# Tagged Questions

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

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### Find prob. to see $5$ people in hairshop

A hair shop, people arrives at the rate $1$ person/hour, and it spend $0.5$ hour to completely cut the hair. Find the probability to see $5$ peoples in the hair shop, including the person who are ...
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### “matrix representation of the gradient”

I'm reading a paper and they say "it's convenient to employ a matrix representation for the gradient of f ". Then they simply give the matrix form, but obviously I'm a bit lost. Here are the equations ...
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### Find the form of $f$

For $f: \mathbb{R} \rightarrow \mathbb{R}$ , the following holds: $\forall x,y \in \mathbb{R} : f(x+y) = f(x)\cdot f(y)$ $\forall y : \lim_{x\to y}f(x) = f(y)$ $f$ is not identically $0$ Find the ...
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### Let $a_n=\frac {(n+1)^{100}}{e^\sqrt n}$ for $n\ge 1$ then the sequence $(a_n)_n$ is ? convergent?

i am finding $$lim_{n\to \infty} a_n=\frac {(n+1)^{100}}{e^\sqrt n}$$ by applying L'hospital rule $\frac {200(n+1)^{99}\sqrt n}{{e^\sqrt n}}$ and applying it doesn't solve it because it will ...
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### Need help with optimization problem involving a triangle and its lengths.

Find the rectangle of maximum area that can be inscribed in a right triangle with legs of length a=43 and b=44 if the sides of the rectangle x,y are parallel to the legs of the triangle, as in the ...
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### How do I approach a question with multiple variables and no set equations?

I am very confused when it comes to related rates. I am not comprehending how I need to go about solve these questions. An example problem is ... The radius of a circular oil slick expands at a rate ...
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### What does $f(x,y) = x + y$ mean in graphing?

All right, so I'm supposed to graph these $5$ things: $$x \geq 3\\ x\leq 6\\ y \geq 3\\ y \leq 6\\ f(x,y) = x + y$$ I was able to graph the first four, but I have no idea what the last one means. I ...
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### Let $\{a_n\} ,\{b_n\}$ be given bounded sequences of positive real numbers.

Let $\{a_n\} ,\{b_n\}$ be given bounded sequences of positive real numbers. Then (Here $a_n\uparrow a$ means $a_n$ increase to a n goes to $\infty$, similarly, $b_n\downarrow b$ means $b_n$ decreases ...
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### Does the existence of the derivative at a point imply the existence of the left and right derivative?

I'm asking this because I've seen "results" in the internet that state: A function $f$ is differentiable at $x = a$ if and only if both the right-hand derivative and left-hand derivative at $x = a$ ...
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### Prove that the sequence $a_1, a, a_2, a, a_3, a,\ldots$ converges iff $a_1,a_2,a_3,\ldots$ converges

Prove that the sequence $a_1, g, a_2, g, a_3, g,\ldots$ converges to $g$ iff $a_1,a_2,a_3,\ldots$ converges to $g$. Obviously, if $a_1, g, a_2, g, a_3, g,\ldots$ converges to $g$, then its ...
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### Investigate for convergence or divergence

$\require{cancel}$ Investigate for convergence or divergence: $$\sum_{i=1}^\infty \frac{3^n+4^n}{4^n+5^n}$$ I'm allowed to use basic tests for convergence or divergence: P-series test Geometric ...
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### How is $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}= \log 2?$ [duplicate]

How is $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}= \log 2?$$ I haven't done sequences in a long time, therefore proving this seems almost impossible. How is this sum gotten. Help very much ...
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### Mean Value Theorem applied on $x^t$

I am trying to proof that when Mean Value Theorem is applied on function $f(x)=x^t$ where $t\geq 2$ on interval $[a,b]$ with $0 < a < b$, then the point $c$ given by the theorem, i.e. such $c$ ...
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### how to simplify $\frac{n}{\sqrt{n^2(1+\frac{1}{n^2})}}$ to $\frac{n}{n\sqrt{\frac{1}{n^2}+1}}$

I was using Symbolab to calculate a limit step by step and I do not understand one of the steps that it used to simply the equation. The calculator simplified ...
I could use some help solving these 2 problems. I've finished the rest, I just need help on these two. I know I'm supposed to use natural log and l'hopitals rule. As the $x$ goes to $0$, find the ...
This is the problem: Design a rectangular milk carton box of width w, length l, and height h which holds 520 $cm^3$ of milk. The sides of the box cost 2 $cent/cm^2$ and the top and bottom cost 3 ...