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

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1
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2answers
41 views

Whether the sequence following is convergent?

$c<-3$ is a real number, $\{x_n\}$ is a sequence of real number, $\displaystyle x_1=\frac{c}{2}$, and $\displaystyle x_{n+1}=\frac{c}{2}+\frac{x_n^2}{2}$. Whether $\{x_n\}$ is a convergent ...
1
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1answer
263 views

Work on Springs using Hooke's Law

I'm currently stuck on parts c and d of this problem. The problem says Suppose a force of 20 N is required to stretch and hold a spring 0.4 m from its equilibrium position (0). I found k constant to ...
0
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1answer
28 views

Inverse functions and their values?

I have these following questions and I managed to do about half of them, but the problem is that I am not confident in the correctness of them. Can someone help me with the ones that are incorrect or ...
0
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1answer
58 views

Finding an expression for a joint probability if two random variables have the same distribution function.

If $X$ and $Y$ are independent random variables with the same distribution function, say $F$, find an expression for $P(X<t, Y<t)$. My attempt: $P(X<t, Y<t) = P(X<t)P(Y<t) = ...
1
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2answers
54 views

Find the derivative of x^1/5 from the definition

I've been trying to figure out how to compute the derivative of $f(x) = x^{1/5}$ at $x=1$ from the definition. Here's what I've done: $$f'(1) = \displaystyle \lim_{\Delta x\rightarrow 0} ...
42
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3answers
2k views

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Show that : $$ \int_{0}^{\Large\frac\pi2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over ...
3
votes
2answers
86 views

Sequence and Limit

If $\lim\limits_{n\to\infty}a_n=1$, $0\leq a_n\leq 1 \; \forall n\in N,$ then what about the $\lim\limits_{n\to\infty}(a_n)^n$. Is this limit exists? If yes then what is the value of this limit?
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1answer
51 views

Finding area between two cosine curves

I must to find the area between these two curves: $$y = 2 \cos 7x, y = 2 − 2 \cos 7x$$ $$0 ≤ x ≤ π/7$$ And this is all I have so far: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
1
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1answer
69 views

Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
3
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3answers
121 views

Evaluating the limits $\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$ and $\lim_{(x,y)\to(\infty,8)}(1+\frac{1}{3x})^\frac{x^2}{x+y}$

I got the following problem: Evaluate the following limits or show that it does not exist: $$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$ and ...
2
votes
3answers
85 views

How do I compute $\displaystyle\lim _{x\to 0} \tfrac{e^x+\sin x -1}{\ln(1+x)}$?

I'm a Calculus I teacher's assistant. One of my students asked me how to compute this limit $$\lim _{x\rightarrow 0} \dfrac{e^x+\sin x -1}{\ln(1+x)}$$ I could not solve it. I need some hint. P.s: ...
0
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2answers
35 views

Limit of |x-2| as x approaches -2

I believe that it equals -4. In the epsilon-delta definition, we can set delta equal epsilon and I become this satisfies the definition. The problem is I can't seem to prove based on this that 0 less ...
1
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0answers
35 views

Explicit solution to a nonlinear equation possible here?

I am looking for a solution in $s$ to $$ \lambda -\frac{1}{s} +K e^t \log(\delta) \delta^s = 0 $$ Mathematica is not best pleased with this equation. If the equation were $$ 0- \frac{1}{s} +K e^t ...
1
vote
2answers
57 views

If $\int_{0}^{x}\{t\}dt = \int_{0}^{\{x\}}tdt$ and $x>0$ and $x\notin \mathbb{Z}$, Then $x$ is

If $\displaystyle \int_{0}^{x}\{t\}dt = \int_{0}^{\{x\}}tdt$ and $x>0$ and $x\notin \mathbb{Z}$ and $\{x\} = $ fractional part of $x$ i.e $\{x\} = x-\lfloor x \rfloor $. Then value of $x$ is ...
1
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1answer
28 views

How do I show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$

I want to show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$ for real-valued $g$ that is continuous for $0\leq x\leq1$ . Is it enough to say that $|g(x)| \leq \sup_{0\leq x\leq1}|g(x)|$ ...
2
votes
1answer
38 views

Sketch parametric curve

An exercise in my textbook asks to sketch the parametrical curve of the following equation: $$x=e^t\cdot\cos(t)\\y=e^t\cdot\sin(t)\\t\ge0$$ I would usually try to solve one of the equations for t and ...
0
votes
1answer
71 views

Parametrize a curve with respect to arc length

“It is often useful to parametrize a curve with respect to arc length because arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system.” Quoted ...
5
votes
3answers
382 views

Limit of iteration of sine function

Let $f(x) =\sin x $, and denote by $f^n = f\circ f\circ ...\circ f$ the n-th iteration of function $f.$ Find the limit (if exist) :$$\lim_{n\to\infty} n\cdot f^n (n^{-1} )$$
0
votes
1answer
42 views

if $\lim _{n\to \infty }\left(b_n\right)=L$ than $\lim _{n\to \infty }\left(\frac{1}{b_n}\right)=\frac{1}{L}$?

it's begginer question i know but very important for me to understand: $b_n\:\ne 0,\:L\ne 0$ if $\lim _{n\to \infty }\left(b_n\right)=L$ than $\lim _{n\to \infty ...
5
votes
3answers
299 views

Intuitively, how do you explain the concept of Flux?

Lately in my physics and mathematics classes, I've come across the concept of Flux. And although I've been able to define them mathematically and figure out how to use them. I'm still not entirely ...
0
votes
2answers
27 views

Substituting logs

If $b=log_3(x),$ what value of $x$ satisfies $log_b(log_3(x^2))=3?$ I just started learning this topic by myself. I wanted to know if my working is correct. If not can someone help me with this ...
0
votes
1answer
85 views

Find the derivative of $x^{2/5}$?

How do you find the derivative of $x^{2/5}$ using the definition of the derivative?
0
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2answers
33 views

$\lim_{h\to0} \frac{\sqrt[3]{x+h+3}-\sqrt[3]{x+3}}{h}\;?$

What is the fraction that have to multiply to calculate the limit $$\lim_{h\to0} \frac{\sqrt[3]{x+h+3}-\sqrt[3]{x+3}}{h}\;?$$
1
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2answers
57 views

Evaluation of $\int\frac{1}{\sin^2 x\cdot \left(5+4\cos x\right)}dx$

Evaluation of $\displaystyle \int\frac{1}{\sin^2 x\cdot \left(5+4\cos x\right)}dx$ $\bf{My\; Solution::}$ Given $\displaystyle \int\frac{1}{\sin^2 x\cdot (5+4\cos x)}dx = \int \frac{1}{(1-\cos ...
0
votes
3answers
37 views

Factoring and algebra

I'm reading Calculus for dummies but got stuck on the following The line "Factor x out of the denominator." is the part I do not understand. How is it possible to go from sqrt(x^2 + x) + x to ...
0
votes
0answers
25 views

Is there a reason why M can't be all summable sequence?

Let M be the set of all summable non-negative sequences $\{x_k\}_{k=1}^\infty$ of real numbers, that is, $x_k \geq 0$ for all k and $\sum_{k=1}^\infty x_k$ converges to a real number. Let $d:M \to ...
2
votes
2answers
64 views

What is $d(\sin(x),\cos(x))$ if d is a distance function in a metric space?

Let $M=\{f:[a,b] \to \textbf{R} | f \,is \,continuous \}$. Let $d:M \to \textbf{R}$ be defined by $d(f,g)=\int_a^b |f(x)-g(x)| \,dx$. What is d represent geometrically, and show that M, d is a metric ...
0
votes
1answer
53 views

When does this integral converge?

So I'd like to find out for which values of $a,b>0$ the following integral is well-defined and how that will change if the absolute value is removed? Thanks! $I = \int_0^\infty ...
2
votes
1answer
64 views

Solving Nonlinear second order ODE

I want to know how to solve this nonlinear second order ODE. This example is based on the option pricing under the CEV model. $$ \frac{1}{2}\sigma^2(x)u''(x)+\mu u'(x)-Cu(x)=-g(x) $$ where $\mu, C$ ...
0
votes
1answer
74 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
2
votes
2answers
96 views

Special values $\psi \left(\frac12\right)$ and $\psi \left(\frac13\right)$

I wonder if it is easy to prove that $$ \begin{align} \psi \left(\frac12\right) & = -\gamma - 2\ln 2, \\ \psi \left(\frac13\right) & = -\gamma + \frac\pi6\sqrt{3}- \frac32\ln 3, \end{align} ...
3
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0answers
38 views

How to solve integrals of the form $\int u^{-\alpha} e^{-\beta u} du$?

I have to simplify an integral of the form $\int u^{-\alpha} e^{-\beta u} du$, where $\alpha, \beta \in \mathbb{R}^{++}$. Is it a standard integral, or a family that subsumes gamma integrals? Is there ...
2
votes
1answer
74 views

How to find limit of this integral???

I had this problem on my last exam,and I couldn't do it: $$\begin{align} \lim_{x \rightarrow \infty}\frac{\int_{0}^{x} e^{t^{2}}dt}{x^{5}\int_{0}^{x^{2}}\frac{e^{t}}{t^{2}} dt} \end{align}$$ ...
0
votes
1answer
38 views

Find the parametric equation of the following parabola?

It doesn't give me $2$ equations this time just $1$ and I have no clue what to do; $y^2 = 4x$ ANSWER IN BOOK: $x = t^2, y = 2t$
0
votes
4answers
68 views

Can someone explain what this question is asking me? (Calculus II)

How to define the area of the region bounded by $y = \sqrt{1-x^2}$ and $y = 0$? I am assuming we will be using Riemann's sum however I am not sure if we are given $[a,b]$ or is that what we are ...
0
votes
1answer
49 views
1
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3answers
47 views

$\frac{\partial \coth ^{-1}(x)}{\partial x}$

I am asked to find $\dfrac{d\coth ^{-1}(x)}{dx}$ I rewrite it to become $x=\dfrac{1}{\tan(y)}$ $\dfrac{\text{dx}}{\text{dy}}=-\dfrac{1}{\frac{\sec ^2(x)}{\tan ^2(x)}}=-\sin^2(x)$ However the ...
0
votes
1answer
28 views

existence of a differentiable function under certain given condition

Does there exist any $f:\mathbb{R}-\mathbb{R}$ be a differentiable function such that exist $x\neq 0$ with $f(x)=0$ and at any neighborhood of $x$ exist infinitely many point where $f$ does vanish ...
1
vote
4answers
84 views

If $f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and if $lim_{x\to\infty}f'(x)=0$ Then $f$ uniformly continuous on $[0,\infty)$

I got this problem: Let $f$ be a continuous function on $[0,\infty)$ and differentiable function on $(0,\infty)$ such that $\lim_{x\to\infty}f'(x)=0$. (1) Prove that for each $0<\epsilon$ there ...
2
votes
1answer
51 views

How to simplify this complex integral? [closed]

How to approximate this integral as a function of a and b? $$\int_0^\pi\int_0^{2\pi}\sqrt{(a-b\sin\varphi\cos\theta)^2+(b\cos\varphi)^2+(b\sin\varphi\sin\theta)^2}d\theta d\varphi$$ where a and b ...
2
votes
3answers
48 views

Showing that $x+ cos x - 1 > 0$ for all $x > 0$

I got this problem: Show that for all $0<x$, $0<x+cos x - 1$ I tried to show it several times but none worked. I showed that $lim_{x\to\infty} (x+cos x - 1) = \infty$ by using the Squeeze ...
0
votes
3answers
120 views

How to solve $ \frac{\left( \sqrt{2} \right) ^a}{a}=1$

I know that $2$ and $4$ solve this problem. I also assume that there real solutions. However I don't know how to bring $$ \frac{\left( \sqrt{2} \right) ^a}{a}=1$$ into a form so that the solutions, ...
1
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1answer
44 views

Volume of a solid bounded by surfaces - is it correct?

Could you check if my calculations and reasoning are correct. And maybe suggest a nicer way of solving this problem? We are given a solid bounded by these surfaces: $y=x^2, \ y=1, \ 2x+y+z = 4, \ ...
2
votes
2answers
87 views

Simplify $\sinh (\log (x))$

$$\sinh (\log (x))=\frac{x^2-1}{2 x}$$ However I do not see how this is done, here is an idea I had but I'm probably way off: $$\sinh \left(\ln \left(\frac{1}{2} ...
0
votes
2answers
76 views

Parallel vectors in $\mathbb{R}^n$.

Def: We say that $\vec{x},\vec{y}\in\mathbb{R}^n$ are parallel vectors if $|\vec{x}\cdot \vec{y}|=||\vec{x}||\,| |\vec{y}||$. (i.e equality holds in Cauchy–Schwarz inequality) I'm having some ...
5
votes
0answers
66 views

If Graham's number used $4$s instead of $3$s, at which $G$ would that number be bigger than Graham's number?

If $3\uparrow \uparrow\uparrow\uparrow3=G_1$, $ \quad G_2=\underbrace{3 \uparrow \ldots\uparrow3}_{G_1 \ \text{times}}, \quad G_3=\underbrace{3 \uparrow \ldots\uparrow3}_{G_2 \ \text{times}}$ , $ ...
1
vote
2answers
34 views

Construct function with known asymptotes

Suppose I have two lines, described by $$ f_1(x)=r_1(x-x_0)+y_0$$ $$ f_2(x)=r_2(x-x_0)+y_0$$ E.g, for $(x_0,y_0)=(1,\frac{1}{2})$, $r_1=\frac{1}{2}$ and $r_2=4$ it looks like: I am looking for a ...
1
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0answers
62 views

Integral of $\displaystyle \int \sin^{-1} \left (e^{\sqrt x}\right ) \mathrm{d}x$ …

How do I evaluate the following; $$\displaystyle \int \sin^{-1} \left (e^{\sqrt x}\right ) \mathrm{d}x$$ $$\displaystyle \int \sin^{-1} \left (e^{-\sqrt x} \right )\mathrm{d}x$$ Is there a closed ...
1
vote
3answers
36 views

$y=\sinh ^{-1}\left(\frac{x}{a}\right)$

I am struggling with two things regarding the derivative of sinh, but also other functions like it. $$y=\sinh ^{-1}\left(\frac{x}{a}\right)$$ $$x=a \sinh (y)$$ Know my question is, if take the ...
2
votes
0answers
26 views

Calculating a limit with exponential terms

Let $a,b,c$ be non-negative numbers. What are the conditions on $a,b,c$ so that $$ \lim_{n\to\infty}\left[1-e^{-na}-e^{-nb}\right]^{\exp(nc)} = 1. $$ Using the fact that $$ \log(1+x)\geq\frac{x}{1+x} ...