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

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1answer
33 views

$\int_0^1(1+\log(x))\sin(x)dx$ How to solve this Integral?

$$\int\limits_0^1(1+\log(x))\sin(x)dx$$ Someone has challenged me to solve this, I solved it without bounds, I have no idea how to do it with those limits.. Is $u=1+\log(x)$ right substituion? or ...
0
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0answers
14 views

Question concerning continuity of composite functions

Consider two functions,a(r) and b(r).If a is continuous at c,and b is continuous at a(c) , then b(a(c)) is continuous at c .(This is a theorem stated in the text Thomas' Calculus) Now consider ...
0
votes
1answer
17 views

Equation Of Common Normal

How to find equation of common normal between two random non-intersecting conic sections (say a parabola and an ellipse) ? What should be the general approach ?
0
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1answer
26 views

Boolean Algebra, stuck

I'm having trouble simplifying this Boolean Algebra equation. Can anyone help? XY'Z + X'Y'Z + XYZ + XY'Z
1
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1answer
18 views

Double integral proving that a function is a probability density

If $$g(x,y)=f(x+y)/(x+y)$$ for $x,y>0$ and $$\int_0^{\infty} f(z) \, dz = 1$$ How do you show that $$\int_0^{\infty} \int_0^{\infty} \frac{f(x+y)}{x+y} dx \, dy = 1$$ as well?
3
votes
3answers
48 views

Give the equations that are a tangent to the parabola $y = x^2 + 5x + 6$ and pass through $(1,1)$

I have been given the question: Give the equations that are a tangent to the parabola: $y = x^2 + 5x + 6$ and pass through the point $(1,1)$ I have tried two different methods for solving this. ...
1
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0answers
17 views

A simple Laplace transformation problem

Could someone please help me with my misunderstanding here? Here's the question: "A mass M is attached to a spring of stiffness $\omega^2M$ and is set in motion at t=0 by an impulsive force P. The ...
2
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0answers
31 views

convergence of general harmonic series

My question is about determining the convergence of a general harmonic series using the integral test. According to the following resource: pg.32 , we can see that for a general harmonic series ...
1
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1answer
33 views

Integrate $\int_{0}^{2}{\int_{y^2}^{4}{ycos(x^2)dxdy}}$

I'm asked to evaluate $\int_{0}^{2}{\int_{y^2}^{4}{ycos(x^2)dxdy}}$. Letting $f(x)=cos(x^2)$ We have have that $\int_{y^2}^{4}{cos(x^2)dx} = F(4)-F(y^2)$ by the FTC. This gives us \begin{align*} ...
1
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2answers
32 views

I need help with this trigonometric integral

I dont know how to do this integral $\int \dfrac{dx}{\sin(x) + \cos(2x)}$ i have tried the fundamental trigonometryc identity $(\sin x)^2 + (\cos x)^2 = 1$ but that does not work out the way i ...
1
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1answer
23 views

Mean value formula integrals

Let $f: B(0,R) \rightarrow \mathbb{R}$ be a continuous function. Then I was wondering whether $$\frac{1}{\text{area}(\partial B(0,r))} \int_{\partial B(0,r)} (f(x)-f(0)) dS(x) \rightarrow_{r ...
0
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1answer
25 views

Arc length of polar curve

I was trying to determine the arc length of the polar curve $r = f(\theta) = a(1 - \cos \theta)$, and it was going well until I got to the definite integral. I know that $f'(\theta) = a \sin \theta$, ...
1
vote
1answer
20 views

Rules for manipulating differential/ Leibniz notation?

What are the rules on manipulating Leibniz Notation? dy/dt = -(y-3)/2 Can I treat it like a fraction and do this? dy = -(y-3)/2 dt can I "convert" the dy into a derivative? (dy/dt)(1/(y-3)) = ...
1
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3answers
42 views

Show that the function is continuous

To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}$ with $f=\left\{\begin{matrix} \frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\ 0 & , (x,y)=(0,0) \end{matrix}\right.$ is ...
2
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1answer
42 views

Taylor series question

I've been struggling with this problem: Find the Taylor series representation for $xe^{2x}$ I was able to find the Taylor series for $e^{2x}$ (centered at a=k) in a previous exercise which I ...
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votes
3answers
39 views

Calculus Lab Problem Differentiation [on hold]

I am kind of confused on how to differentiate this function. Here is the problem: $$P(t) = \frac{1}{(1 + e^{-t})} $$ Use the differentiation rules to compute the exact value of $P'(2)$. Certain ...
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0answers
26 views

Deriving $u=g(x) \implies du = g'(x)dx$

$u=g(x) \implies du = g'(x)dx$ How does this work? If I think about differentials as fractions, then I can easily see that $\frac{du}{dx} = g'(x)$ and then seperate the differentials as $du = ...
1
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1answer
35 views

Differentiable at endpoints?

If $f$ is continuous on [$a,b$] then the area function $A(x)=\int_{a}^{x} f(t)dt,$ for $a \le x \le b$, is continuous on [$a,b$] and differntiable on $(a,b)$. My question is: why is it not ...
-1
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1answer
39 views

Finding area between curves [on hold]

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)" ...
2
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3answers
88 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
28 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
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3answers
26 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 ...
1
vote
1answer
27 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
44 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
1answer
45 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
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0answers
20 views

Infinity sumramanujan mellin transform [on hold]

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 ...
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votes
2answers
25 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
32 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
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2answers
37 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
49 views

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

I have no idea to find the below limit $$\lim_{n \rightarrow +\infty}\frac{2\sqrt{9n^2+20n+10}-6n-5}{\sqrt{9n^2+20n+10}-3n-5}=?$$
3
votes
1answer
83 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
45 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 = ...
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3answers
55 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
41 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
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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
59 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
52 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
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0answers
33 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
54 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
28 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 ...
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votes
3answers
80 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 $e^{\frac{1}{3}}$? $$\lim_{n\to\infty} \left(\frac{3n-1}{3n}\right)^{n} = e^{\frac{1}{3}}$$
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vote
1answer
24 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 ...
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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 $x=−12y^2+2y$
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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.
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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 ...
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0answers
38 views

hard work of mathematician [on hold]

how many hours did all greatest mathematician work ?
2
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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
17 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
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0answers
11 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: ...