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

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0
votes
2answers
52 views

The supremum of set $A=\left\{ \left(\frac{m+n+1}{m+n}\right)^{m+n};\ m,n\in\mathbb{N}^* \right\}$.

Let $A=\left\{ \left(\dfrac{m+n+1}{m+n}\right)^{m+n};\ m,n\in\mathbb{N}^* \right\}$. Calculate the supremum of $A$ ($\sup A$). I tired To find the first few elements of $A$ ...
-3
votes
0answers
63 views

why is $\frac {dy}{dx} dx = dy$? [duplicate]

if $y$ is a function of $x$, why is $\frac {dy}{dx} dx = dy$? This has been bugging me. Why is it you can treat that as a fraction? I would like the traditional calculus view first if possible, then ...
-1
votes
1answer
77 views

find $\mathop {\lim }\limits_{n \to + \infty } \frac{1}{{n^2 }}\sqrt[n]{{\frac{{3n!}}{{n!}}}}$ [on hold]

help me please $\left(3\right)$ $$\mathop {\lim }\limits_{n \to + \infty } \frac{1}{{n^2 }}\sqrt[n]{{\frac{{3n!}}{{n!}}}}$$ $\left( 4 \right)$$$\mathop {\lim }\limits_{n \to + \infty } \left( ...
6
votes
2answers
123 views

Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?

My professor says that writing this is convenient $$\int \frac 1x \mathrm{d}x = \ln|x| + C\tag{1}$$ but wrong, since it should be written as: $$\int \frac 1x \mathrm{d}x = \begin{cases}\ln x + C ...
4
votes
2answers
81 views

How solve $\int \frac{dx}{(x^2-x)^x}$ [on hold]

I want solve $$\int \frac{dx}{(x^2-x)^x}$$. thanks for help
0
votes
0answers
43 views

basic calculus/analysis question. why does the multivariable chain rule work?

Say $f$ is a function of $x(t)$ and $y(t)$ $$\frac{ df}{dt} = \frac{ \partial f}{\partial x} \frac{ dx}{dt} + \frac{ \partial f}{\partial y} \frac{ dy}{dt}$$ why is it so additively symmetric? (The ...
1
vote
1answer
22 views

If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?

Let $f\in S_\infty\subset L_1(\mathbb{R},\mu)$ with $\mu$ as the Lebesgue linear measure be a Lebesgue-summable function such that $$\forall (p,q)\in\mathbb{N}^2_{\ge 0}\quad\exists C_{pq}>0: ...
0
votes
1answer
26 views

volume of solid of revolution about y-axis of region bounded by $x=1-y^2$ and the y-axis

Find the "volume of the solid that results when the region bounded by $x=1-y^2$ and the y-axis is revolved around the y-axis" This is from a worksheet that my teacher gave me. my attempt: ...
3
votes
2answers
254 views

basic calculus/analysis question. why is $\frac {dy}{dx} dx = dy$?

if $y$ is a function of $x$, why is $\frac {dy}{dx} dx = dy$? I have not learnt real analysis, but have done a bunch of math method courses at university and this has been bugging me. Why is it you ...
3
votes
2answers
52 views

Improper Integral of $\int\frac{dx}{(2x-1)^3}$

Improper Integral of $$\int_{-\infty}^0\frac{dx}{(2x-1)^3}$$ from Anton Calculus 8th Edition, page 576, question 9. Answer is $-\frac{1}{4}$ but I'm finding $-1$ The integral, substituting ...
0
votes
1answer
17 views

Subset bounded under linear transformation

Let $T:\mathbb R^n\to\mathbb R^n$ and let $B_r[0]=\{x\in \mathbb R^n : \left \| x \right \|\leq r\}$. Show that $T(B_r[0])$ is bounded. My proof is: $T$ is continuous and $B_r[0]$ is compact (by ...
1
vote
1answer
19 views

Volume of revolution about the line $y = 2$ of region bounded by $x=y^3$ and $y=x^2$

I need to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $x=y^3$ and $y=x^2$ about the line $y=2$. My Attempt: $\pi \int_0^1 ...
0
votes
0answers
38 views

Systems of First Order Linear Equations - Differential Equations

Consider the vectors $x^{(1)}(t) = (t,1)$ and $x^{(2)}(t) = (t^2, 2t)$ I computed the Wronskian which is t^2. But I was wondering how to solve the following questions: 1) In what intervals are ...
0
votes
2answers
21 views

Find the area bounded by the graphs $y^2=4+x$ and $y^2 + x = 2$

My attempt: $\int 2-y^2 + y^2-4 dy $ I changed the sign of $y^2-x=2$ as I think that the distances must be converted to absolute value. e.g. let $y = 0$, then x must be a positive?
1
vote
1answer
34 views

Can I use Fubini's theorem on this function?

Given function is $f(x,y)=\sin(x)e^{-xt}$. Problem says "use" Fubini's theorem and the fact that $\int_0^\infty e^{-xt} dt=\frac{1}{x}$ to prove that $$\lim\limits_{A\rightarrow\infty}\int_0^A ...
2
votes
1answer
38 views

''Differential equation'' with known solution $\sin$ and $\cos$

I am given the following two two equations $f,g : \mathbb{R} \to \mathbb{R}$ are differentiable on $\mathbb{R}$ and they satisfy $\forall x,y \in \mathbb{R}$ $$f(x+y) = ...
0
votes
1answer
45 views

Integrate $\frac{\lambda y^{2}}{\sqrt{2\pi}} e^{-(\frac{1}{2}+ \lambda x)y^{2}}$ with respect to $y$

$$F(x,y)= \frac{\lambda y^{2}}{\sqrt{2\pi}} e^{-(\frac{1}{2}+ \lambda x)y^{2}}$$ Please show that the function when integrated with respect to $y$ is $F_X(x)= \frac{\lambda}{(2\sqrt{2}(\frac{1}{2}+ ...
2
votes
0answers
93 views

Question about solutions of $y''+(w^2+b(t))y=0$ .

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
2
votes
2answers
54 views

Evaluate $\lim_{n\to\infty}\prod_{k=1}^{n}\frac{2k}{2k-1}$

How can I calculate the following limit: $$ \lim_{n\to \infty} \frac{2\cdot 4 \cdots (2n)}{1\cdot 3 \cdot 5 \cdots (2n-1)} $$ without using the root test or the ratio test for convergence? I have ...
0
votes
4answers
53 views

Find the limit as x approaches 0 of $4/x^2$ +$2 /(1-\cos(x))$

Find the limit as x approaches 0 of $4\over x^2$ +$2 \over (1-\cos(x))$ I get stuck on the third l'Hopital where one takes the limit as x approaches 0 of $4\cos(x)-4 \over 2-2\cos(x) +4x\sin(x) ...
0
votes
1answer
17 views

Possible Points of Local Extrema

I know that we should check critical numbers (points where f'(x) is either zero or not defined) and endpoints (for a closed interval) as possible points of local extrema of f(x). Obviously, all these ...
1
vote
2answers
106 views

How $\sqrt{x}$ is an function? [duplicate]

How $\sqrt{x}$ can be a function when $\sqrt{4}$ is equal to $-2$ and $2$?
0
votes
2answers
30 views

Solving a limit with taylor

I'm stuck in solving this limit $$\lim_{x\to0} \frac{(1+x)^{\frac1x} - e}{x}$$. Here I can must use Taylor expansion. My idea is to obtain the form $e^y-1$ on the numerator and then use Taylor ...
1
vote
3answers
40 views

Inflection point of $\,f(t) = \frac{1}{1+e^{(-t)}}$

I am trying to calculate the inflection point of the logistic function $f(t) = \dfrac{1}{1+e^{(-t)}}$. According to the definition given in Wikipedia, "A differentiable function has an inflection ...
1
vote
3answers
60 views

Using mathematical induction to show that for any $n\ge$ 2 then $\prod_{i=2}^n\left(1-\frac{1}{i^2}\right)=\binom{n+1}{2 \cdot n}$

I'm trying to work through some practice problems but I've been stuck on this for god knows how long now and I've no idea where to even start. Just wondering if it would be possible for someone to ...
0
votes
1answer
40 views

What is the difference between these two?

I have two sentences that looks almost the same but they differ a bit: $$a_n > 0$$ A) If $a_n\to\infty$, then $\sqrt[n]{a_n} > 1$ for almost any n B) If $a_n\to\infty$, then there is $c>1$ ...
1
vote
3answers
85 views

$ \mathop {\lim }\limits_{n \to + \infty } \frac{{v_{n + 1} }}{{v_n }} = 2$

help me please true or fulse (1)$$\mathop {\lim }\limits_{n \to + \infty } \sqrt[n]{{n\left( {n + 1} \right) \cdots \left( {n + n} \right)}} = 1? $$ \begin{array}{l} u_n = \sqrt[n]{{n\left( {n + ...
1
vote
1answer
33 views

$I_{n}=\left[a+\dfrac{(k_n-1)}{2^n};\ a+\dfrac{k_{n}}{2^n}\right]$

Let $ \mathcal{P} \subset \mathbb{R}$,\ $\mathcal{P}\neq \emptyset $ and let $b$ be an upper bound of $\mathcal{P}$. Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
0
votes
1answer
20 views

Find $dw/dv$ using the chain rule

DIFFERENTIATION find $dw/dv$ using the chain rule. if: $$ w = 4/(x^2 +y^2 +z^2) $$ $$ x = u^2 + v^2 $$ $$ y = u^2 + v^2 $$ $$ z = 2(uv) $$ $$ {dw\over dv} (u, v) = ????? $$ I've been at it for ...
0
votes
6answers
49 views

Evaluate $ \lim _{x\to 0}\left(\frac{\sin (3x)}{3x}\right)^{1/x}$

How can I find the following limit? Is it possible using L'Hospital's rule? $$ \lim_{x \to 0} \left (\frac{\sin(3x)}{3x} \right)^{\large{1/x}}$$
1
vote
2answers
41 views

Math floor and limits? $\lim \limits_{n \to \infty}\frac{n}{2}\left\lfloor \frac{3}{n}\right\rfloor$

I have these equations: $$\lim \limits_{n \to \infty}\frac{n}{2}\left\lfloor \frac{3}{n}\right\rfloor$$ $$\lim \limits_{n \to \infty}\frac{2}{n}\left\lfloor \frac{n}{3}\right\rfloor$$ What is ...
3
votes
0answers
22 views

Evaluating $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$ without Complex Numbers [duplicate]

Alright, so the standard way to evaluate $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$, is to respectively take the real and imaginary part of $$\sum_{k=0}^n{\rm e}^{ikx}={\frac ...
5
votes
3answers
79 views

Why is the area under $\frac1{\sqrt{x}}$ finite and the area under $\frac1x$ infinite?

If this integral is calculated the normal way, $$\int_0^1 \frac1{\sqrt{x}} dx = 2\sqrt{1}-2\sqrt{0}=2$$ However, the graph of $\dfrac1{\sqrt{x}}\to \infty$ as $x\to 0$, so the area under the graph ...
1
vote
1answer
33 views

Limits to infinity - some wonders

$a_n \to 0$ , $b_n \to \infty$ Is it true that: A) $$\lim \limits_{n \to \infty}(a_n - b_n) = -\infty$$ B)$$\lim \limits_{n \to \infty}\frac{a_n}{b_n}=0$$ Well about A I couldn't find a counter ...
1
vote
3answers
64 views

Why base of a logrithim function must be greater than one?

I'm in domain section of my textbook. It says that for logarithm functions, the base must be greater than one. I can understand why base shouldn't be one but what is problem with negative numbers? ...
2
votes
1answer
39 views

Using $\log$ and $\ln$ in Integration [duplicate]

I found in some integral equations where they use $\log(n)$ and in some other with $\ln(n)$. Suppose $$ \int_{n_0}^{\large\frac{n_0}{2}} \frac{1}{n}dn $$ Which formula should I use ? $$ \log(n)\ ...
-2
votes
0answers
15 views

Simplify the following Boolean expression (a ⋀ b ) ⋁c) ⋀ (a ⋁ b ) ⋀c) [on hold]

Simplify the following Boolean expression (a ⋀ b ) ⋁c) ⋀ (a ⋁ b ) ⋀c)
1
vote
1answer
36 views

definition of derivative for complex analysis

How can I use the definition of derivative to find the derivative of $\dfrac{\bar{z}^2}{z}$. My attempt, $\dfrac{\dfrac{\overline{z+\Delta z}^2}{z+\Delta z}-\dfrac{\bar{z}}{z}^2}{\Delta z}= ...
0
votes
0answers
25 views

How to prove the convexity of $f$ if the strict epigraph of $f$ is convex

I have trouble to prove the equivalence of the following two definitions of convex function. For convenience, I list them as follows: Def-A. Let $f : I\to \mathbb{R}$ be a function, where $I$ is (and ...
2
votes
4answers
107 views

Evaluating $\displaystyle4\int \frac{\tan^2x\:\sec\:x}{\sec\:x\:+1}dx$

I was solving following integral $$\int \frac{\sqrt{x^2+4}}{\frac{x}{2}+1}dx$$ I think I need do a trigonometric substitution but I eventually end up with $$4\int ...
1
vote
1answer
65 views

Demostrate $\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$ [on hold]

Demonstrate: $$\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$$ $A,B$ are undetermined coefficients
4
votes
3answers
105 views

The proof use intermediate value theorem?

Let $x_1, \dots, x_n$ distinct points in $[\alpha, \beta]$, and $y_1, \dots, y_n$ reals with same sign. Assume that $f: [\alpha, \beta] \rightarrow \mathbb{R}$ is continuous. Then, prove that exists ...
1
vote
2answers
50 views

For every $n$ there exists $k_n \in \mathbb{N}$ such that $a+k_n/2^n$ is an upper bound while $a+(k_n-1)/2^n$ is not

Let $ \mathcal{P} \subset \mathbb{R}$,\ $\mathcal{P}\neq \emptyset $ and let $b$ be an upper bound of $\mathcal{P}$. Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
0
votes
1answer
30 views

redefining already defined variables in integration

I have a question about redefining variables. In some proofs a variable is defined and later it is defined otherwise. This doesn't make sense to me. Consider the following proof for example: ...
0
votes
0answers
24 views

“Write about accelerating convergence using Aitken's $\Delta^2$ process”

My try to solve it: I need someone who understands this field of maths well to correct my answer if any mistakes are found
1
vote
2answers
38 views

Bounding a real integral involving complex constant

Is this integral finite $$|\int_{-\infty}^{\infty} e^{-i\pi x^2}\ dx|$$ can we use the fact that $e^{-\pi x^2}$ have compact support to estimate the above integral?
0
votes
2answers
70 views

show $\exists\ m\in\mathbb{N} \text{ such that: } \quad a+\dfrac{m}{2^n}\geq b$

Let $ \mathcal{P} \subset \mathbb{R}$, $\ \mathcal{P}\neq \emptyset $ et let $b$ an upper bound of $\mathcal{P}$ Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
0
votes
1answer
30 views
1
vote
0answers
31 views

Finding the area inside the rose

Calculate the area inside the rose:$$r(\theta)=a\cos(n\theta)$$ where $n$ is a positive integer and $a$ a positive constant. Why I'm making a mistake ...
0
votes
2answers
31 views

Spivak GENERAL limit law proof

Suppose $f(x) \le g(x)$ for all real $x$ Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$ Let limit for $f(x)$ be denoted by $L$ Let limit for $g(x)$ be denoted by $M$. ...