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

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0
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0answers
26 views

Recursive definition of cosine

Can anybody define $cos(x)$ in a recursive way? Similar to factorial. function fact(n) return n==0?1:n*fact(n-1)
0
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0answers
22 views

Understanding Cauchy's mean value theorem

We studied in class today about the Cauchy's mean value theorem, but in somewhat more complicated version, and i find it difficult to prove. here the theorem: Let $f,\ ...
0
votes
1answer
7 views

Checking when an $a$-dependent function is continuous, differentiable.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$. Hints firstly are preferred. b. ...
0
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0answers
6 views

Derive inverse Laplace Transform using two given trigonometric transforms (5.2-13)

I am not certain how to begin this problem. Someone please point me in the right direction. Problem Using the two given formulas ($1$ and $2$ below) show that: ...
1
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2answers
30 views

Antiderivative of $\frac{\sqrt{4-x}}{x\sqrt{x}}$

I need help to find the antiderivative of the function $\displaystyle x \, \mapsto \, \frac{\sqrt{4-x}}{x\sqrt{x}}$ on $]0,4[$. I have tried the change of variables $u = \sqrt{4-x}$ but it didn't ...
0
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1answer
24 views

Determine when $f_{a}(x)$ is bounded.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$ What should $a$ be in order for ...
0
votes
2answers
31 views

Finding the limit of the following expression

How can I evaluate: $$\lim_{n \rightarrow ∞} \bigg \{ \frac{n+3}{n+1}\bigg\}^n\quad ?$$ I know the answer is $e^2$, as this is a practice problem from a textbook. However I cannot understand how ...
-1
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0answers
11 views

equivalence between hermition positive definite and norm [on hold]

How to prove the following two are equivalent: $A$ is a hermition positive definite, $\sqrt{x^TAX}$ is norm
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0answers
20 views

Antiderivative of an even function

I'm faced with an issue in terms of antiderivatives of even and odd functions. Define $f \in C[-a,a]$ where $a>0$. Let $f$ be an even function on $[-a,a]$. We wish to show that $$\int_{-a}^a ...
1
vote
2answers
13 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
0
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1answer
17 views

Determine average rate of change of function [duplicate]

How to determine the average rate of change of $f(x)= x^5-3x^4$ on the interval $[-2,4]$
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0answers
7 views

Need Help with this conical container word problem [on hold]

A conical container (r= 7 ft, h = 28 ft) is filled to (h=24 ft) of a liquid weighing 62.4 ft/lb^3. How much work will it take to pump the contents to the rim? r = radius h = height
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3answers
50 views

Calculus: continuous

Q: if $f$ is continuos on $[0,1]$ with $0\leq f(x) \leq 1$ 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 following proof works: ...
1
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2answers
22 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 ...
1
vote
1answer
18 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 ...
0
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2answers
27 views

How to find the equation of tangent line? [duplicate]

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
15 views

Unit normal at a point to the surface [on hold]

Question States: Find the unit normal to the surface $z$=$x^2$+$y^2$ at a point (-1,-2,5)
2
votes
1answer
24 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 ...
0
votes
1answer
23 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 ...
0
votes
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 ...
3
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0answers
21 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
31 views

Proving a sequence converges using epsilon-N definition.

I'm stuck with what to do next in my homework problem please help.
-3
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1answer
24 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.
0
votes
1answer
28 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
33 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 ...
0
votes
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
17 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 ...
-1
votes
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) = ...
1
vote
1answer
16 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 ...
0
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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 $\partial A=\{(0,y,z)|y^2+z^2\le1\}\cup\{(x,y,z):x^2+y^2+z^2=1,x\ge0\}$. ...
0
votes
2answers
59 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 ...
0
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1answer
23 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 ...
1
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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 ...
2
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0answers
34 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 ...
2
votes
3answers
35 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. ...
0
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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 ...
0
votes
1answer
28 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.
0
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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 ...
3
votes
4answers
128 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} ...
-1
votes
1answer
16 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 ...
0
votes
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 ...
0
votes
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. ...
-1
votes
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 $y=R$ ...
1
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0answers
17 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 ...
3
votes
0answers
23 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 $[0,\infty)$)
1
vote
3answers
58 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 ...
5
votes
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 (?)