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

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1answer
104 views

cause of differentiation only on an open set.

In my school calculus textbook,derivative is introduced by saying 'Let f(x) be a real valued function defined on an open interval (a,b) & let c belongs to (a,b).Then f(x) is to be differential at ...
0
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1answer
46 views

Transform unconstrained optimization problems into constrained ones?

I want to formally show that the following minimization problem $$ \min_\theta||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ is equivalent to $$ \min_{\beta, \{w_i \}^{n}_{i=1}} ...
1
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1answer
50 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
1
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1answer
54 views

Use continuity to show that a function is continuous on an interval

Use the definition of continuity and the properties of limits to show that the function $g(x)= 2\sqrt{3-x}$ is continuous on the interval $(-\infty,3]$. I know that for a function to be continuous, ...
0
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1answer
62 views

basic thing that I always get confuse about

LEt $\epsilon > 0$ be given. if $a - b < \epsilon $ Why does it follow that $a = b $ ??? This bothers me a lot. Why does it follow? shouldn't be $a < b $ ?
1
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1answer
71 views

Is this possible physically?

The length x of a rectangle is decreasing at the rate of 5 cm/min and the width y is increasing at the rate of 4 cm/min. When x=8 cm and y=6 cm,find the rates of change of a)the perimeter and b)the ...
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2answers
56 views

Find general solution of first order DE using integrating factor

I have the equation $$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$ And am asked to find the general solution using the integrating factor. I am a bit confused as I have been shown two ways to do it. ...
3
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4answers
159 views

How to find the integral for $\int 2^{\sin{x}}\cos{x}\;\mathrm{d}x$?

What would be the ideal approach in finding the integral for: $$ \int 2^{\sin{x}}\cos{x}\;\mathrm{d}x $$
6
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2answers
119 views

Help with integral

I seem to be stuck trying to prove the following integral $$ \int\frac{\cos^mx}{\sin^nx}dx = -\frac{\cos^{m+1}x}{(n-1)\sin^{n-1}x}-\frac{m-n+2}{n-1}\int\frac{\cos^mx}{\sin^{n-2}x} dx + C\,\,(n \neq 1) ...
0
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1answer
49 views

Rate of change question about Volume

A large vase has a square base of side length $6 \text{ cm}$, and flat sides slopingoutwards at an angle of $120^{\circ}$ with the base. Water is flowing in at $12 \text{ cm}^3/\text{s}$. Find, to ...
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2answers
73 views

How to find $\frac{dy}{dx}$ for $\sqrt{xy} = 1$? [closed]

What approach would be ideal in finding $\frac{dy}{dx}$ for $\sqrt{xy} = 1$?
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2answers
219 views

Evaluation of Integral $ \int\ln(\tan x)dx$

Evaluation of Integral $\displaystyle \int\ln(\tan x)dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\ln(\tan x)dx = \int \ln(\sin x)dx - \ln (\cos x)dx$ Now Using $\displaystyle \sin x = ...
3
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1answer
129 views

Fundamental Theorem of Calculus when Integrand is a Function of the Bounds

I have a function, $$F(r) = \int_0^r |c x^2 + {(2 a + b - 4 a r - 3 b r - 2 c r) x^2\over2 r} + b x^3 + a x^4| dx$$ a, b and c are constants. I want to determine r such that $f=F'(r) = k$. ...
3
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5answers
96 views

How to integrate $\frac{\sqrt{x}}{1-\sqrt{x}}$?

How to integrate $\frac{\sqrt{x}}{1-\sqrt{x}}$? I tried by using integration by parts, but always got sucked. Should be very easy...
-3
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1answer
37 views

How to solve for $k$ when the area about the $x$ axis and under the graph of the $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?

What approach would be ideal in solving for a number $k$ when the area about the $x$ axis and under the graph of the function $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?
2
votes
2answers
183 views

“Slow” and “fast” rates of convergence

I have recently read about convergence and divergence. However, I am having trouble understanding how something can converge/diverge "slowly" or "fast". If you sum up two series (that converge to the ...
1
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2answers
101 views

Series: Let $S=\sum\limits_{n=1}^\infty a_n$ be an infinite series such that $S_N=4-\frac{2}{N^2}$.

Let $S=\sum\limits_{n=1}^\infty a_n$ be an infinite series such that $S_N=4-\frac{2}{N^2}$. (a) Find a general formula for $a_n$. (b) Find the sum $\sum\limits_{n=1}^\infty a_n$. Can you explain ...
3
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1answer
36 views

Is this proof about limit involving e correct?

I know that this is a pretty basic limit, I found this limit in this forum but not the way I did, so I need to know if this is right: We know that $$\lim_{x \to \infty}\left(1 + \frac{1}{x} ...
2
votes
2answers
64 views

Limit as $x$ goes to $0$ of $x^x$ [duplicate]

$$\lim_{x \to 0}x^x$$ I know the answer is one but I have no idea how to get there. I tried taking a natural log and I think I need lhopitals rule but I keep going In circles.
1
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1answer
50 views

area in polar coordinates

Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this particular question. To be completely honest I am not sure how to even approach this problem, ...
0
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1answer
49 views

Why is My answer Wrong?

I have a question here $\frac{d}{dx}\left(\frac{8}{e^{1-4x}}\right)$ I simplify this $8\left(\frac{1}{\left(e^{1-4x}\right)^2}\right)\left(-4e^{1-4x}\right)$ to $\left(\frac{32}{e^{1-4x}}\right)$ ...
0
votes
2answers
144 views

Series approximation for Pi up to n decimal places.

I have the following series which gives me Pi. I need to figure out how many terms of the series I need to be accurate (with respect to Pi) up to 4 decimals. I also need a formula to figure out ...
1
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1answer
55 views

Finding volume of the solid of revolution?

Can anyone help me with finding the volume of a solid of revolution of f(x) about the x axis for the interval [1,6]. It's supposed to be able to be done without needing calculus but I am having ...
7
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1answer
72 views

inequality $ \prod_{n=2}^{\infty}n^{\zeta(n)-1} <\frac{\pi^2+6}{6}$

Let $ \zeta(s) $be the riemann zeta function, then $$ \prod_{n=2}^{\infty}n^{\zeta(n)-1} <1+\frac{\pi^2}{6}$$ The problem is difficult, I don't know how to go started Thank you very much!
2
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4answers
210 views

What is the answer to the paradox of the infinitesimal?

I just read this article on npr, which mentioned the following question: You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're ...
0
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1answer
52 views

polar coordinates

Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this problem. I was trying to use the polar coordinates (d,a)with the equation of the line thus ...
0
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1answer
43 views

I have a couple questions concerning a proof of the chain rule I found

http://kruel.co/math/chainrule.pdf I found this proof of the chain rule, and it looks very thorough and legitimate. I have a few questions concerning it though. Is this in fact a legitimate proof ...
1
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1answer
115 views

How to evaluate $\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$

How to evaluate $$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$$, where n is integer > 0? I know the gamma function formula will give $$ ...
6
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1answer
205 views

Is the standard part function another devil's staircase?

The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means). I was incredibly amazed when I found out ...
4
votes
4answers
242 views

Rigorously prove that the tangent line is indeed tangent?

Let $f$ be a function of $x$ and $f'$ be the derivative, at a point $(x_0, f(x_0))$ the slope is $f'(x_0)$, we know from any calculus book that the line $g(x) = f(x_0) + f'(x_0)(x - x_0)$ is tangent ...
6
votes
1answer
172 views

Log Sine: $\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta.$

Hi I am trying to calculate $$ I:=\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta. $$ Here is a related Integral...$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d ...
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2answers
59 views

Convergence of $\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$

Does this diverge or converge ?? $$\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$$ where $H_n$ is the nth harmonic number, $p_n$ is the nth prime. My impression is that it diverges, but I don't ...
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2answers
29 views

Fourier coefficients assume a maximum and minimum?

Let $f:\mathbb R\to\mathbb R$ be continuously differentiable and periodic with period $2\pi$. The Fourier coefficients are defined by $$\hat f_n=\int_{-\pi}^\pi f(x)\exp(-inx)dx$$ My questions: Is ...
0
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1answer
87 views

If a series converges does its sequence of partial sums converge?

By Definition, a series $\displaystyle \sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $\displaystyle S_n = \sum_{k=1}^n a_k$ converges. My question is, is the converse true? If ...
0
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1answer
69 views

Determine the exact value of the expression cos(Arcsec(4/3))

Ok Sec theta is 4/3 which is r/x. So then y = 1 (I think). But I don't know how to get the exact value from the question. My answer wound up being Cos theta = 3. I'm not seeing something
9
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2answers
212 views

Computing $\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$

I don't know how to compute: $$\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$$ I have tried substituting $t=\tan ^{2} x$ but got nothing out of it. I know there's some trick involved, but ...
0
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1answer
33 views

Can a non-linear polynomial have integers values at only integer points?

P(x) be a polynomial with real co-efficient s,deg(P) greater than or equal to 2.Prove that it is not possible that whenever P(x) is an integer,x is also an integer. I tried it in various way and I ...
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2answers
25 views

Find the derivative with respect to $x$ of $y=\log_4 (x^3)$

I got $\frac{3x^2}{x^3(\ln 4)}$. Then $x$ cancels and left with $\frac{3}{x \ln4}$ ? That felt too easy so I'm sure its wrong. Or am I actually correct on this one?
3
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3answers
42 views

using substitution wrongly

Solving integral, first way: $$\int \frac{du}{u^2-9}=-\int \frac{du}{9-u^2}$$ $$u={3\sin v}$$ $$du=3\cos vdv$$ $$-\int \frac{3\cos vdv }{9-9\sin^{2}v}=-\frac 13\int\frac{dv}{\cos v}=-\frac ...
3
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1answer
90 views

UNBEATABLE recurrence relation

Hi I don't know where to start to solve this reccurence relation: $g(1)=2$ $ g(2n)=3g(n)+1$ $ g(2n+1)=3g(n)-2$ of coures I can make it: $ g(1)=2$ $ g(n)=g(2n)/3-1/3$ $ ...
3
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5answers
234 views

Integral: $\int \frac{dx}{\sqrt{x^{2}-x+1}}$

How do I integrate this? $$\int \frac{dx}{\sqrt{x^{2}-x+1}}$$ I tried solving it, and I came up with $\ln\left | \frac{2\sqrt{x^{2}-x+1}+2x-1}{\sqrt{3}} \right |+C$. But the answer key says that the ...
2
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1answer
71 views

How can i solve $\sum_{n=1}^{\infty} \frac{n 3^n}{ (n+3)!}$

I have to solve this sum: $$\sum_{n=1}^{\infty} \frac{n 3^n}{ (n+3)!}$$ but I really don't have any idea.According to wolfram-alpha I should get $\frac{1}{2}$. Any help will be greatly appreciated.
1
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0answers
35 views

Total derivation

$$\pi_1(p_2)=\max_{x_1}p_1(x_1;p_2)x_1-MC_{x_1}x_1.\tag{2}$$ The first order condition is $$\phi(x_1;p_2)=\dfrac{\partial p_1}{\partial x_1}(x_1;p_2)x_1+p_1(x_1;p_2)-MC_{x_1}=0$$ and the second ...
5
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1answer
113 views

eulers original derivation for the Euler–Maclaurin formula?

Please does someone know a good description of how Euler did derive his summation formula? Thank you!
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3answers
99 views

How to determine the convergence of the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt[n]{\log n}}$?

Any ideas on how should I solve this sum? $$\sum_{n=2}^{\infty} \frac{1}{\sqrt[n]{\log n}}$$
1
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1answer
70 views

Conditions yielding a unique fixed point of a continuous differentiable function.

Let $f$ be a function defined on $[0,1]$ which is continuous for each point in $[0,1]$ and differentiable for each point in $(0,1)$. Suppose that $f^\prime (x) \neq 1$ for every $x \in (0,1)$. ...
3
votes
4answers
76 views

How is $ \lim_{t \to \infty}\left(1+\frac{r}{t}\right)^{tn} = e^{rn} $.

How is this limit computed? $r$ is a constant. $$ \lim_{t \to \infty}\left(1+\frac{r}{t}\right)^{tn}$$
0
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2answers
67 views

Evaluate $\lim_{x\to0}\left(x^2\left(1+2+3+\ldots+\left[\frac{1}{|x|}\right]\right)\right)$ [closed]

Evaluate$$\lim_{x\to0}\left(x^2\left(1+2+3+\ldots+\left[\frac{1}{|x|}\right]\right)\right)$$ For any real number $a, [a]$ is the largest integer not greater than $a$.
0
votes
1answer
31 views

Differentiation (Transformation)

This is related to transforming $\infty$ to the origin using the transformation $x=\frac{1}{t}$. So, $dx=-\frac{1}{t^2}dt$, which implies that $\frac{d}{dx}=-t^2\frac{d}{dt}$. I cannot figure out how ...
2
votes
2answers
129 views

Sum convergence test problem.

I’ve come across this sum, and I need to test the convergence, but I really have no idea how to get started. This is the sum: $$ \sum_{n=1}^{\infty} \frac{n^{\sqrt{6}+1}+n+5}{n^{\sqrt6+2}+2n+10} $$ ...