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

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0
votes
2answers
30 views

Finding area between curves

A) Find area between $x^3-15x^2+50x$ and $-x^3+15x^2-50x$. B) Decide whether to integrate with respect to x or y. Then find the area of the region. $y=1/x, y=1/x^2, x=7$ C)" " $x+y^2=2 , x+y=0$ D)" ...
1
vote
2answers
40 views

Primitive of $\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $

How do I evaluate the integral of $$\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $$ in a simple way? The way I could do the question, was by multiplying and dividing the fraction by $\cos ...
2
votes
3answers
22 views

Intuitive explanation of second derivative test for functions of two variables.

I will be teaching multivariable calculus again this semester, and I am not so happy with the explanation I have for the second derivatives test for functions of two variables. QUESTION: What is a ...
0
votes
3answers
23 views

Find the area between the given function , and two tangents off of the point (2,-2)

So here is a general graph of the first couple directions. $T_1$ and $T_2$ are supposed to be the points where the tangent line intersects the parabola. The tangent lines and points where the ...
0
votes
0answers
9 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
3
votes
2answers
32 views

Calculating in closed form another digamma alternating series

Is there any clever way of finish it fastly? $$\sum _{n=1}^{\infty } (-1)^{n+1} \left(\psi ^{(0)}\left(\frac{5}{8}+\frac{3 n}{8}\right)-\psi ^{(0)}\left(\frac{1}{8}+\frac{3 n}{8}\right)\right)$$ ...
0
votes
2answers
40 views

Simplifying $f(x) = \left(x^{3} + 2x^{2} + O(x)\right)\cdot\left(1 + \frac{1}{x} + O\left(\frac{1}{x^{2}}\right)\right) $

Simplify $$f(x) = \Big(x^{3} + 2x^{2} + O(x)\Big)⋅\Bigg(1 + \frac{1}{x} + O\bigg(\frac{1}{x^{2}}\bigg)\Bigg) $$ as $x \to +\infty$. I am a bit stuck as to what to do with the three sets of ...
0
votes
0answers
17 views

Infinity sumramanujan mellin transform

i was reading about the mellin transform ans i found the following $$\sum _{k=1}^{\infty } \left(\frac{e^{-k x}}{e^{-2 k x}+1}-\frac{\pi \text{sech}\left(\frac{\pi ^2 k}{x}\right)}{2 ...
-4
votes
2answers
22 views

how to get XAU in usd formula if i knew the USD value [on hold]

how to get XAU(troy ounce gold) value if i knew USD value i want to know from the formula 1 XAU = ? USD if i knew 1 USD = 0,000889276 XAU the question mark is the number i need like today 1 XAU= ...
1
vote
1answer
28 views

Integral of polynomial related to prime divisors

Given the following integral $I_{m,n}=\int_{0}^{1}(1-x^n)^m \mathrm{d}x$. Prove that for any fixed $n$ and for any $m$ $I_{m,n}$ is a rational number and when written in the form $\frac{p}{q}$ with ...
0
votes
2answers
26 views

Given $\phi \in C^{1,b}(R)$, find $\phi_n$ countably piecewise affine functions whose derivatives converge to $\phi'$ uniformly where differentiable

Let $\phi \in C^{1}(\mathbb R)$ with bounded derivative. I am trying to build $\phi_n$ a sequence of countably piecewise affine functions, s.t. $\phi_n'$ converges uniformly to $\phi'$ on $N^c$, where ...
0
votes
3answers
46 views

How to find this type limit which has polynomial in sqrt?

I have no idea to find the below limit $$\lim_{n \rightarrow +\infty}\frac{2\sqrt{9n^2+20n+10}-6n-5}{\sqrt{9n^2+20n+10}-3n-5}=?$$
2
votes
1answer
59 views

Calculating $\int_0^{\pi/4} \frac{\cot (x)}{\cot ^2(x)+\sqrt{\cot (x)}} \, dx$

This is not really one of that kind of integrals that Mathematica cannot handle with, but given the case of a contest, how would we like to handle with it? I would like so much to know your ideas ...
3
votes
1answer
43 views

Find the sum of the roots of the floor equation

How to find the sum of the roots of the following floor equation? $$[\frac{x}{2}]+[\frac{x}{3}]+[\frac{x}{5}]=x$$ I found the following solutions by Mathematica: $\{\{ x= 0\},\{x = 6\},\{x = ...
1
vote
3answers
54 views

Why does this sum converge $\sum\limits_{k=1}^\infty\left (\frac{k\sin k}{2k+1}\right)^k$

I don't understand why this sum converges. $$\sum\limits_{k=1}^\infty \left(\frac{k\sin k}{2k+1}\right)^k$$ $$\lim_{x\to\infty} \left(\frac{k\sin k}{2k+1}\right) = diverge$$ I don't find any other ...
-1
votes
1answer
38 views

Is this function bounded or not?

$f(x) = \left(1-\frac ax\right)^2$ where both $x>0$, $a>0$ Is this function bounded? i.e. is there an M such that $f(x) ≤ M < \infty$ ? How can I figure this out? Thanks very much in ...
0
votes
0answers
20 views

how can I prove that a derivative of an implicit function is bounded?

I have the following implicit function $V(\tau,\mu)$. The function is bounded and continuous and differentiable on $\mathbb{R}$. What other properties or assumptions should I make or what conditions ...
1
vote
1answer
51 views

Closed form of this sum

$$\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { ...
3
votes
2answers
49 views

Sum involving zeta functions

Find closed form of the following - $$ \displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)} $$ I don't know how to approach to it - Using the integral definition? I cannot use ...
1
vote
0answers
31 views

Integrals with error function and exponentials

I'm trying to solve the integrals below: $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{x}{\sqrt{x^2+y^2}}\cdot \operatorname{erf}\left(m\cdot\sqrt{x^2+y^2}\right) \cdot \exp(-a\cdot ...
-1
votes
1answer
53 views

Are all derivatives of sinc function bounded on real axis?

It seems that all derivatives of $sinc$ function ($sinc(x)=sin(x)/x$) are bounded on real axis. Is it true or no?
2
votes
1answer
23 views

Asymptotic behavior of elliptic integral (first kind)

I came accross some obstacles in proving that the time $T(\delta)$ taken by a pendulum to travel from $\theta=\pi-\delta$ to a considerably distant angle $\theta=\theta_0\in(0,\pi/4)$ diverges ...
-1
votes
3answers
78 views

Limit of $\left(\frac{3n-1}{3n}\right)^{n} $ as $n\to\infty$

How the $\lim_{n\to\infty} \left(\frac{3n-1}{3n}\right)^{n}$ is equal to $e^{\frac{1}{3}}$? $$\lim_{n\to\infty} \left(\frac{3n-1}{3n}\right)^{n} = e^{\frac{1}{3}}$$
1
vote
1answer
21 views

Find multi-variable function that will make the statements true.

Let x and y denote the concentrations of two proteins encoded by the genes A and B respectively. Let f(x, y) be the rate of change of the concentration of protein A. Find a formula for f(x, y), given ...
-1
votes
0answers
14 views

Find the volume of the solid of revolution using cylindrical shell method [on hold]

Find the volume of the solid of revolution using cylindrical shell method. The solid obtained by revolving about the x-axis the region enclosed by the curves $x=12y^2−4y+5$ and $x=−12y^2+2y$
0
votes
1answer
12 views

Limit with épsilon and delta , multivalues

Hi i need help with about the how to apply the definition of limit correctly, the following problem $ \lim \limits_{(x,y)\to (5,6)}x^{2}+6y^{2}-7$ I appreciate your sugerences.
-1
votes
0answers
25 views

Dot Product with multiple components [on hold]

Im currently working with dot product, I cant find any examples that emulate problems 2 and 3. I understand the gist of the concept, I am just coming up short finding a good example to help me. Can ...
-4
votes
0answers
37 views

hard work of mathematician [on hold]

how many hours did all greatest mathematician work ?
2
votes
2answers
66 views

Derivative of any $x$ which is not zero.

I'm studying derivatives and came across this example. The exercise doesn't mention if x is a constant or any function. As mentions that x is different from zero. Why the derivative that x is not zero ...
0
votes
1answer
16 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
0
votes
0answers
10 views

If the solution of the following ODE unique with given initial value?

I am considering the following ODE: $$t\frac{d}{dt}f(t)=F(f,g)$$$$t\frac{d}{dt}g(t)=G(f,g)$$. F,G are polynomials. For given an initial value $f(0)=f_*,g(0)=g_*$ satisfying ...
2
votes
2answers
22 views

Finding Tangent Line Using Limit Definition

I'm supposed to get the equation of the tangent line to the graph of $f(x)= \frac{8}{x}$ at the point $(2,4)$. I started with $$\frac{\frac{8}{x+h} - \frac{8}{x}}{h},$$ then I cross multiplied: ...
1
vote
2answers
54 views

Explain why continuity along straight lines is not enough to conclude continuity

Consider the function with domain $A = \{ (x,y) \in \, \mathbb{R}^2: (x,y) \neq (0,0)\}$ given by $$\frac{2x^2y}{x^4+y^2}$$ Letting $(x,y)$ approach $(0,0)$ along the straight line $y=ax$ , where ...
3
votes
2answers
74 views

evaluating some limits with $\ln(x)$

I don't understand how to prove these results. $\lim\limits_{x \to +\infty}\dfrac{\ln{x}}{x} = 0$ $\lim\limits_{x \to 0^{+}}x\ln{x} = 0$
7
votes
6answers
414 views

What allows us divide/multiply dx in calculus?

I've read nearly all of the threads on this topic but none seem to answer my question or lead me in the best direction. When performing U-substituion or even in it's most basic form: $y = 2x$, ...
-2
votes
1answer
54 views

I need help to solve this function [on hold]

given that $f(x,y,z)=xy^2-y^2+z^2$ solve $$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0 $$
4
votes
3answers
72 views

Evaluate $\int \theta\sec\theta \tan\theta \ d\theta$

integral of $\int \theta\sec\theta \tan\theta \ d\theta$ my work $\frac{d}{d\theta}\sec(θ) = \sec(\theta)\tan(\theta)$ So if we let $u = \theta$ and $v' = \sec(\theta)\tan(\theta)$, then we get: ...
-2
votes
0answers
47 views

$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ [on hold]

I need help, I dont understad how it do $ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ please please ...
1
vote
1answer
47 views

Evaluation of $\int\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx$

Evaluation of $\displaystyle \int\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx\;,$$ Now Put ...
3
votes
1answer
56 views

An exponential equation

Need to solve the equation $$(x+1)^{x-1}=(x-1)^{x+1}$$ After applying logarithm on each side one obtains the following equation: $$f(x+1)=f(x-1)\text{, where }f(x)=\ln x/x $$ which doesn't seem to ...
0
votes
1answer
28 views

A function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.

I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous. I've seen from a ...
-1
votes
1answer
61 views

Do we say that $\frac{1}{0}$ has no solution or $=\infty$? [duplicate]

If someone has come up to you, and just asks you directly to compute $$ \frac{1}{0} $$ Would we say that no solution exists or that it equals infinity?
1
vote
2answers
32 views

Cauchy's root test for series divergence

Just a question regarding determining the divergent in this example :$$\sum{ 1 \over \sqrt {n(n+1)}} $$ is divergent. It explains the reason by saying that $a_n$ > $1 \over n+1$. If I am not wrong it ...
3
votes
1answer
52 views

Find: $\lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n}dx$

Find: $$\lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n}dx$$ Probably, no recursive form could be found, and elementary tools (integration by parts, change of variable, etc.) are ...
-3
votes
2answers
31 views

prove that if $\lim_{n\to \infty} a_n=l$ then $\lim \sup a_n=\lim \inf a_n=l$ [on hold]

Let $(a_n)_{n\in\mathbb{N}}\subset\mathbb{R}$, prove that if $\lim_{n\to \infty} a_n=l$ then $\lim \sup a_n=\lim \inf a_n=l$. Any Hint? i'm lost
1
vote
1answer
34 views

Solid of revolution how to set the regions

I am stuck in this exercise, I cannot get the right answer. The exercise is the following: Rotate around $y = 1$ the region that is between $y=1$, $x=3$, $y=x^\frac{3}{2}$ and the x-axis. As far as ...
4
votes
2answers
212 views

How to compute fraction sums?

For example, $$\sum\limits_{k=1}^{n}\frac{1}{(2k-1)(2k+1)}=\frac{n}{2n+1}$$ Is there an easier way to evaluate fraction sums (without using partial sums)?
6
votes
1answer
108 views

Evaluating $\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$

Evaluating $$\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$$ I had read the similar problem, but it doesn't work.
0
votes
1answer
34 views

1 Equation 4 Unknowns values

I have to establish a budget and target for 4 months, up to August we have the Year to Date which is the total sales we had until August included. For a POS (They all have different ratio that is why ...
2
votes
1answer
28 views

Error estimation for the Wallis product

From the Wallis product we know $$\prod_{k=1}^{\infty} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot ...