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

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2
votes
1answer
43 views

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 ...
2
votes
2answers
51 views

“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 ...
2
votes
2answers
23 views

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 ...
2
votes
2answers
33 views

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 ...
2
votes
1answer
37 views

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 ...
2
votes
2answers
20 views

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 ...
2
votes
4answers
38 views

How to take an integral using half angle trigonometric substitution.

So i have this question which is asking to take the integral using a predefined trigonometric substitution which is $$u=\tan\frac{x}{2}$$ and the integral equation is $$\int\frac{\sin x\ dx}{(6\cos ...
2
votes
1answer
52 views

Convergence/divergence of the series $\sum_{n=4}^\infty\frac{(-1)^n}{\log(\log(n))}$?

We have the following sum: $$\displaystyle \sum_{n=4}^\infty \dfrac{(-1)^n}{\log(\log(n))} $$ I have a hunch this series is conditionally convergent, but I get nowhere using the ratio test. What ...
2
votes
3answers
44 views

Indefinite Integral Using Substitution

can someone please help? I need to evaluate this indefinite integral: $$\int \frac{x}{\sqrt{x^2+2}}dx$$ I tried using substitution for letting u = x, but I can't get past finding the antiderivative ...
2
votes
1answer
76 views

Oscillation of a function and continuity

I would be very grateful if somebody could verify my proof. Assume that $f$ is continuous at $p$. We can always choose $\delta_1>0, \delta_2>0$ such that $d'(f(x),f(p))\le\frac{1}{2n}$ ...
2
votes
1answer
57 views

Use the change of variables to evaluate the double-integral $\iint_R4(x+y)e^{x-y}\;dA$

Use the change of variables indicated to evaluate the double-integral $\iint_R4(x+y)e^{x-y}\;dA$ where $R$ is the interior of the triangle whose vertices are $(-1,1)$, $(0,0)$, and $(1,1)$; ...
2
votes
1answer
47 views

Series of $1/\ln(n!)$

How do I start with this series? $$\sum_{n=2}^\infty \dfrac{1}{\ln(n!)}$$ I can use any method to solve this problem. When I try using Ratio Test I get stuck with: $$\lim\limits_{n \to \infty} ...
2
votes
2answers
67 views

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 ...
2
votes
2answers
52 views

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 ...
2
votes
2answers
47 views

Substitution of variable in DE

For the ODE $\epsilon \frac{d^2u}{dt^2}-a(t)\frac{du}{dt}+b(t)u=0$, what does one have to do in order to express it in terms of $T=\frac{1-t}{e^\nu}$? Here's my attempt: $$\epsilon ...
2
votes
2answers
38 views

Proving the following limit statements

I need to prove those If $f(x)\ge 0$ and $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}\sqrt{f(x)}=\sqrt{L}$. If $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}|f(x)|=|L|$. If $f(x)\ge g(x)$ for ...
2
votes
1answer
40 views

A fence of $y$ ft is $x$ ft from a wall, find shortest ladder using trigonometry

A fence $y$ ft high is $x$ ft from a wall. Find the length of the shortest ladder that will rest with one end on the ground and the other end on the wall. This is a classic problem that is ...
2
votes
2answers
48 views

is there a function f:R→R , differentiable on (a,b) but not on [a,b] ?(f is continuous on [a,b])

is there a function like f:R→R such that : 1) f is continuous on [a,b] 2) f is differentiable on (a,b) 3) f(a)=f(b) but f is not differentiable on [a,b]??? (if the answer is no , then prove that f is ...
2
votes
2answers
32 views

How to compute $ \lim\limits_{t\rightarrow0}\frac{\int_w^{w+tv}f(y)dy}{t} $?

$w,v\in R$ and $f\in C^\infty(R)$ . How to compute $$ \lim\limits_{t\rightarrow0}\frac{\int_w^{w+tv}f(y)dy}{t} $$ I feel it should be $vf(w)$, but I don't know how to prove it
2
votes
2answers
52 views

using symmetry or geometry for a double integral $\iint_D (4-\sqrt{9-x^2-y^2}+\sin(xy))dA$

Using either geometry or symmetry, evaluate: $$\iint_D \left(4-\sqrt{9-x^2-y^2}+\sin(xy)\right)dA$$ where $D$ is the disk with $r=3$ centered at $(0,0)$. [What I did] I separated them to 3 ...
2
votes
3answers
38 views

Let $A\subset R^{n}$ . Then $A$ is disconnected iff there exists a continuous and surjective functon $f:A\to${0,1}

Let $A\subset R^{n}$ . Then $A$ is disconnected iff there exists a continuous and surjective function $f:A\to${0,1} How can I prove this? To prove $\rightarrow$, I know that if $A$ is disconnected, ...
2
votes
1answer
119 views

Why is this a fake proof?

I am aware of the "definition" of the total differential as follows: $$\mathrm{d}f = \frac{\partial f}{\partial x} \mathrm{d}x + \frac{\partial f}{\partial y} \mathrm{d} y.$$ Now, assume we wished ...
2
votes
2answers
48 views

Can anyone help me understand Lagrange Multipliers?

I'm currently trying to understand the method of Lagrange Multipliers. The explanation I'm currently looking at says something along the lines of "Suppose we wish to minimise the function $f(x,y)$ ...
2
votes
1answer
93 views

What are the roots of $ x^{2} = 2^{x} $? [duplicate]

What are the roots of $ x^{2} = 2^{x} $? I drew the graphs and found $ x = 2 $ and $ x = 4 $, and there is one other root in $ [-1,0] $. Can anyone describe an algebraic method to obtain all roots?
2
votes
1answer
39 views

Solve the differential equation for obtaining $x$ as a relation of $t$: $\frac{d^2x}{dt^2}=\alpha\sqrt{x}$

Question: Solve the differential equation for obtaining $x$ as a relation of $t$: $$\frac{d^2x}{dt^2}=\alpha\sqrt{x}$$ My attempt: $$\frac{d^2x}{dt^2}=\alpha\sqrt{x}$$ $$\Rightarrow ...
2
votes
1answer
55 views

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 $x = a$ ...
2
votes
1answer
58 views

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 ...
2
votes
1answer
37 views

Inverse Trigonometry Question (Stuck in Algebraic Simplification)

The original question: If x and y are of same sign, then find the value of $$\cfrac{x^3}{2} \csc^2 \left(\cfrac{1}{2}\tan^{-1} \cfrac{x}{y}\right) + \cfrac{y^3}{2} \sec^2 ...
2
votes
1answer
33 views

contradicting examples for series statements

I have tried to find counter examples for the following but without any success. Would appreciate your help: $a_n \to \infty$ then $a_n$ is monotonically increasing $|a_n| \to a$ then $a_n$ ...
2
votes
4answers
30 views

Maximum of a function defined via a definite integral

exploring a problem I have introduced a function: $$I(x)=\int_0^{\pi/2}xe^{-x\sin t} dt.$$ To my surprise the maximum of the function appears to be achieved at a value seemingly equal to $e$. I have ...
2
votes
2answers
57 views

What is a intuitive way to look at substitution?

Substitution can be a very powerful tool in mathematics sometimes simplifying problems, as an example: $\lim\limits_{n \to \infty} {(1-\frac{1}{n})}^n = {(1+\frac{1}{n})}^{-n} = {(1+\frac{1}{n})}^{n ...
2
votes
2answers
51 views

How would you differentiate this? I can't get anywhere

Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative? $\frac{\partial}{\partial t} \left\{ \int_{0}^{t} \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) d\xi ...
2
votes
1answer
34 views

Find a power series representation for the function. (Assume $a > 0$.)

I'm down to my last attempt (my teacher allows $5$ tries per question)! Thank you!!
2
votes
1answer
81 views

Maximize inner product of two vectors of which end points are on two separate circles

$$A=\{(x,y)\mid(x-3)^2+(y-4)^2=1\}$$ $$B=\{(x,y)\mid(x-5)^2+(y-2)^2=1\}$$ $$\text{Maximize}\space \overrightarrow{OA}\cdot \overrightarrow{OB}$$ What I tried so far is: For a given $A$, the product ...
2
votes
1answer
47 views

Laplace transform of $\cos^2(\omega t)$

Find the Laplace Transform of $\cos^2(\omega t)$, where $\omega$ is a constant. From a cosine identity: $cos^2(\omega t) = \frac{1}{2}(1+\cos(2\omega t))$. So then I get: \begin{align} ...
2
votes
1answer
42 views

Does the function exist?

Does the function which satisfies the following conditions exist: It is continuous on the segment $[a, b]$; It is not equals to the identical zero (it means that $f(x)$ may equal $0$, but not on the ...
2
votes
2answers
86 views

Product of two sinusoidal functions model

I'm trying to make a model of the rise and fall of sea levels. According to this explanation and image in the textbook, the product of two sinusoidal functions should look something like this: ...
2
votes
4answers
60 views

Can I write: $\lim \limits _{x \to x_0} (f(x) + g(x)) = \lim \limits _{x \to x_0} f(x) + \lim \limits _{x \to x_0} g(x)$?

If $\lim \limits _{x \to x_0} (f(x) + g(x))$ exists, can I write: $\lim \limits _{x \to x_0} (f(x) + g(x)) = \lim \limits _{x \to x_0} f(x) + \lim \limits _{x \to x_0} g(x)$ ? I mean, to write this do ...
2
votes
2answers
71 views

Determine the value of $x$ that is for $x^3-6x+7=0$ is false such that $x\in\mathbb{R}$

Determine the values of $x$ such that the following equation $x^3-6x+7=0$ is false, where $x\in\mathbb{R}$ My thoughts: I think that I should solve the equation $x^3-6x+7=0$ for ...
2
votes
2answers
43 views

Initial values problem with absolute value

I've some doubts about initial values problems involving differential equation with absolute values. For example if I have a differential equation like $y'=|x+1|$ with initial condition $y(3)=-2$, ...
2
votes
2answers
71 views

Non-rigorous limits to infinity trouble

I had to solve this problem: $$ \lim \limits_{x \to ∞} {x\over \sqrt {3x^2+2}} $$ and I had no idea how to get rid of the square root from the denominator. I googled for some time and found out that ...
2
votes
2answers
38 views

Integration of hypergeometric functions?

I would calculate the following integral \begin{equation} I_x = \int_{0}^{1} y^{b+\mu-1} (1-y)^{\nu-1}\, _2F_1(a,b+\nu +\mu;c; xy) \, dy. \end{equation} Such that $\quad \Re a,\Re b,\Re \mu, \Re \nu ...
2
votes
4answers
49 views

Calculate $\lim_\limits{x \to 0} \frac {\cos(xe^x)-\cos(xe^{-x})}{\arcsin^3x}$

$$\lim_{x \to 0} \frac {\cos(xe^x)-\cos(xe^{-x})}{\arcsin^3x}$$ Formula of difference $\cos$ have not helped me. Assuming that L'Hopital is forbidden but you can use asimptotical simplifications ...
2
votes
1answer
41 views

$\lim_{(x,y)\rightarrow (0,0)}\dfrac{xy^2}{x^2-y^2}$

I want to solve the following question. $$\lim_{(x,y)\rightarrow (0,0)}\dfrac{xy^2}{x^2-y^2}$$ I am going to use polar coordinates. $x=r\cos\theta$ and $y=r\sin\theta$ $$\lim_{(x,y)\rightarrow ...
2
votes
3answers
34 views

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 ...
2
votes
1answer
101 views

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 ...
2
votes
1answer
88 views

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 $c$ ...
2
votes
1answer
33 views

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 ...
2
votes
2answers
75 views

Find these limits using l'Hopital's rule

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 ...
2
votes
2answers
142 views

Word problem , Design a rectangular milk carton box

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 ...