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Questions tagged [calculus]

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

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How to Evaluate $\int_{0}^{\infty}\frac{x^2+2}{x^4+4} \ dx$ given $\mathcal{F}_c(e^{-x}\cos(x))=\sqrt{\frac{2}{\pi}}\frac{k^2+2}{k^4+4}$

I have previously shown that ($\mathcal{F}_c(f(x))$ denotes the Fourier cosine transform of $f(x)$) $$\mathcal{F}_c(e^{-x}\cos(x))=\sqrt{\frac{2}{\pi}}\frac{k^2+2}{k^4+4} \tag{1}.$$ Using this ...
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3answers
28 views

How to integrate problem

I've tried u substitution and partial fraction doesn't seem to work either. $\int \frac{1}{x^2+3}$ Can I take $\int \frac{1}{x^2+\sqrt(3)^2}$ in order to use $x=atan\theta$
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Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.

If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by: $f(x,y)= \begin{cases} (x,y-x^{2}) & if & x^{2} \leq y \\ (x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
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1answer
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How do I determine the interval over which my error calculation should be conducted?

I've been instructed to find the values of x for which the function $f(x) = e^{-2x}$ may be approximated by the Maclaurin series $1-2x+2x^2-\frac{4}{3}x^3$ with an error of less than 0.001, but no ...
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1answer
32 views

Show that $\psi(n)$ has finitely many roots

Define $\psi(n)=\pi(n)-\phi(n)$ where we have the prime counting function and totient function respectively. I'm interested in where $\psi(n)=0$. Specifically is it possible to prove that there are ...
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1answer
21 views

a Mean Value Theorem Proof

Disclaimer: this is not homework for marks Question: Does there exist a differentiable function $f$ with $f(0) = 2$, $f(2) = 5$, and $f'(x) ≤ 1$ for all $x ∈ (0,2)$? If not,why not? My attempt (...
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1answer
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Determining the values for which a Maclaurin Polynomial approximates $f(x)=\cos(x)$ within an error of 0.001

I'm tasked with determining the values of $x$ for which $f(x)=\cos{x}\approx 1-\frac{x^2}{2!}+\frac{x^4}{4!}$ has an error no greater than 0.001. Using the error for Taylor polynomials $E = |R_n(x)| =...
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1answer
41 views

How do you evaluate this integral $\int \frac{6x^{3}+7x^2-12x+1}{\sqrt{x^2+4x+6}}dx$

$$\int \frac{6x^{3}+7x^2-12x+1}{\sqrt{x^2+4x+6}}dx$$ I have met this complex integral today, how do you evaluate this? Can you do this in more steps? Is this an elliptic integral? I don't know how ...
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Solving $\frac{dy}{dx}=\sqrt{3x+2y}-\frac{3}{2}$ without stuff from higher-order differential equations

I'm trying to solve this equation: $$\frac{dy}{dx}=\sqrt{3x+2y}-\frac{3}{2}$$ without using stuff from higher-order differential equations. I've tried using substitution $ w=\frac{y}{x} $, but ...
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Prove that $f(x,y)=f(x)f(y)$ and $h(r)=f(x)f(y)$ (Hint: Use A2 and Exercise 1)

A1: $f(x,y)=f(x_1,y_1)$ whenever $x^2 +y ^2=x_1^2 + y_1^2$ A2: $f(x,y)=f(x,0)f(0,y)$ whenever $x≠0$ and $y≠0$ Let $f(x): f(x,0)$ if $x≠0$, Let $f(x): 1$ if $x=0$ Exercise 1: Prove that $f(x,0)=f(0,x)$...
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Let $I_n=\int_0^{\pi/2}x^2\cos^n(x)dx$ show that $\lim \limits_{n \to \infty}\frac{(2n)!!}{(2n-1)!!}I_{2n}=0$ and find $I_{2n}\frac{(2n)!!}{(2n-1)!!}$

Let $I_n=\int_0^{\pi/2}x^2\cos^n(x)dx$ show that $\lim \limits_{n \to \infty}\frac{(2n)!!}{(2n-1)!!}I_{2n}=0$ and that $I_{2n}\frac{(2n)!!}{(2n-1)!!}=\frac{\pi^3}{24}-\frac{\pi}{4}\sum_{k=1}^n\frac{...
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Prove that f(x,0)=f(0,x) for all X (Hint:Use A1) Let f(x):{ f(x,0) if x≠0, Let f(x):{ 1 if x=0

A1: f(x,y)=f(x1,y1) whenever x^2 +y ^2=x1^2 + y1^2 Prove that f(x,0)=f(0,x) for all X (Hint:Use A1) Let f(x): f(x,0) if x≠0, Let f(x): 1 if x=0 [NOTE: I have about 5 exercises that need help with, ...
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2answers
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Finding the position $s(t) $ of a particle that has acceleration defined as $a(t)=3t+5$.

I am a Calculus 1 student, and we're learning about antiderivatives. I've run into a problem I'm not sure how to solve. A particle moves with acceleration defined by $a(t) = 3t+5$. Find the ...
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1answer
21 views

Picture flow of ODE

Consider the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0, \quad x \in \mathbb R^2 \\ \Phi(x,0) = x, \quad x \in \mathbb{R}^2 \end{cases}$$ Suppose that the flow $\Phi$ ...
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1answer
48 views

gaussian-like integral $\int_{-\infty}^{\infty}\frac{1}{x^2+t^2}\exp^{-\frac{x^2}{2t}}dx$

I'm trying to figure out how to evaluate the integral $\int_{-\infty}^{\infty}\frac{1}{x^2+t^2}\exp^{-\frac{x^2}{2t}}dx$ where $t\geq 0$. I have tried a lot of methods but without success. Please ...
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1answer
33 views

How to evaluate this integral $\int \frac{6x^{3}+7x^2-12x+1}{\sqrt{x^2+4x+6}}dx$

I encounter this complex integral today and I don't know how to evaluate it. I have only learnt u-substitution this semester and it seems it is not sufficient to find the primitive function for this ...
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1answer
23 views

Finding the first three nonzero terms in the Maclaurin series: $y=\frac{x}{\sin(x)}$

As the title says I would like to find the first three nonzero terms in the Maclaurin series $$y=\frac{x}{\sin(x)}$$ I have the first few terms for the expansion for $\sin(x)=x-\frac{x^3}{6}+\frac{x^...
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1answer
26 views

Infinite series question involving integrals

Can someone help me solve this tricky infinite series problem? I tried to find the indefinite integral of the nth term but my solution didn't make sense at all. I suspect there must be an ...
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2answers
51 views

Typical Calculus BC Separation of Variables Question

I was told that I have a cylindrical water tank $10$ ft tall that can store $5000 $ ft$^3$ of water, and that the water drains from the bottom of the tank at a rate proportional to the instantaneous ...
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1answer
15 views

What is the general solution of $xy=(x^2+4y)(dy/dx)?$

I have tried to use u-substitution to separate the variables, as well as distributing $dy/dx$ as y' to $(x^2+4y)$. However, I just can't seem to separate the variables no matter what methods I've ...
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3answers
41 views

How to find the general antiderivative of $f(x)=x(6-x)^2$?

I want to find the general antiderivative of $f(x)=x(6-x)^2$. However, I keep getting it wrong. I am new to antiderivatives, but I think the first thing I should do is differentiate. $$\frac{d}{dx}(...
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2answers
34 views

Sum and Product inversion

Under what conditions on $a_{i,j}$ when $i\in\{1,2...,n\}$ and $j\in\{1,2...,m\}$ this relation holds: $$\sum_{i=1}^{n}{\prod_{j=1}^{m}{a_{i,j}}}= \prod_{j=1}^{m}{\sum_{i=1}^{n}{a_{i,j}}}$$ Addendum ...
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Is this proof that $f$ is Riemann integrable correct?

Let $f:[a,b]\rightarrow \mathbb{R}$ be a bounded, non constant function. We know that f is integrable in $[c,b]$ $\forall c \in [a,b]$. Prove that $f$ is Riemann integrable in $[a,b]$. I have tried ...
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2answers
33 views

Find the sum of the infinite series in Calculus [on hold]

How do I find the sum of the following infinite series? $$11 + 2 + \frac{4}{11} + \frac{8}{121} + \cdots$$
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how to prove an f^2n(x) equation when given f(x) using maclaurin series?

So I have found the equation for f^n(0) = (1/n!)*x^2n. Then I plug in 2n for n, but this is where i get confused. That would cause the equation to be 1/(2n)!x^4n and when you plug in 0 for x you get 1/...
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Line integral: intersection of a plane and a 'special' cylinder

Calculate $\int_{\Gamma} \vec{F} \cdot d\vec{s}$, with $\vec{F}(x,y,z)=(x^2,y^3,x^2)$ and $\Gamma$ the intersection curve of $x^2+y^4=1\,(x \ge 0)$ and the plane $x+y+z=1$, oriented from $(0,1,0)$ to $...
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1answer
66 views

Request for info about Real Analysis to a beginner [on hold]

So I'm a junior HS student very interested in mathematics and I think I have pretty good chances to get admitted into Stanford online HS and take a university level real analysis course that's worth 5 ...
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0answers
12 views

Question about the gradient of a composite function

I am new to calculus and am trying to work out the following question, with no success so far… Any feedback would be great! Within function $f(x,y)$, variable $y$ is a function of $(x,z)$, in other ...
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0answers
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proving chandrasekhar-wentzel lemma

We were asked to prove the Chandrasekhar-Wentzel lemma using any method different to the one given in Wikipedia. $$\oint_cr \times (dr \times n) = -\iint_s(r \times n)\nabla\cdot n ds $$ I started ...
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4answers
34 views

How to solve non-fractional limits like $\lim_{x \to \infty} x^{(\ln5) \div (1+\ln x)}$?

I have the limit $\lim_{x \to \infty} x^{(\ln5) \div (1+\ln x)}$. I am trying to figure out how to solve this, but I only know how to handle limits when they can be made into fractions. Is there some ...
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1answer
45 views

Is the solution wrong?

Using cylindrical coordinates, calculate $$ \int_0^2dx\int_0^{\sqrt{2x-x^2}}\sqrt{x^2+y^2}dy\int_0^a zdz \quad (a>0).$$ I found that $0 \le r \le 1, 0 \le z \le a, 0 \le \theta \le \pi$. Thus $$ \...
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1answer
15 views

Question about gradient in a complex function

I am new to calculus and cannot see the logic of the following question… Any feedback will be really appreciated! The function $f(x,y,z)$ is differentiable at all points, and satisfies $f(x,y,2x^2+y^...
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Find the volume of the solid enclosed by the surface $x=a\cos^3 u, y=a\sin^3 u, z=v, (0 \le u \le \pi, v \ge 0)$ and the planes $z=0$ and $x+y+z=a$

My attempt: $$\operatorname{vol}(V) = \iint_D dxdy\int_0^{a-x-y}dz = \iint_D(a-x-y)dxdy,$$ where $D$ is the region that represents the given part of an asteroid ($x,y$). Now I don't know how to find ...
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1answer
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Question about calculating the gradient of a composite function

I am new to calculus and was asked to answer the following question. I have added my answer, although I am not sure at all whether it is correct. Any feedback would be amazing! Thanks in advance $\...
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0answers
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Three statements regarding a normal and a tangential plane

I am new to calculus, and was given the following question to answer. I have worked out an answer, but am not 100% sure about the details. Any feedback would be great! Many thanks in advance. Given ...
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2answers
37 views

How do I write the Taylor expansion of this function $z = f(x,y)$

I have an assignment where I need to write the Taylor expansion for the function $$ z = f(x,y)$$ which is given by the following formula: $$ z^3 − 2xz + y = 0$$ for the point $A(1,1)$. I know how ...
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0answers
27 views

Derivation of correlation coefficient using bell curve

You can use python to add up all those distances between the data point and the line of best fit, given by the fact that -x/a is perpindicular to ax. I plugged the sum of all the distances in to the ...
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2answers
80 views

Evaluate $\lim_{n\rightarrow \infty }\frac{1}{n}\int_{1}^{n}\frac{x-1}{x+1}dx$

$$\lim_{n\rightarrow \infty }\frac{1}{n}\int_{1}^{n}\frac{x-1}{x+1}dx$$ My approach is not correct, I think. I took $f(x)=(x-1)/(x+1)$ which is continuous so there is a $F(x)$ a primitive of f(x) ...
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State of the Mean Value theorem for inetgrals

I am getting confused by the Mean value theorem for integrals, differently stated in different sources. The idea is that for any continuous functions $f$ on ana interval $[a,b]$, there exists some $...
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Limits and derivates [on hold]

What do u mean by limit and derivatives ...and where they are used ?
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0answers
37 views

Proving a function is stable if and only if $|f'(x)| \geq c$.

A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is stable provided that $$|f(x) - f(y)| \geq c|x - y|$$ for all $(x, y) \in \mathbb{R}^{2}$, where $c > 0$ is called the stability ...
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3answers
73 views

Interpret $\int_0^1 \sqrt{4 - x^2}\,dx$ geometrically and find the area

Sadly I'm stuck on this one. Now I now I could resort to finding the antiderivative $F$ and apply the fundamental theorem of calculus (i.e. $F(b) - F(a)$), but that isn't asked (or introduced by the ...
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0answers
23 views

Spivak “Calculus on Manifolds” remarks about Fubini Theorem

On Spivak "Calculus on Manifolds" p.58 he provides a general version of Fubini theorem. Which I present below (I omit parts that are not important to the question): Let $A\subset \mathbb{R}^n$ and ...
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2answers
15 views

variable co-efficient 2nd order linear ODE

I am trying to solve a variable co-efficient 2nd order linear ODE by using a transformation for the independent variable: $y'' + \frac{2}{4x} y' + \frac{9}{4x} y = 0$ with transformation $t = \sqrt{x}...
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0answers
14 views

calculus and statistics combined course curriculum

My school is looking for non-AP options after Algebra 2 (common core) so they've decided to offer a combined introductory course of calculus and statistics. Not calculus-based statistics/probability, ...
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1answer
26 views

Reduction of order method for second order linear ODE

Given the following differential equation $$x^2 y'' + x(2x^2 + 1)y' + (2x^2 -1)y =0$$ solve it using the reduction of order method. The given solution is $$y_1(x) = \frac 1x.$$ I have been ...
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0answers
23 views

2D Heat equation with Initial Data

I have the following 2D heat equation: $$u_t - \Delta u = e^t$$ where $(x_1, x_2) \in \mathbb{R}^2, t > 0, u(x_1, x_2, 0) = cos(x_1) sin(x_2)$ I am looking to find the general solution $u(x_1, ...
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0answers
42 views

Perfect Square for Trig Substitution

I start off with this perfect square and I end up having to use two different trig subs depending on the way the square is made. Original: $3-2x-x^2$ --Option A-- = $-x^2-2x+3$ = $-(x^2+2x-3)$ = $...
2
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1answer
13 views

Correct algorithm for finding regions of increasing / decreasing and the relationship to critical points.

Yet again, I find myself confused about something that seems basic: critical points and regions of increasing / decreasing. Previously, I thought that to identify regions of increasing / decreasing, ...
2
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2answers
52 views

Power Series expansion of $x\over(1+x-2x^2)$

I am unable to solve this specific problem. The only "notable series expansion" I can use (and know) is $\sum^{+\infty}_0 x^n =$$1\over(1-x)$ I tried several things but none worked. Writing $x\over(...