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

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1
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1answer
47 views

Why Does The Taylor Remainder Formula Work?

I've been studying calculus on my own and have come across Taylor series. It is very intuitive until I came across the remainder part of the formula where things got fuzzy. I understand why the ...
0
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2answers
37 views

Derive the solution to $\frac{dQ}{dt} = kQ$

Derive the solution to $\frac{dQ}{dt} = kQ$ in terms of $Q_0$ Here is my work: $\frac{dQ}{dt} = kQ$ $\frac{dQ}{Q} = kdt$ $\int\frac{dQ}{Q} = \int kdt$ $lnQ = kt + C$ $Q = e^{kt}e^{C}$ Did I ...
0
votes
1answer
40 views

Application of Complex Variables

By considering the integral of: $$\left(\dfrac{\sin\alpha z}{\alpha z}\right)^2 \dfrac{\pi}{\sin{\pi} z},\quad \alpha \lt \dfrac{\pi}{2}$$ around a circle of large radius, prove that ...
0
votes
1answer
17 views

Population Growth Word Problem Using the Law of Natural Growth

The problem is included in the image below. There are three parts to the problem, and all three are on the same page. I am looking for solution verification on all three parts, but I have a specific ...
6
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3answers
138 views

A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$

I came across an interesting problem that I do not know how to solve: Find $x>0$ such that $$\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}.$$ Could you ...
-3
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0answers
19 views

Find moment of inertia of truncated cone about its axis [closed]

Question : Show that the moment of inertia of a truncated cone about its axis is $\frac{3M(a^5-b^5)}{10(a^3-b^3)}$, where $a$ and $b$ are the radii of the two ends and $M$ is the mass of the ...
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0answers
17 views

Calculus Story Problem Involving Right Triangles [closed]

A highway patrol plane flies 3 miles above a level straight road at a steady 120mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance is 5 miles. ...
0
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1answer
13 views

Law of Natural Growth World Problem: How many years will it take to sell 100 franchises?

Pizza Unlimited is a national pizza firm and is selling franchises throughout the country. The president estimates that the number of franchises N will increase at a rate of $15$% a year, that is, ...
0
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1answer
28 views

Why the square root canceled here?

My problem in the following example I want to know why the square root canceled in the formula of distance
1
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3answers
46 views

How to find differential equation

$$\frac{dy}{dx}-8x=2xy^2\quad y=0\,x=1$$ I separated $x$ and $y$. \begin{align*} \color{red}{\frac{dy}{y^2}}&=\color{red}{2x+8x dx}\\ \frac{dy}{y^2}&=\color{red}{10x dx}\\ \color{red}{\ln ...
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4answers
56 views

Proving a subsequence converges

Lately, I was looking for a proof to show that the subsequence of a sequences converges to the limit but I can't find any formal ones online. And, I was wondering if anyone has a concrete proof or a ...
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3answers
33 views

Logarithmic differentiation of $y=(x^2(7x-14)^{1/3})/(1+x^2)^4$

Honestly I have no clue how to rewrite then start it. I know you have to Ln both sides but how would you Ln the right side?
9
votes
3answers
82 views

Integration of $\frac{1}{\sin x+\cos x}$

I'm given this $\int\frac{1}{\sin x+\cos x}dx$. My attempt, $\sin x+\cos x=R\cos (x-\alpha)$ $R\cos \alpha=1$ and $R\sin \alpha=1$ $R=\sqrt{1^2+1^2}=\sqrt{2}$, $\tan\alpha=1$ ...
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7answers
64 views

Find $\frac{dy}{dx}$ of $y=\frac{\ln (x)}{x}$ [closed]

I just need help figuring out this problem! I have no clue what to do. It's probably easy but I don't know. Would I use implicit differentiation or what?
10
votes
1answer
92 views

Something Isn't Right With My Parking

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like ...
0
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2answers
27 views

How to use the definition of supremum to prove that $\sup{S} = 2$?

Let $S$ $=$ {$2-\frac{3}{n+1} | n\in \mathbb{N}$}. Use the definition of supremum to prove that $\sup{S} = 2$. Here's what I have so far. We want to show that $2$ is an upper bound, and it is also ...
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3answers
33 views

How can I find for which values of $p$ this series converges?

The series is $\sum_{k=2}^\infty\frac{1}{k(\ln(k))^p}$. I know $\sum_{k=2}^\infty\frac{1}{k(\ln(k))}$ diverges but don't know how to go about finding where it's convergent.
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0answers
26 views

Partial derivatives of the exponential-multivariable

Consider the function $u(x,y)=e^{x^4y^6}$. How do I calculate $\frac{\partial^{35}u}{\partial x^{20} \partial y^{15}}(0,0)$ using Taylor series?
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0answers
33 views

Need proof of integration of sine parametrized functions [duplicate]

Yesterday, i encountered an integral formula (actually it's a generalization, i think). This : $$\int_0^\pi x f(\sin x)\,dx = \frac{\pi}{2} \int_0^\pi f(\sin x)\,dx$$ For simple functions like ...
0
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2answers
40 views

changing order of integration

I was tasked with changing the order then drawing and calculating the integral $\int _0^2 dy\:\int _{y}^{y+2}\:\frac{x}{y+2}dx$...which got very complex. I understood that the D area had to be split ...
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0answers
16 views

Multiplicative structure of a family of functions and $\sigma$-algebra

I have a question about a multiplicative structure and $\sigma$-algebra. Let $S$ be a set and $\mathcal{J}$ be a familly of $\mathbb{R}_{+}(:=[0,\infty[)$-valued bounded functions on $S$, having the ...
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0answers
38 views

Integral: $\int \frac{1}{1+x^4}dx$ [duplicate]

I asked my teacher how to do it, he answered I had to use series but I won't learn this method in highschool. But I want to know how to solve it. I read online about series but I don't see how ...
1
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0answers
17 views

Arc length in curvilinear coordinates - reference request [closed]

Can someone recommend a website or a book with solved problems regarding (advanced mathematics): Transformation from cartesian to curvilinear coordinates Parametrization of the spatial curve Arc ...
39
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4answers
4k views

What's wrong with l'Hopital's rule?

Upon looking at yet another question on this site on evaluating a limit explicitly without l'Hopital's rule, I remembered that one of my professors once said something to the effect that in Europe ...
0
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3answers
40 views

How to solve this two limits

Can someone help me to solve this limit please? it always lead to A/0 such that A is any number belong to R. also the left side part lead to same answer:
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2answers
36 views

exponential functions.

I am confused of solving expnential functions they look easy but cant solve it. 1: $$\large e^{8\cdot\ln(b^{1/4})}$$ and this one solving for x: 1: $$\ln(6x-2) = 5$$ FYI : Its not an assignment. i ...
4
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2answers
40 views

Check my general solution to the differential equation?

Given differential equation: $$y' = \frac{(2xy^{3}+4x)}{(x^{2}y^{2}+y^{2})}$$ This is the general solution that I got for the above differential equation: $$\frac{1}{3} \ln{\lvert ...
2
votes
1answer
27 views

What does it mean take the determinant of the Jacobian in: $ V_{k+1} = \int_{M_{k}} \Bigg\vert det(\frac{\partial y}{\partial x}) \Bigg\vert dx$

In this Lecture, in the subsection Evolution of Volumes tell us: Let $M \subset D$ be a compat subset of phase space. We can define its volume by a usual Riemann integral: $$ Vol(M) = ...
0
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1answer
23 views

Calculate angle of inclination in NE direction [closed]

A man is on the hill in a point $(-100, -100, 430)$, the hill is given by an equation $z=500-0.003x^2-0.004y^2$. What is the angle of inclination in NE direction? (i guess ne direction on the ...
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0answers
39 views

How to link these two equations? Any ideas? [closed]

How do I link these two equations? ∫▒〖x^n e^ax dx〗 ∭_D▒〖f(r,θ,z)r dr dθ dz〗
5
votes
3answers
73 views

Prove that $f: [a,b] \rightarrow \mathbb{R}$ is strictly monotone

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b$ be real numbers, and let $f: [a,b] \rightarrow \mathbb{R}$ be a function ...
0
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0answers
23 views

relation between direct and indirect BFGS algorithm

I am trying to verfiry the calculation of $B^{-1}_{k+1}$ from the Wikipage. I tried using Sherman–Morrison formula twice: ...
0
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6answers
95 views

Why $\int 2 \cdot \frac{\ln(x)}{x} dx$ is $\ln(x)^2 + C$?

Why the integral of $2 \cdot \frac{\ln(x)}{x}$ is $\ln(x)^2 + C$ (where $C$ is of course a constant) ? After some years of my high school math classes, I am again doing derivatives and integrals, but ...
1
vote
4answers
101 views

The integral of Gaussian function of three variables [closed]

How do I solve this $$\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty} e^{-x^2-y^2-z^2}\ dx \ dy \ dz$$
1
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1answer
41 views

Roots of a functionwith condition $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$

Let $f:[0,\pi] \rightarrow \Bbb R$ be a continuous function which satisfies $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$ Show that $f$ has at least two roots.
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0answers
22 views

Differentiate $g\circ f$ transformation

Differentiate $g \circ f$ of the following functions: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$f(x,y)=(x-y,x+y)$$ $$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$g(x_1,x_2)=(e^{x_1} \cos ...
1
vote
1answer
36 views

$\sum_{n=1}^\infty a_n<\infty$ if and only if $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$

For $a_n$ positive sequence. I think I can prove one direction, but not both.
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votes
3answers
68 views

derivative of $\frac{1}{\cos^2(2x)+\sin^2(2x)}$ [closed]

Calculate the derivative of: $$\frac{1}{\cos^2(2x)+\sin^2(2x)}$$. How to calculate such derivative?
0
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0answers
9 views

Building an integral model to estimate variable based on population — where to start?

I'm trying to build a model that provides an estimate of cumulative pollution output based on the population level of a city. The pollution level for this city increases exponentially with ...
1
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1answer
22 views

First order approximation of $F(x)=\int_0^x f(t) dt$ in the neighbourhood of $\infty$

Let $f(x)$ continuous on the real line. Then the first order approximation of $$F(x)=\int_0^x f(t) dt$$ in the neighbourhood of $0$ is: $$F(x)=\int_0^x f(t) dt\sim 0 + x f(0), \ \ \ (x\rightarrow 0)$$ ...
0
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2answers
34 views

When using the Integral test, why is the value of the integral different from the sum of the series?

According to my textbook, the value of the improper integral is not always equal to the sum of the series. But why is that?
2
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1answer
37 views

Is the Bolzano-Weistrass theorem the same as as the Extreme Value Theorem?

I know the theorems do not state exactly the same thing, but, are they stating the same thing in general? BW Theorem: A bounded sequence of real numbers has a convergent subsequence. EV Theorem: In ...
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1answer
18 views

Geometric interpretation of $\frac{dr}{d\theta}$ in Cartesian Coordinates

You'll have to excuse me if this questions is extremely trivial; it's been years since I went back to elementary calculus and I humbly accept that I haven't really gotten deep into polar ...
3
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1answer
104 views

Clarification on Implicit Derivatives steps

I have been attempting to wrap my head around this problem for a couple days now. I've attempted numerous different iterations to try and find how the answer is derived, but I just don't see the ...
3
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3answers
73 views

How to prove $\lim\limits_{(x,y)\to(0,0)}\frac{{x^3{y^2}}}{{{x^4} + {3y^4}}} = 0$?

To prove that $$\lim\limits_{(x,y)\to(0,0)}\frac{{x^3{y^2}}}{{{x^4} + {3y^4}}} = 0$$ I start with $$\left| {\frac{{{x^3}{y^2}}}{{{x^4} + 3{y^4}}}} \right| \leqslant \left| ...
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0answers
27 views

Laplace transformation with circuits

Im confused about the significance of inversing laplace transformations. What is the interpretation of s compared to t? Why is each Laplace transform only defined for some values of s? hopefully ...
2
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0answers
40 views

A silly problem on critical points?

A critical point or stationary point of a differentiable function of a single real variable, $f(x)$, is a value $x_0$ in the domain of $f$ where its derivative is $0$. Until a few seconds ago, I ...
1
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1answer
91 views

Finding the surface area of a spheroid

I'm asked to evaluate this: What is the surface area of the surface defined by $\frac{x^2}{3} + \frac{y^2}{3} + \frac{z^2}{4} = 1$? I first parameterized it with spherical coordinates and then I ...
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0answers
45 views

Cyclindrical coordinates and Spherical coordinates [closed]

Consider the solid body which lies above the upper half of the cone $x^2 +y^2= 3z^2$ and below the sphere $x^2 + y^2 + z^2 = 4z$. Assume this body is of constant density. (a) Use cylindrical ...
1
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1answer
35 views

integrating something from a partial derivative $v=\int \frac{2x}{x^2+y^2}\,dy$

i am trying to learn harmonic analysis, and i have$$\frac{\partial u}{\partial x}=\frac{2x}{x^2+y^2}=\frac{\partial v}{\partial y}$$ and i want to get $v$. so what i do is: $$v=\int ...