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

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0
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0answers
19 views

There is a rational number between ever 2 real numbers

I have encoutered the following proof Let $r_1<r_2$ both in $\mathbb{R}$ such that $0<r_2-r_1=\epsilon$. By Archimedean principle there is $q\in Q$ such that $0<q<r_2-r_1$. ...
0
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0answers
8 views

Work done by force field

Problem, (a) Verify that the force field F(x,y) = (cosx − cosy)i + xsinyj is con- servative. (b) Find a potential function φ(x,y) for F(x,y). (c) Find the work done by the force field F(x,y) ...
1
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1answer
30 views

How can I find the sum of the series $\sum_{n=0}^\infty {(-1)^n \over 4^n}$ or show that it diverges using the geometric series test?

First, I reindexed it: $$\sum_{n=0}^\infty {(-1)^n \over 4^n} = \sum_{n=1}^\infty {(-1)^{n-1} \over 4^{n-1}} = \sum_{n=1}^\infty {\left(-1 \over 4\right)}^{n-1} $$ So now I'm pretty sure it's in the ...
1
vote
1answer
21 views

An explanation for sketching the graph of the family of circles: $x^2+y^2=2cx $

I found in the internet a graph for the family of circles: $x^2+y^2=2cx $ and I'm not sure I quite understood if its true or I would be able to draw it by myself. this is the graph: Aren't the ...
2
votes
3answers
121 views

Evaluating $\int\frac{x^4+1}{x^6+1}dx$

I have problem with this integral: $$\int\dfrac{x^4+1}{x^6+1}dx$$ I guess it is easy, but I was trying for quite a long time and the only thing I got is headache. Thanks for help
0
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1answer
45 views

Is this equality correct?

I am working on a problem and stuck at some point. By intuition I believe that the equality below should hold. Then the bigger problem makes sense. However, I could not prove it. Does anybody prove or ...
-1
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1answer
23 views

Exponential Function Equation and inverse Pre-Cal

I am not completely sure if I wrote the equation correctly. For A I wrote: m(t)=100(b^x) Not sure it is correct...but how do I find the inverse? That doesn't make sense to me. Do I use log?
1
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1answer
26 views

Proof of Implicit function theorem

I was trying for a simple proof for implicit function theorem on two variables.I came across a book by Dipak Chatterjee.It says as follows : $f(x,y)$ be a function of two variables and $(a,b) be a ...
-5
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1answer
26 views

Numbers with variable power is poitive. [on hold]

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a<0$.
1
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1answer
17 views

Integral of polynomial using substitution

I have an integral problem that I'm working on, it's a polynomial which I imagine either can't be factored or needs to be completed, and then substituted using a trig identity: ...
-1
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0answers
10 views

Minimising the surface area of a rectangular prism [Solution Verification]

A packaging company is going to make open topped boxes, with square bases that hold $100$ centimetres$^3$. What are the dimensions of the box that can be built with the least material?
1
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1answer
31 views

Corollary of Green's Identities/Formulas

Green's Identities in $\mathbb{R}^3$ are as follows \begin{align} \int_\Sigma (\nabla f \cdot \nabla g + f \Delta g) \, dV &= \int_{\mathcal{S}} f \nabla g \cdot \mathbf N \, d \mathcal{A}\\ ...
-2
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0answers
14 views

5 point estimation

I want to perform Probabilistic load flow using 5 point estimation method based on method of moments. To estimate five points from a specific probability distribution along with their corresponding ...
1
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1answer
21 views

Evaluating complex functions integrals over closed curves

I recently evaluated the following two integrals: $\int_\gamma \dfrac{\bar z\,dz}{2i}$ where $\gamma$ is a circle with radius $r$ around some point. $\int_\gamma \dfrac{\bar z\,dz}{2i}$ where ...
1
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2answers
88 views

Evaluate the integral $\int \sin(x)\cos(3x^2)dx$

I am looking for a solution for the following integral problem. $$\int \sin(x)\cos(3x^2)dx$$ Passed over these integral things long time ago. I cannot see how to go for a solution.
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2answers
33 views

What is e^e^x? Also, what is log e^e^x to the base e, i.e, ln(e^e^x)? [on hold]

What is e^e^x? Also, what is log e^e^x to the base e, i.e ln(e^e^x)? Thank you.
0
votes
1answer
47 views

$\sum_{i=1}^n\frac{1}{y_i^2}=1$ and $y_{min} \rightarrow \infty$, prove $\lim_{n \rightarrow \infty }\sum_{i=1}^n e^{-y_i}=0$

we have two conditions, with $y_{n,i}>0$: $$\sum_{i=1}^n\frac{1}{y_{n,i}^2}=1$$ $$min(y_{n,i}) \rightarrow \infty \quad as \;n \rightarrow \infty$$ do we have $$\lim_{n \rightarrow \infty ...
0
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1answer
48 views

Prove: $\int_a^b e^{z_0t}dt=\frac{1}{z_0}e^{z_0t}|_a^b$

From a complex variables online course, and I need to prove that $$\int_a^b e^{z_0 t}dt=\frac{1}{z_0} e^{z_0 t}|_a^b$$ For every $0\neq z_0\in \mathbb{C}$ and for every $a,b\in\mathbb{R}$. Do I need ...
-1
votes
0answers
59 views

Why can't we differentiate constants like variables?

I understand why we the derivative of $x$ is $1$. Similarly, the derivative of a constant function $y=a$ is $0$, because it's slope is flat, since for every $x$, $y=a$. But can you please tell me why ...
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0answers
33 views
0
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1answer
31 views

Taylor series and Maclaurin series problems

Im currently working on these two problems, and Im getting really confused with them. Can someone walk me through them? I will post the work I have so far. http://imgur.com/qXj7zC1 Here is my ...
0
votes
1answer
18 views

Nature of acceleration from $x$ vs $t$ graph

The figure approximately shows the $x$-coordinate of a particle as a function of time. How can we decide whether the accelerations at time $t_1,t_2,t_3$ are positive or negative?
8
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1answer
68 views

If $\lim_{n\to\infty} (a_{n+1}-a_n)=0$ and $|a_{n+2}-a_n|<\frac{1}{2^n}$ then $(a_n)$ converges

Let $(a_n)$ be a sequence such that $\lim_{n\to\infty} (a_{n+1}-a_n)=0$ and $|a_{n+2}-a_n|<\frac{1}{2^n}$ for all $n$. I have to decide whether or not $(a_n)$ converges. My attempt: I think it ...
2
votes
2answers
22 views

Calculus by Apostol Exercise 5.5 Problem 17

There is a function $f$, defined and continuous for all real $x$, which satisfies the equation of the form $$\int_0^x f(t)\,dt = \int_x^1 t^2f(t)dt + \frac{x^{16}}{8} + \frac{x^{18}}{9} + c,$$ I ...
2
votes
3answers
45 views

Find the value of the integral $\int_0 ^\sqrt2 \sqrt{4-x^2} \ dx$

Find the value of the integral $\int_0 ^\sqrt2 \sqrt{4-x^2} \ dx$ I was unsure how to do this question so I looked at the mark scheme, and it said use $x=2\sin\theta$ and so $dx=2\cos\theta \ ...
-4
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0answers
26 views

Integration everywhere [duplicate]

Find this $\displaystyle\int\frac{e^ x}{x}\;dx$
2
votes
2answers
28 views

unsure how to rearrange $f(x)$ into suitable $p(x)/q(x)$

Consider the function $f(x)= (x^3 + 2x - 3) / (x^2 + 3x + 4)$ by dividing the numerator and denominator by the highest power of $x$ present, convert $f(x)$ into the form $P(x)/Q(x)$ where both $P(x)$ ...
2
votes
4answers
69 views

How to find $\int \frac{\ln(x)}{x^2}dx$

I need to find $$\int \frac{\ln(x)}{x^2}dx.$$ I have tried substitution with $u=\ln(x)$, then $du = 1/x dx$, but this only takes care of one of the $x$ on the bottom: $$ \int \frac{u}{x} du. $$ I ...
0
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0answers
22 views

Find the maximum value of $x^{\alpha}y^{\beta}$ subject to the constraints $x+2y \le 2$ and $x > 0$ and $y > 0$.

The Statement of the Problem: Given real numbers $\alpha > 0$, $\beta > 0$, $\alpha + \beta \le 1$, find the maximum value of $x^{\alpha}y^{\beta}$ subject to the constraints $x+2y \le 2$ and ...
0
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0answers
22 views

proof that Riemann integrals is extended by Lebesgue integrals

After reading a proof sketch somewhere (forgot the link) I've written a proof in my own words. I'm not quite sure if I got the details right, since there were variants of this floating around that any ...
1
vote
3answers
54 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .

Note: $x_n$ is a sequence which is not necessarily convergent. The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also ...
0
votes
1answer
8 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
1
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1answer
41 views

What kind of differential equation is $(x^2+2y^3)y'=xy$?

what kind of differential equation is $(x^2+2y^3)y'=xy$?, I think its an inexact differential first order, its surely not linear, I tried to check if its separable also didn't work, its not also ...
2
votes
2answers
38 views

Integral by using substitution (How to proceed?)

Using the substitution $x=a\sin\theta$, or otherwise, find $\int\frac{1}{x^2\sqrt{a^2-x^2}}dx$. My attempt, $x=a\sin\theta$ $dx=a\cos (\theta)d\theta$. Then $\sqrt{a^2-x^2}=\sqrt{a^2-a^2\sin ...
4
votes
1answer
35 views

Convergence of the following sequence:

It could be exhaustion from the amount of work that I've done today, but I'd like to prove for myself that $$\lim_{n\to \infty} e^{-t\sqrt{n}}(1-\frac{t}{\sqrt{n}})^{-n}=e^{\frac{1}{2}t^2}$$ Here's ...
2
votes
2answers
28 views

Given $\frac{x_n}{ x_{n+1}} \leq \frac{y_n}{ y_{n+1}}$ and $\sum y_n$ converges, prove $\sum x_n$ converges

With $x_n>0, y_n>0$, $\exists N $ s.t. $\forall n>N \frac{x_n}{ x_{n+1}} \leq \frac{y_n}{ y_{n+1}}$. Want to prove: If $\sum y_n$ converges, $\sum x_n$ converges. I think this is about ...
1
vote
1answer
65 views

Can a simple but rigorous argument be found to prove that this function is strictly increasing?

I have a problem here that asks to show that the function $ f: [0,\infty) \to \mathbb{R} $ defined by $$ f(x) \stackrel{\text{df}}{=} \begin{cases} \dfrac{1}{x} \left( 1 + \dfrac{x^{2}}{4} \right) ...
1
vote
1answer
14 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
0
votes
1answer
65 views

$f(x) =\ln(2x^2 + 1)$ is continuous on $\mathbb{R}$

True or False The function $f : \Bbb R \to \Bbb R$ defined by $f(x) = \ln(2x^2 + 1)$ is continuous on $\Bbb R$. I know this condition that The function $f$ is continuous at some point $c$ of its ...
0
votes
0answers
7 views

Prove of disprove: If f is twice differentiable on (-1,1) and f"(0)>0, then there is δ>0 such that f is convex on (- δ, δ) [duplicate]

First of all I feel like this statement is true. The way I'm thinking about it is that f"'(x)>0 implies convexity. The fact that they used f"(0)>0 makes no difference I think. Even if we consider it, ...
1
vote
2answers
29 views

To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as ...
0
votes
1answer
49 views

Integral of a total derivative

I have seen the "total differential" $$ d \ln A = -d \ln B/c $$ Representing how infinitesimal changes in $A$ are related to infinitesimal changes in $B$. Someone then took the integral of this ...
3
votes
1answer
33 views

Volume of a Solid of Revolution Rotated Around the Y-Axis

Sorry to post an obvious homework question here, but my daughter's calculus teacher isn't much on "teaching" and left a problem like this one out of the notes. I can't find much on the internet to ...
1
vote
2answers
48 views

Antiderivative of $ (x^2 + c)^{-3/2} $ [on hold]

What method should be used to determine the antiderivative of this expression? Edit: I have $ c > 0 $ in the problem I'm working on.
0
votes
1answer
27 views

For differentiable function where $f'(0)=a$ and $f'(1)=b$ we have that for all $c\in(a,b)$ there exists a $y$ such that $f'(y)=c$.

So what I'm trying to prove: Assume a function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable and $f'(0)=a$ and $f'(1)=b$. Prove that for any $c\in(a,b)$ there exists a $t\in\mathbb{R}$ such that ...
0
votes
1answer
45 views

Fourier series of $\cos^2x$

This is my first step with Fourier series and I'm stuck at the beginning. So my solution: The function $f(x)=\cos^2x$ is an even function. Thus I use formulas: $a_0 = \frac{2}{\pi} \int _0 ^\pi ...
0
votes
2answers
34 views

Determine if the sequence- $a_1 = 2$ , $a_{n+1} = 72/(1+a_n)$ is convergent.

I've tried to prove that the sequence is convergent by using the monotonic sequence theorem but after computing the first few terms, I realized that the sequence is not monotonic thus, it isn't ...
-1
votes
2answers
23 views

Integrating to find deceleration, and finding ball height? [on hold]

1) A ball is thrown straight up from a height of 8 ft with an initial velocity of 40 ft/sec. How high will the ball go? (Take g = 32 ft/sec2.) How would I do this? Wouldn't I need to find a velocity ...
0
votes
1answer
22 views

Determine if the sequence an = ln(n)/(n^(1/n)) is convergent.

I have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!
1
vote
2answers
20 views

Why is $\frac{d^m}{d(-x)^m}=(-1)^m \frac{d^m}{dx^m}$

Im not a mathematician so dont judge me with my following "proof". I want to show that a Legendre polynomial is odd if its index is odd, and even if its index is even. This in order to prove that ...