# Tagged Questions

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

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### How to compute the derivative of $\sqrt{x}^{\sqrt{x}}$?

I know have the final answer and know I need to use the natural log but I'm confused about why that is. Could someone walk through it step by step?
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### Compute the derivative of the function $f(x)=\left(1-\frac1x\right)^x$, $x > 1$, and conclude that $f(x)$ is monotone increasing

So for this question the derivative for this function is $$f'(x)= \left(1-\frac1x\right)^x\left[\log\left(1-\frac1x\right)+\frac{x^2}{(x-1)}\right]$$ but I am not sure how to use the derivative to ...
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### How to solve the Few Scientists Problem (big word problem) in its general form?

I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve the general form. Let me start with the concrete ...
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### Is the path with the highest average value the same as the path with the lowest total difference from the function's maximum?

If you have a function $f(x, y)$ and you draw two paths (curved lines) from points A to B where: The first path is the path with the highest average value (if the value at distance $d$ from the ...
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### How can the limit exist for a function on an interval (a,b) but not be continuous on that interval?

This is from a practice test true/false question. The statement I was given is that If $\lim_{(x,y)→(a,b)} f(x, y)$ exists, then $f(x, y)$ is continuous at $(a, b)$. I put True but the answer ...
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### What is the radius of the next convergence

What is the radius of convergence of: $$\sum_{n=2}^\infty \frac{(1-x)^{5n}}{n5^n\ln(n)}$$ I know that I should use power series but how?
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### Points in a given volume/Area

I have a rectangular prism(3D bounding box) for which i have the point(i.e center of gravity) and the height,width,depth dimensions . Given these parameters, is it possible to find all the points that ...
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### Issues with solving PDE

It's been a while since I've had to solve the heat equation, and so I am having a slight issue. The question is as follows: A long, hollow, rigid tube, of length $L$ and constant cross section is ...
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### Finding coefficients of a function, given a list of points on the function

Given $f(x) = ax^n + bx^{n-1} + ... + cx + d$, a list of points, and a specification of a tangent line (point $p_t$ and equation) find $a, b, ..., c, d$ s.t. $f(x)$ passes through each point the ...
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### How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$

I'm trying to prove this sequence converges: $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$ I noticed that this is continuous function which its derivative is always less than $0$ for $x \gt 1$, so I ...
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### Uniformly converge

fn(x) is a sequence of continuous function at [a,b] that uniformly convergent to f(x) at [a,b]. Prove that for all p>0: Lim(integral |fn(x)-f(x)|^p from a to b)dx when n to infinity =0 Sorry for my ...
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How to show this? $x\in[a,b]$, $f,g$ are continuous. $$1)\quad\int_a^x |f(t)-g(t))|dt \leq (x-a) \max_{a\leq t\leq b} |f(t)-g(t)|$$ Someone suggested: $$2)\quad\int_a^x |f(t) - g(t)| dt \leq \int_a^... 2answers 38 views ### The volume of a cone is 18π m^3 Find the minimum length of the slant edge Using pythagoras theorem, I received..$$l (Slant)=\sqrt{r^2+h^2}$$Using the volume of a cone formula in terms of h..$$h=\dfrac{54}{r^2}$$I then subbed this into the 1st equation and diffrenciated ... 2answers 83 views ### Comparing the greatest values of two functions (Derivatives) I've tried doing this task, and for this kind of task I should be using derivatives. When I done all the calculus, everything I got were some weird result which I do not know how to compare. Task ... 1answer 38 views ### Find antiderivative of 8\sin^3(2x)\cos(2x) I was tasked with finding the antiderivative of 8\sin^3(2x)\cos(2x) This is what I have$$4\sin^4(2x)-\int24\sin^3(2x)\cos(2x)\,dx$$I don't know the step after that. 1answer 29 views ### Implication Of limit f(x)=L>0 I know that there is a theorem saying that if lim_{x\to a}f(x)=L>0 there is \epsilon neighborhood that is >0. Than I came across the following: if lim_{x\to a}f(x)=L>0 there is \... 1answer 32 views ### what is the limit of the following sequence when n \longrightarrow \infty? what is the limit of the sequence a_n=\frac{n!(0.5)^{n-1}}{\sqrt{n}\{[(n-1)/2]!\}^2} when n \longrightarrow \infty. It seems that the right answer is \frac{1}{\sqrt{2\pi}}, but I cannot figure ... 2answers 160 views ### Slow decreasing function that exhibits asymptotic behaviour. I am currently doing some work on modelling the effects of treated nets usage on mosquito populations. Nets do not retain their maximum efficacy forever. They lose their chemical efficacy after about ... 2answers 67 views ### calculate the volume There is a triangular prism with infinite height. It has three edges parallel to z-axis, each passing through points (0, 0, 0), (3, 0, 0) and (2, 1, 0) respectively. Calculate the volume within ... 4answers 376 views ### How to compute \int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx? How to compute the integral,$$\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$$where, \varphi = \dfrac{\sqrt{5}+1}{2} is the Golden Ratio? 1answer 707 views ### Find dy/dx by implicit differentiation: x = \sec (1/y) I tried to solve it by: Taking the derivative of both sides using Chain Rule: 1 = \sec(\frac{1}{y})\tan(\frac{1}{y}) \frac{1}{y'} Multiplying both sides by the derivative of y' to isolate y': ... 2answers 69 views ### evaluate \int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx  This is supposed to be a very easy integral, however I cannot get around. Evaluate:$$\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx$$What I did is:$$\int_{0}^{2\pi}\frac{dx}{\cos x + \sin x +...
I am computing Fourier coefficients for some function f and have a question about how to treat the integral of |x|*cos(nx), over the interval [-$\pi$, $\pi$]. Is there symmetry to apply here, ...