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

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Calculus: continuous

Q: if f is continuos on [0,1] with 0<=f(x)<=1 for all x in [0,1], prove that there exists C in [0,1] such that f(c)=c I don't understand why the proofing is below showed: g(x)=f(x)-x 0
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1answer
11 views

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$.

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$. What I have: Assume that $k<0$, need to show that this contradicts the ...
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1answer
9 views

Compute integral given 2 other integrals

I want to know which solution is correct. The question states: If f is an integrable function on [1,3], and if $$\int_1^2f(x)dx=4 \space\space\space\space\space\space and \space\space\space\space ...
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2answers
17 views

How to find the equation of tangent line?

Q: find the equation of the tangent line to the graph of $f$ at the indicated point. Then verify your answer by sketching both the graph of $f$ and the tangent line. [PS: the point of tangency (x,y)] ...
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1answer
11 views

Unit normal at a point to the surface

Question States: Find the unit normal to the surface $z$=$x^2$+$y^2$ at a point (-1,-2,5)
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0answers
20 views

How do we know which variable to substitute in integration by substitution?

Often times, I encountered questions that requires Integration by substitution; however, I am still somewhat confused regarding the choice of values that should be substituted by u since it differs by ...
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1answer
20 views

Probability that sum of two uniformly distributed random variables is less than some constant

I am trying to find a way of determining the probability that the sum of two continuously uniformly distributed random variables is less than some constant $C$, formally: Let $A \sim ...
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1answer
19 views

Can a Function have Multiple Valid Indefinite Integrals

Working with U-substitution, I have to integrate the following. $\int x\cos(x^2)\sin(x^2)dx$ From my understanding you can take the integral by substituting $u$ for either $\cos(x^2)$ or ...
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1answer
28 views

Differentiating both sides with respect to time.

So I have this problem: An active volcanic mountain grows in the shape of a cone while maintaining its base diameter equal to its height. The volume of the mountain increases at a rate of ...
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0answers
18 views

Multi-variable function is undefined at every point, then limit still may exist

The following question was posed; If a multi-variable function $f(x,y)$ was undefined at every point on a curve, then could a limit exist of a point $(x_0, y_0)$ on this curve for this function? i.e ...
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2answers
30 views

Proving a sequence converges using epsilon-N definition.

I'm stuck with what to do next in my homework problem please help.
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1answer
23 views

Show that the sequence $\{b_j\}$ given by $b_j = j$ as $j$ approaches infinity is not bounded [on hold]

Show that the sequence $\{b_j\}$ given by $b_j = j$ as $j$ approaches infinity is not bounded by using the definition of boundedness of a sequence. Help please.
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1answer
26 views

Founding maxima or minima to a function

$g(x)=e^{x-1}+x^{2}-3+2x$ How can I find when this function has maxima and minima? I found the derivative but I can't understand how find the solution when $g'(x)=0$. It's high school material.
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2answers
30 views

Level curves for “unsolvable” integral

Problem: Sketch the level curves of g defined by $$g(x,y)=\int_x^y{e^{-t^2}dt}$$ (The error function does not need to be used here). Attempts at solution: (1) Apparently we could take $y=x$, then ...
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0answers
9 views

Proving that the derivative of the LRL vector $=0$ [on hold]

How to prove that the derivative of the Laplace–Runge–Lenz vector $=0$? $$A=\dot{x}\times(x\times\dot{x})-\dfrac{k}{\mu}\cdot\dfrac{x}{||x||}$$ $$\dot{A}=0$$
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0answers
15 views

Basis of a linear operator

If $\phi$ is linear operator and consider three cases: $\phi(X)=X^t ;\phi(X)=AXB;\phi(X)=AX+BX$. If $A,B$ are given find the basis $E_{11},E_{12},E_{21},E_{22}$. There are 3 answers 4x4 matrices. My ...
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1answer
30 views

How to prove this equation about derivatives?

I'm currently studying derivatives, and I saw some equations but this one just not seems much trivial to me: $$\lim_{h\to 0}\left(\frac{f\left(x_0-2h\right)-f\left(x_0+3h\right)}{h}\right) = ...
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1answer
15 views

Volume of Revolution Verification

Question: A region in the $xy$-plane is bounded by the $x$-axis, the lines $x=1$, $x=2$ and the curve $y=2x^2 +1$. Find the volume obtained by rotating the region around the $x$-axis. I did ...
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1answer
11 views

determine outer normal unit vector of $\{(x,y,z)|y^2+z^2\leq1\}$

I want to calculate the outer normal unit vector $n$ for the boundary of $$ A=\{(x,y,z)|x^2+y^2+z^2\le 1,x\ge0\} $$ So I have $\partial A=\{(0,y,z)|y^2+z^2\le1\}\cup\{(x,y,z):x^2+y^2+z^2=1,x\ge0\}$. ...
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2answers
58 views

How to find the integral $\sin^2\sqrt2x$

I need help finding the integral of $\sin^2\sqrt2x$ I started to integrate it using integration by parts: $u=sin^2\sqrt2x$ and $dv=dx$ $\int u \,{\rm d}v = uv - \int v\,{\rm d}u$ But ...
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1answer
22 views

Finding the value of constants that make a function continuous

$$ f(x) = \begin{cases} x^{-1} & \text{for $x<-1$} \\ ax+b & \text{for $-1\le x\le \frac 12$} \\ x^{-1} & \text{for $x>\frac 12$} \\ \end{cases}$$ I don't understand how I am ...
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0answers
13 views

Understanding the behaviour of $F(x)=e^{(1/(x-b)-(1/(x-a))}$

I'm trying to understand the behaviour of the function $F(x)=e^{(1/(x-b)-(1/(x-a))}$ over the open subset $(a,b)$ of $R^{1} $and Zero otherwise ( we let $0<a<b$ ). Is this function ...
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0answers
32 views

Is this a Hilbert space? If not, is it reflexive?

Let $E$ be a Banach space. Let $L^2(\Omega, E)$ denote the space of random variables taking values in $E$ with second order moment. Is $L^2(\Omega,E)$ a Hilbert space? or at least, reflexive? 1) I do ...
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3answers
33 views

If $\sum a_n$ converges absolutely , then so does, $\sum \frac {a_n^2} {1+a_n^2}$

If $\sum a_n$ converges absolutely , then so does, $\sum \dfrac {a_n^2} {1+a_n^2}$ Attempt: Given that $\sum a_n$ converges absolutely $\implies \sum |a_n|$ converges. ...
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0answers
22 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
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1answer
27 views

Area of Lemniscate of Bermoulli

I need to find out area of one loop of Lemniscate $r^2 = \sin(2\theta)$. I have tried taking square root and substitution but those haven't led to anything.
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0answers
8 views

Parametric equation of a particle moving around a circle at known speed

A runner is running around a circular track of radius $r$ meters at $q$ meters per minute. The track is oriented on a Cartesian coordinate system with center at the origin and such that the runner ...
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4answers
124 views

Evaluate the sum $x + \frac{x^3}{3} + \frac{x^5}{5} + … $

Evaluate the sum $$x + \frac{x^3}{3} + \frac{x^5}{5} + ... $$ I was able to notice that: $$ \sum_{n=0}^\infty \frac{x^{2n-1}}{2n-1} = \sum_{n=0}^\infty \int x^{2n-2}dx = \lim_{N\to\infty} ...
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1answer
15 views

How do I find the convergence of this summation using the comparison test? (∑(1/√(n^3-n)))

How do I find the convergence of this summation using the comparison test? \begin{equation} \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3 - n}} \end{equation} I am not sure what the comparison sequence ...
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4answers
57 views

How many solutions has this third degree equation?

how many solutions has this equation: $$ {x}^{3}+4\,{x}^{2}-1=0 $$ i tried ruffini so far and it is not working, now i'm stuck and no idea of how to aproach this.
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1answer
30 views

Which of these statements about a continuous function is true? [on hold]

A function $f$ is continuous on the interval $[0, 2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true? (A) There exists a $y$ in the interval ...
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1answer
39 views

Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges. [duplicate]

Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges. Attempt: $\sum |a_n|$ converges $\implies \sum |a_n|<M$. If $\sum |a_n|$ converges, then $\sum a_n $ also converges. ...
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0answers
7 views

Integrate $(\frac{y}{R})^{3/7}\, dA$

How do I find the integral for: $\displaystyle \bigg(\frac{y}{R}\bigg)^{3/7}\, dA$; where $R =$ pipe radius, $r = $radius from centerline, and $y = R-r$ ? I know I'm supposed to integrate from $y=R$ ...
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0answers
16 views

Calculate $R_n$ for $f(x)=-\frac{x^2}3+7$ [on hold]

$R_n$ is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval. Calculate $R_n$ for $f(x)=-\frac{x^2}3+7$ on the interval $[0,4]$ and write your answer ...
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0answers
22 views

I want to know that the supremum function continuous [on hold]

Let $g(y)=\sup_{x\in[0,y]}f(x)$ for $y\ge0$. I want to know that the function $g(y)$ is continuous on $[0,\infty)$. (here we suppose $f(x)$ is continuous on $[0,\infty)$)
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3answers
55 views

Use integration by substitution

I'm trying to evaluate integrals using substitution. I had $$\int (x+1)(3x+1)^9 dx$$ My solution: Let $u=3x+1$ then $du/dx=3$ $$u=3x+1 \implies 3x=u-1 \implies x=\frac{1}{3}(u-1) \implies ...
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1answer
87 views

Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$

Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$ Abel's/Dirichlet's tests cannot be applied here. I guess it's something more tricky involving integration maybe (?)
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0answers
26 views

For the following integrals find a and find b [on hold]

In the following picture, what is a=? what is b=?
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2answers
51 views

convergence of $ \sum_{n=1}^{\infty} (-1)^n \frac{2^n \sin ^{2n}x }{n } $

Find values of $x$ for which the following series converges $$ \sum_{n=1}^{\infty} (-1)^n \dfrac{2^n \sin ^{2n}x }{n } $$ Attempt: (a) Check for Absolute Convergence If we consider $ ...
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4answers
68 views

Limit involving sum of fourth and 5th powers of Natural numbers [on hold]

$$ \lim_{n \to \infty}\frac{\left(1^4+2^4+3^4+\cdots+n^4\right)^3}{n^3\left(1^5+2^5+3^5+\cdots+n^5\right)^2}$$
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1answer
24 views

Derivative of bilinear forms

I want to solve the following problems: Let $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form. Prove that it's differential is $$ Df_{(x,y)}(a,b) = f(x,b) + f(a,y).$$ ...
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1answer
18 views

How do I put a summation chart like the one attached into my TI-84 calculator? [on hold]

This is the link to my book: http://www.calcchat.com/book/Calculus-8e/ It's Chapter 4 Section 2 Problem 45 Part (E).
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1answer
18 views

Plane equation x units from point

I'm trying to find the equation of a plane normal to a certain vector $<x_1, y_1, z_1>$, and x units from a given point, $(a,b,c)$. Normally this question would be trivial, and I would simply ...
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0answers
15 views

Basis of a recurrance subspace [on hold]

In a linear dimension $R^n$ we look at U={$a_1,a_2.......a_n$} which belongs to $R^n$ and $a_1=2a_2=2^2.a_3=......=2^{n-1}a_n$.Prove that U is a subspace of $R^n$ and find the basis.I really have no ...
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2answers
22 views

Convergence of $\int_0^1 \frac{dx}{(\cos(x)-1)\sqrt{1-x^2}}$

Study the convergence of $$\int_0^1 \frac{dx}{(\cos(x)-1)\sqrt{1-x^2}}$$ Well, we can observe the $$\left| \frac{1}{(\cos(x)-1)\sqrt{1-x^2}} \right| \le \left| \frac{1}{(\cos(x)-1)} \right|$$ ...
3
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2answers
223 views

Deriving formula for derivative

I have a formula in my book for differentiating numerically. $$f'(x_0)=\frac{1}{12h}[-25f(x_0)+48f(x_0+h)-36f(x_0+2h)+16f(x_0+3h)-3f(x_0+4h)]+\frac{4}{5}f^{(5)}(\xi)$$ I was wondering if anyone ...
3
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0answers
38 views

Solutions for large $x$

Given the differential equation $$y''(x)-x*y(x)+y^3(x)=0,$$ look voor the two following solutions with $x$ large and positive. Look for: a) A oscilatory solution with two arbitrary coefficients b) ...
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1answer
117 views

Does there exist continuously differentiable function $f:\mathbb{R}\longrightarrow\mathbb{R}$?

Does there exist continuously differentiable function $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that for all $x\in \mathbb{R},\,\,f(x)>0$ and, $f'(x)=(f\circ f)(x)$? I see this question in ...
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0answers
35 views

Problem about limit of an integral

I came across this question while doing some exercises on integrals, and I was wondering if I could get some help. a) Show that for $n < -1$, $\int_1^N x^n dx$ converges as $N \to\infty$, and for ...