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

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1
vote
2answers
64 views

If $f(X+a) = f(X-a)$ where $a$ is infinitesimally small then… [on hold]

If $f(X+a) = f(X-a)$ where $a$ is infinitesimally small then does the result $f'(X) = 0$ follow, where $X$ is a point along the $x$ axis?
-4
votes
2answers
61 views

Why is $n^\sqrt n - 2^n \to - \infty$ and $\sqrt n ^n - 2^n \to +\infty$ [on hold]

Can you explain technically why the following limits are correct? $$n^\sqrt n - 2^n \to - \infty$$ and $$\sqrt n ^n - 2^n \to +\infty$$
2
votes
0answers
26 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
-1
votes
2answers
48 views

How do I calculate the following limit? $\lim_{n \to \infty} n ((8 + \sin (2^\frac{1}{n}))^\frac{1}{3} -2)$

How do I calculate the following limit? I'm short on ideas for this one: $$\lim_{n \to \infty} n ((8 + \sin 2^\frac{1}{n})^\frac{1}{3} -2)$$
1
vote
0answers
21 views

References for the following functional

In many of the types of problems Ive looked at the following quantity keeps arising and I was wondering if anyone knew any references I could look at to learn some its properties. Take any function ...
3
votes
4answers
74 views

How do I calculate $\lim_{n \to \infty} n^\frac{1}{n} (n+1)^\frac{1}{n+1} … (2n)^\frac{1}{2n}$

How do I calculate the limit of the following sequence? $$\lim_{n \to \infty} n^{\frac{1}{n}} (n+1)^\frac{1}{n+1} ...... (2n)^\frac{1}{2n}$$
0
votes
1answer
12 views

Find limit inferior and limit superior of $[1+\sin n]$ and $n - [\sqrt n]$

I have to find the limit inferior and limit superior of the following sequences: $$[1+\sin n]$$ and $$n - [\sqrt n].$$ I have done similar exercises before, but never with the integer part function ...
0
votes
1answer
50 views

$\frac{1}{2}\int\frac{1}{1-x^2}d(1-x^2)$

I think the result should be $ln(|1-x^2|)^\frac{1}{2} + C$. But the answer is $ln(1-x^2)^\frac{1}{2} + C$. Could you please tell me where the wrong is? I just used the formula table.
1
vote
0answers
34 views

Primes with the first $k$ digits of the solution of the equation $e^{-x^2}=x$

Let $s$ be the solution of the equation $e^{-x^2}=x$ The first $1000$ digits are : ...
2
votes
3answers
43 views

Prove $\lim_{n\to\infty} \left(\sin \frac{\pi n}{2} \cdot \cos \left(\sin \frac{1}{n} \right ) \right)$ doesn't exist

I must prove that the limit $$\lim_{n\to\infty} \left(\sin \frac{\pi n}{2} \cdot \cos \left(\sin \frac{1}{n} \right ) \right)$$ doesn't exist and also find all of its partial limits. Apparantely ...
-1
votes
1answer
46 views

Prove serial representation of the integral $\int_0^1 x^x \,{\rm d}x$ [duplicate]

I have to prove the serial representation of: \begin{equation*} \int^1_0 x^x\,{\rm d}x=\sum^{\infty}_{n=0}\frac{(-1)^{n-1}}{n^n}. \end{equation*} It obtains: \begin{equation*} ...
2
votes
1answer
25 views

derivative of $f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi)$

Let $$ f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi) $$for some $r\in(0,\sigma)\subset\mathbb R$ and $\phi\in (0,\rho)\subset(0;2\pi]$. How do you calculate $Df=(\partial_1 f,\partial_2 f)$ ? I thought ...
0
votes
2answers
57 views

integration substitution what am I doing wrong?

integrate $\frac{x^3}{(4+x^2)}$ Let $u = 4+x^2$ so $\frac{du}{2}=xdx$ Then I need to integrate $\frac{(u-4)}{u}$ which comes out as $u-4\ln u$ converting back to $x$, $4+x^2-4 \ln(4+x^2)+C$ But I ...
-2
votes
2answers
71 views

$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1 }{ \ln { x } } dx } $ [duplicate]

How do I solve the following integral: $$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1}{ \ln { x } } dx } $$
0
votes
0answers
24 views

Where does the $kΔt$ go?

Frame 142 out of Quick Calculus, 2nd Edition: Suppose the position of an object is given by $S=f(t)=kt^2+lt+S_0$, where $k$, $l$ and $S_0$ are constants. Find $v$. I've worked the problem to: ...
3
votes
2answers
62 views

Finding line that divides an area into equal halves.

My question is simple, but I am not getting the answers for some reason. The question is: Consider the area enclosed between the graph of $y = 1 - x^2 $and the $x$ axis. Which line parallel to the ...
11
votes
2answers
500 views

How to avoid stupid mistakes in calculus exams without checking the whole process?

Few days ago I failed my Calculus exams. And again it was mostly due to simple mistakes such as forgetting about minus in front of fraction, switching y coordinates of two points etc. The assignments ...
-2
votes
0answers
28 views

A problem of Taylor series [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...
3
votes
2answers
36 views

Limit of cos function in a sequence

In my assignment I have to calculate to following limit. I wanted to know if my solution is correct. Your help is appreciated: $$\lim_{n \to \infty}n\cos\frac{\pi n} {n+1} $$ Here's my solution: ...
1
vote
1answer
23 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
0
votes
0answers
25 views

Find Taylor-Maclaurine expansion of function

Find Taylor-Maclaurine expansion of function: $f(x)=\sin(x)\cdot \cos(x) \cdot \arctan x^2$ my try: $f(x)=\frac{1}{2}\sin{2x} \cdot \arctan x^2$ and we have $\displaystyle \sin{2x} = \sum _{n=0} ...
-3
votes
0answers
19 views

Maximizing production [on hold]

Henry Ford has been called the Father of Industry with his use and development of the assembly‐line process of mass production. The field of Industrial Ecology views Ford’s approach to maximizing ...
0
votes
2answers
46 views

Determine uniform convergence of series

I have problems with checking if series $\displaystyle S_n(x)= \sum _{n=1} ^{\infty} \frac{1}{n(1+(x-n)^2)}$ is uniform convergent at $(0 , +\infty)$ My try: consider $\displaystyle |S_n(x) - ...
-4
votes
0answers
31 views

calculus continuous or discontinous [on hold]

Can you please help me with these questions? I need to find the discontinuity points of the following functions in their natural, maximal domain of definition.: $$f(x)=\frac{\sin^2x}{x|x(\pi-x)|}$$ ...
0
votes
1answer
21 views

Find the height, $h$, for the maximum area of the curved surface of a right circular cylinder

Find the height, $h$, for the maximum area of the curved surface of a right circular cylinder with base radius $r$ which can fit inside a sphere of radius $R$.
2
votes
1answer
39 views

Can critical point that $f''$ has different sign in its every neighborhood be a local extreme point?

Suppose that $f$ is a second order derivable function on $[0,1)$, and $f'(0)=0$. It is true that: If there exits $\delta>0$ such that $f''(x)\geq0$ for all $x\in[0,\delta)$, then $0$ is a local ...
2
votes
1answer
37 views

Can a function be differentiable at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a,b]$. Is it possible that $f$ is differentiable on the closed interval $[a,b]$, or must the maximal domain for $f'$ be $(a,b)$?
0
votes
2answers
40 views

Can someone help me prove these two limits? I need them for probability.

$$1.\lim_{n \to \ +\infty} \binom{n}{k} \frac{1}{n^k}\left(1- \frac{1}{n}\right)^{n-k}= \frac{e^{-1}}{k!}$$ $$2.\lim_{n \to \ +\infty} \binom{r}{k} \frac {(n-1)^{r-k}}{n^r}= \frac{2^{k}}{k!}e^{-2}$$ ...
0
votes
1answer
37 views

Solving the integral $\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$ with $\sinh$, $\cosh$?

I want to solve the following integral: $$\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$$ I thought maybe it's possible with $\sinh$ or $\cosh$ or something similar, but I can't figure it out. Thanks in ...
0
votes
2answers
65 views

Solve $x^2=\cos x$ using Taylor series for cosx

I have the following equation:$x^2=\cos x$ and calculating the Taylor series of $3rd$ degree around $0$ I've got: $x\approx \pm\sqrt{\frac{2}{3}}$ However, now I need to prove that if x is a ...
2
votes
3answers
48 views

How do I go about solving the integral of csc x?

So here is a solution provided by another user a couple of years ago and I've seen this solution before but I'm not clear on how or why? What would make me think or tip me off that I should multiply ...
1
vote
0answers
28 views

Intersecting lines in sectors of a circle.

Good day everyone, I'm trying to simulate a Laser Range Finder (LRF for short) in a corridor environment. I'm including a small fast sketch I did of this. I can't upload images yet, so I include just ...
3
votes
2answers
65 views

When is the function Continuous?

In my assignment I have to determine when is the function continuous. This is the function: \begin{equation} g(x) = \begin{cases} \left\lfloor {\sin\frac{1}{x}}\right\rfloor&\text{if} \space ...
1
vote
1answer
18 views

How do I figure out how many snacks to put in a vending machine each day, given a function that predicts how many will be sold in one instant?

Not a homework question, just kind of studying calculus on my own. Suppose I have to fill a vending machine with M&M packages tomorrow. To do that, I need to know how many M&M packages will ...
-1
votes
3answers
63 views

Confused about transcendental numbers [on hold]

I'm little confused about the type of numbers that had been known, for example, consider a polynomial equation with rational and irrational coefficients of a degree p-prime number that is greater than ...
4
votes
3answers
59 views

Tangent of a Straight Line

I just got back a math test in my EF Calc class, and I disagree with my teacher on this one problem. We are using derivatives to determine equations of lines tangent to a given equation. The equation ...
5
votes
3answers
161 views

What is the $\lim_{n\rightarrow \infty }(1+\frac{1}{n})^{n^n}$

What is the $$\lim_{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^{n^n}$$ I know that the $\lim_{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^{n}=e$, so I wanted to find the limit by the same ...
-1
votes
4answers
89 views

Evaluate $\lim_{x\to\infty}\left(1+\frac 4x\right)^{\frac x8}$

Evaluate $\lim_{x\to\infty}\left(1+\frac 4x\right)^{\frac x8}$ I think the end result is $1^\infty$, so the answer is undefined?
2
votes
4answers
147 views

How to evaluate this indefinite integration $\int \frac{\tan^4 \theta d \theta}{1-\tan^2 \theta}$?

I have to solve this indefinite integration $$\int \frac{\tan^4 \theta d \theta}{1-\tan^2 \theta}$$ I tried it as follows $$I=\int\frac{(\sec^2 \theta-1)\tan^2 \theta d \theta}{1-\tan^2 ...
2
votes
2answers
37 views

Find the plot of $y=1+\cos t$, $x=\sin^2t$.

I'm trying to find the plot for the following : $$y=1+\cos t, x=\sin^2t$$ I'm trying to get ride off variable $t$. This is what I done for some reason is incorrect : ...
15
votes
2answers
127 views

$xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ implies $\int_0^1 f(x) \,dx\leq\frac{\pi}{4}$

I want to show that if $f\colon [0,1]\to\mathbb{R}$ is continuous and $xf(y)+yf(x)\leq 1$ for all $x,y\in[0,1]$ then we have the following inequality: $$\int_0^1 f(x) \, dx\leq\frac{\pi}{4}.$$ The ...
4
votes
1answer
83 views

The infinite series $\sum_i a_i \prod_{j=0}^{i - 1}(1 - a_j)$ has sum equal to $1$

Suppose we have an infinite sequence of constants $\{ a_i \}_{i=0}^{\infty}$, where $0 < a_i < 1$ for every $i$, and we define $b_i \equiv a_i \prod_{j=0}^{i - 1}(1 - a_j)$. How do we prove ...
7
votes
2answers
140 views

Feynman technique of integration for $\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$

I've been learning a technique that Feynman describes in some of his books to integrate. The source can be found here: ...
0
votes
1answer
116 views

How to solve this integral: $\int \frac{\sqrt{-x^2 - x + 2}}{x^2}dx$?

Question is self explanatory. I have an exam and our professor gave us questions. This is the one I couldn't do. Any ideas would be very helpful: $$\int \frac{\sqrt{-x^2 - x + 2}}{x^2}dx$$
2
votes
0answers
76 views

Evaluate $\lim_{n\to\infty}\int_0^{\infty}\cos^n(x)dx$ [on hold]

How can I solve that $\lim_{n\to\infty}\int_0^{\infty}\cos^n(x)dx$?
1
vote
0answers
24 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
-4
votes
2answers
50 views

Calculate $\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$ [on hold]

Help with this integral, please! $$\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$$
6
votes
1answer
131 views

Integration of $\frac{e^{\cos^2x}+\ln(1+x)}{10^{x^3}\arctan(\sqrt{x})}$, possibly numerical

A couple of days ago I came across the following integral: $$\int_{0.02}^{0.08} \frac{e^{\cos^2x}+\ln(1+x)}{10^{x^3}\arctan(\sqrt{x})}\,{\rm d}x$$ The funny thing is, I found this integral written in ...
0
votes
2answers
41 views

Integrating $\sin^3(x)/(2+\cos(x))$

I could use some help solving the following integral: $$\int \frac{\sin^3(x)}{2+\cos(x)} dx$$ So far I tried using the equality: $$\sin^3(x) = \frac{3}{4} \sin(x) - \frac{1}{4}\sin(3x)$$ which ...
2
votes
4answers
63 views

limit of sin function as it approches $\pi$

In my assignment I have to find the Classification of discontinuities of the following function: $$f(x)=\frac{\sin^2(x)}{x|x(\pi-x)|}$$ I wanted to look what happens with the value $x=\pi$ because ...