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

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0
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1answer
11 views

Use the form of the definition of the integral to evaluate the integral (4-2x)

Use the form of the definition of the integral to evaluate the integral. $$a=2, b=5, (4-2x)dx$$ $$\lim_{n \to \infty} \sum_{i=1}^{n}(4-2x)\Delta x$$ $$\Delta x = \frac{3}{n}$$ $$x_i=2+i\frac{3}n$$ ...
2
votes
2answers
58 views

What is a French curve, as mentioned by Feynman?

I'm reading "Surely You're Joking, Mr. Feynman!", he says: I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a ...
0
votes
1answer
27 views

Limit Definition to find derivative

Given that $$f(x) = \begin{cases}x\left(1-\cos(\frac1x)\right) & x \ne 0 \\ 0 & x=0 \end{cases}$$ Use the limit definition of the derivative to find $f'(0)$. Not sure how to do it but this ...
0
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2answers
48 views

show that $\lim_{k\to\infty}\frac{ k!} { k^k}= 0$ using stirling's formula [on hold]

Prove that $\lim _{k\to \infty \:}\left(\frac{k!}{k^k}\right)$ = 0:
9
votes
0answers
72 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
0
votes
1answer
49 views

Calculus in an abstract space

This is page 302 of PDE Evans, 2nd edition. THEOREM 2 (Calculus in an abstract space). Let $\textbf{u} \in W^{1,p}(0,T;X)$ for some $1 \le p \le \infty$. Then $\text{(i)}$ $\textbf{u} \in ...
0
votes
1answer
53 views

Integral involving the zeta function

What is the solution to the following integral? $$\int_0^\infty\frac{x^n}{e^{ax}-e^{bx}} dx$$
1
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3answers
83 views

Prove the limit doesn't exist using basic Calculus

I was given this problem by a friend: $$ \def\limit{\lim_{x\to5}} \def\top{\sqrt{x}-2} g(x) = \frac{\top}{x-5}\\ \limit g(x) = \quad? $$ This caught me by surprise, because I can't remember how to ...
2
votes
1answer
39 views

Volume of revolving $ y = \sin(x) $ about a line $ y = c $

Consider the surface formed by revolving $y=sin(x)$ about the line $y=c$ from some $0\le{c}\le{1}$ along the interval $0\le{x}\le{\pi}$. Set up and evaluate an integral to calculate the volume V(c) ...
0
votes
1answer
29 views

Find the rate of change of the balloon's radius, given the rate of change of its volume

So I have this balloon right? I am blowing air into it and it's volume is increasing at 4cm^3/s. I want to know what the rate of change will be when the balloon's radius is equal to 10cm (assuming ...
0
votes
1answer
18 views

Radius of convergence - Maclaurin series

Given $f(x)$ a function that has derivatives of all orders in $\Bbb R$, and $R_n(x)$ the $n^{th}$ order Lagrange form of the remainder, Prove or disprove: if $$lim_{x \to 0} {\frac {R_n(x)} {x^n}} ...
0
votes
1answer
29 views

how to differentiate an indicator function?

I'm reading this paper and I arrived at this part when they introduce a formula for what they call 'an indicator function'. Here is a shot: what I understood from the first two formulas is that I ...
3
votes
0answers
42 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
1
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0answers
23 views

Determination of a volume.

Consider, in the Cartesian plane, the square Q having vertices in the points $(-1, 0), (1, 0), (0, -1)$, and $(0, 1)$. The sections of a solid with planes orthogonal to $y=0$ are squares having two ...
1
vote
4answers
38 views

Question about tangent and slope

Given the graph of $y=-e^{-x}$ and that there is a tangent to the graph that crosses the x axis at $(-4,0)$ determine the slope of that line. So this seems like a simple question but I don't know why ...
0
votes
1answer
33 views

Volume of the solid with given base, whose sections with the planes orthogonal to $y = 0$ are rectangles of height $4$

Please help me to solve the following problem: Determine the volume of the solid having as base the portion of cartesian plane limited by $y = 0$ and by $y = x^{3}$ in the closed interval ...
1
vote
1answer
30 views

Question about the Fundamental Theorem of Calculus

So I have studied the FOTC, but not really sure of what I read so this question is just to help me learn the FOTC and understand how to do problems like it. $$ if $$ $$F(x)=\int_0^x\sqrt{sin^3(t)}dt$$ ...
0
votes
0answers
18 views

Sequence of integers in arithmetic progression and convergent sequence

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...
4
votes
2answers
44 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
0
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0answers
24 views

How to evaluate the following integral? $∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$

I'm trying to calculate the following integral: $$∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$$ I tried by parts with no success and also by writing $\exp (ix)$ in terms of $\sin$ and $\cos$, with no ...
0
votes
0answers
11 views

abstract conceptual usage of power series, advice on how to approach similar problems

This problem is a bit strange, I have the solution for this particular one, I just think that it a very ambiguous question. How would you go about solving it? My answer is that b is larger because ...
2
votes
1answer
59 views

Evaluating $\int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$

I'm trying to find the general formula for the following: $$I_n = \int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$$ I remember doing it a while back but for the life of me, I can't remember right now. ...
3
votes
0answers
66 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
1
vote
1answer
35 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
1
vote
2answers
25 views

Convergence, interval, radius of power series, conceptual explanation please [on hold]

Could someone explain how to solve the problem. A very basic and broad understanding is what I am looking for so that if I were to have to approach this problem with different numbers I would know ...
5
votes
4answers
321 views

Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
0
votes
0answers
9 views

Heuristic Algorithm for integrating algebraic functions

Is there any heuristic "algorithm" or a good technique for integrating algebraic functions? The general algebraic case was solved by Trager and Davenport. But their algorithms are rather complicated ...
1
vote
2answers
42 views

Help on an integration by substitution

In a proof to show that $\int_{0}^{1} f \left(\left\{1/x\right\}\right) \frac{ \mathrm{d}x}{1-x}=\int_{0}^{1} f(v) \frac{ \mathrm{d}v}{v}$, i found this line : ...
1
vote
4answers
78 views

How many solutions $k>1$ does the equation $\exp ((k-1)/( k+1))=\sqrt{k}$ have?

I have the following equation: $e^{\frac{k-1}{k+1}}=\sqrt{k}$. The question is: how many solutions does it have? ($e$ is Euler's constant and k is a positive real number >1).
1
vote
1answer
30 views

Elevation of 3D function

$f(x,y) = \begin{cases} x^2/y & y \neq 0 \\ 0 & y = 0\end{cases}$ I need to draw the elevation (or you may call it Equivalent curve) of this function and I don't know how to draw them. Can ...
3
votes
1answer
30 views

Evaluate Derivative $\lim_{x \to 1}\frac{10x-1.86x^2 - 8.14}{x - 1}$

Evaluate Derivative $\lim_{x \to 1}\frac{10x-1.86x^2 - 8.14}{x - 1}$ I've already evaluated the limit using the $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ definition of a limit, but now I'm curious as ...
0
votes
2answers
35 views

Is there a continuous compact supported function $f: \mathbb{R}^n\rightarrow \mathbb{R}^{2n}$ such that $f^{-1}$ is continuous differentiable

Is there a continuous compact supported function $f: \mathbb{R}^n\rightarrow \mathbb{R}^{2n}$ such that $f^{-1}$ is continuous differentiable? I don't know which theorem is related to this question, ...
0
votes
0answers
36 views

Prerequisites of Vector Calculus for Freshman

I'm doing BSc as Physics major, i had studied Calculus and Vectors in 12th grade. For now i want to study Mathematical physics myself, i had decided to follow Arfken but it starts from the topics ...
-2
votes
0answers
33 views

Equilibrium in supply and demand [on hold]

The demand for a commodity linearly decreases by 0.5 unit for each unit increase in price & it vanishes when the price is set at 60 . The supply of the commodity vanishes when the price is set ...
0
votes
1answer
27 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
0
votes
0answers
21 views

What is the sign of the generalized Stieltjes constants $\gamma_{k}(a)$?

Recall that the Stieltjes constants $\gamma_{k}$ appear as the coefficients in the regular part of the Laurent expansion of the Riemann zeta function about $s = 1$: $$ \begin{align} \zeta(s) = ...
0
votes
0answers
20 views

How to write f(z) as $\int_0^z g(z,t) dt$? [on hold]

how to write $f(z)$ as $\int_0^z g(z,t) dt$ ? For clarity no trivial cases such as $g(z,t) = h(t)$ , $g$ must really depend on $z$. For instance if $f$ is the Lambert-W function. What techniques ...
1
vote
3answers
89 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
0
votes
0answers
31 views

Method of Characteristics for a PDE

I'm working through a problem at the moment, and I've got an answer, but it seems far too complicated... I've been asked to use the method of characteristics to solve the following PDE; $$x^2 u_x ...
0
votes
2answers
24 views

Linearization of a function using sqrt ln, confused with the denominator

Can someone please explain why the answer to this question is $\frac{9}{7\cdot169}$ rather than $\frac{9}{7\cdot\sqrt[]{169}}$? I understand this may seem simple, but the concept is not at all ...
0
votes
1answer
53 views

Integration of powers: nested $dx$?

How do we solve the likes of the following expression: $$ \int_0^2 \frac{x\,dx}{\sqrt{1 + 2x^2}}\,dx $$ I'm bothered by the nested $dx$ in the numerator. How is this solved using the general power ...
0
votes
2answers
26 views

Problem understanding an implicit differentiation

Here is a general budget constraint: $p{_1}x{_1}+p_{2}x_{2}=M\Leftrightarrow \frac{p_1}{p_2}x_1+x_2=\frac{M}{p_2}\Leftrightarrow {p_{1}}'x_1+x_2=M{}'$. The main idea is that since prices are given, ...
0
votes
2answers
31 views

Solve nonliner equations

We are trying to find intersection of hyperbolas and we ended up in five equations $$\begin{align} A_1X^2+B_1Y^2+C_1XY+D_1X+E_1Y+F1&=0\\ A_2X^2+B_2Y^2+C_2XY+D_2X+E_2Y+F2&=0\\ ...
0
votes
0answers
17 views

Differential of stochastic process

How do I find the dynamics of $X_t=\int_0 ^t \sigma (s,t) dW_s$? It seems that the simple solution of $dX_t = \sigma(t,t)dW_t$ is not correct since I get $X_t = \int _0 ^t \sigma(s,s) dW_s$ if I ...
0
votes
1answer
47 views

Is it true that $\int_{-C} f(x, y)ds = -\int_C f(x, y) ds$ [on hold]

I think it is more of a convention question, right ? $$\int_{-C} f(x,y)ds = -\int_C f(x,y) ds$$
3
votes
1answer
24 views

Minimum and Maximums involving Partial Derivatives

Hi all. I was wondering if someone could help me learn to approach this problem. the partial derivatives I have are: 4x-8 for fx & 2y for fy. fxy = 0. How does the constraint affect the ...
2
votes
3answers
78 views

The sum of geometric series $e^{k-1}/\pi^{k+1}$

Let $T_n=\sum _{k=1}^{n}\dfrac{e^{k-1}}{\pi ^{k+1}}$ calculate the $\lim_{n\to\infty}T_n$ Note $T_n$ is a geometric series: \begin{align*} T_n&=\sum _{k=1}^{n \:}\dfrac{e^{k-1}}{\pi ...
0
votes
1answer
89 views

How to find the derivative of the function $ \int_{x}^{x^2}\frac{t}{\ln(t)}dt$? [on hold]

The problem is to find $\displaystyle\frac{d}{dx}\int_{x}^{x^2}\frac{t}{\ln(t)}\,dt$ I could do this if I had the first clue on how to integrate $\dfrac{t}{\ln(t)}$ but even wolframalpha is giving ...
0
votes
2answers
27 views

Finding the Value of K in an Integral Function

Given the function $$f(x)\begin{cases} -2(x+1), & \text{x $\le0$} \\ k(1-x^2), & \text{x $\gt0$} \\ \end{cases}$$ Find the value of k for $$\int_{-1}^1f(x)dx=1$$ Wasn't really sure how to ...
6
votes
0answers
122 views