lim_{x \to a} g(x)$ My attempt: Couldn't come up with a counter example. Also, solving ...
1
vote
2answers
31 views

Stationary points of $ \ln(x + 1)$?

I'm trying to find the stationary points of $f(x) =\ln(x + 1)$. When I differentiate, I get $f'(x) =\dfrac{1}{ (x + 1)}.$ I then set that to zero and end up getting $1 = 0$? I'm not sure what this ...
1
vote
1answer
40 views

Find $f(x)$ given $f, g$ such that $\,f(0) =2,\, g(0) =1, \, f'(x) = g(x),\, g'(x) = f(x)$.

Let $f$ and $g$ be functions satisfying: $$\begin{align} f(0) & =2\\ g(0) &=1 \\ f'(x) &= g(x) \\ g'(x) & = f(x) \end{align}$$ Find $f(x)$.
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votes
0answers
45 views

Mathematics doubt. [on hold]

What is mathematics. I need a proper definition mathematics.
0
votes
0answers
10 views

Limit of a matrix valued function

The limit of a vector valued function $f:\mathbb{R} \to \mathbb{R^n}$ is defined as: $$\lim_{x \to a} f(x)=(\lim_{x \to a}f_1(x),\dots,\lim_{x \to a}f_n(x))$$ provided that the limits of $f_i(x)$ ...
1
vote
1answer
51 views

Explanation for absolute value

So $f_a:R\rightarrow \:R,\:f_a(x)=\:\frac{1}{\left|x-a\right|+3}$, and we have to evaluate $\lim _{a\to \infty }\int _0^3\:f_a\left(x\right)dx$. But $\left|x-a\right|\:$ is equal with: ...
16
votes
2answers
600 views

Is it mathematically valid to separate variables in a differential equation?

I read the following statement in a book on Calculus, as part of my mathematics course: Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting ...
0
votes
3answers
32 views

Calculate the radius of convergence of $\sum \frac{\ln(1+n)}{1+n} (x-2)^n$

Calculate the radius of convergence of the following: $$ \sum \frac{\ln(1+n)}{1+n} (x-2)^n $$ Will you please help me figure out how to calculate: $$ \lim_{n\to \infty} \frac{\ln(2+n)}{2+n} ...