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

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0
votes
1answer
130 views

Solids of Revolution Question (Method of Cylinders vs Disc/Washers)

Find the volume of the solid formed by revolving the region bounded by y=x^2+1, y=0, x=0, and x=1 about the y-axis. I was practicing this concept and I came across this problem. I did it using the ...
0
votes
1answer
47 views

Explicit form for $\left(e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)\right)^2$

Basically I have been working with polynomials of the form: $$P_n(x)=e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)$$ I do realize that an explicit form for $P_n(x)$ has been asked for on this site ...
3
votes
4answers
199 views

A calculus proof for the general term of the Fibonacci sequence [duplicate]

Let $a_0=1$,$a_1=1$ and $a_n=a_{n-1} + a_{n-2}$ for $n \geq 2$, I would like to prove: $$a_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n + 1}- \left(\frac{1-\sqrt{5}}{2}\right)^{n + ...
2
votes
1answer
91 views

Clarifying the elementary calculus used in this statistics problem

Let $X \sim N(\mu, \sigma^{2})$ and $Y = \alpha X + \beta$ for $\alpha > 0$. I'm looking at a demonstration that $Y = \alpha X + \beta \sim N(\alpha\mu + \beta, (\alpha\sigma)^{2})$, and find ...
0
votes
1answer
40 views

Why is the euclidean norm not differetiable at $0$?

I denote $N(x)$ as the norm-function, although in the denominator it stays $\|x\|$. $$\lim_{x\to 0} \frac{N(x)-N(0)}{\|x\|} = \lim_{x\to 0} \frac{N(x) - 0}{\|x\|} = \lim_{x\to 0} 1 = 1 \ne 0$$ 1) ...
2
votes
2answers
237 views

Differential equation type

How can I solve this differential equation $$(1 + x^2)(1+y^2)\mathrm dx +xy\mathrm dy = 0$$ It doesn't look like separable and I don't think it's neither homogenous. Maybe I need to use the ...
0
votes
3answers
36 views

A problem in calculus mean value theorem

Hi tried to solve this for hours, any idea how to approach this question: prove for every $x>0$ $$2x\times\arctan(x)>\ln(1+x^2)$$
25
votes
2answers
438 views

How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
0
votes
1answer
11 views

Determining at what points multiple variable functions are continuous

With a two variable function what is the procedure to figure out at what points it is continuous? Do I basically just look at what points it would be undefined and anywhere between those points it is ...
2
votes
1answer
23 views

Fourier series: Show that $f$ is a trigonometric polynomial

Let $N\in\mathbb{N}$ and $f_m:\mathbb{R}\to\mathbb{R}$, continuous functions and periodic, $T=2\pi$. Let's assume that $f_m \to f$ uniformly and for all $m\ge 1$: $$\left| \hat{f_m}(n)\right| \le ...
2
votes
0answers
67 views

Prove that $\underline{\int_{a}^b} f \leq 0 \leq \overline{\int_{a}^{b} f}$

My question is just to make sure my proof is on the right track. Problem: Suppose that the bounded function $f\colon [a,b]\rightarrow \mathbb{R}$ has the property that for each $x\in \mathbb{Q}$, ...
2
votes
2answers
40 views

[Proof Verification]Prove that if f is differentiable at $c \in I$ and $f'(c) = 0$, then g is not differentiable at $d:=f(c)$.

Proposition. Let I be an interval, and let $f: I \to \mathbb{R}$ be a strictly monotone and continuous on I. Let $J := f(I)$ and let $g:J \to \mathbb{R}$ be the inverse function of f. Prove that if f ...
2
votes
2answers
111 views

Is $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ is rational number?

Can anyone help with this: Is $\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}$ is rational number?
9
votes
5answers
782 views

Issue with Spivak's Solution

Here was the problem: Here is the solution from his solutions book: This is barely a proof. How can he just say let $f(c) = 0$? How do you prove that $f(c) =0$ and how do you prove that $f(d) ...
1
vote
1answer
34 views

Volume using triple Integrals, cylindrical coordinates

I want to calculate the volume of a solid with $z+1\ge x^2+y^2$ and $3\left(z-1\right)\le -\left(x^2+y^2\right)$. After cylindrical coordinates x=rcosϕ, y=rsinϕ I got $r^2-1\le z$, and $z\le ...
0
votes
1answer
34 views

Minimum value of $F(a,b)$.

Let $$F(a,b) = \sum_{i=1}^n \left[ y_i - (ax_i+b) \right]^2$$ Find the minimum of $F$. Evaluating the dirctional derivatives: $$\frac{dF}{da} = \sum_{n=1}^n 2\cdot (y_i - (ax_i+b))(-x_i) \\ ...
1
vote
4answers
78 views

$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$

$$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$$ I tried to write it as $$I=\int \sqrt{\frac{\cos^6(x)}{\sin^7(x)}}\,dx$$ And $$I=\int \sqrt{\frac{1}{\tan^6(x)\sin(x)}}\,dx$$ but it seems to go ...
0
votes
2answers
26 views

Find $\lim_{b\to a}\frac {1}{b-a}\ln\left[\frac{a(b-x)}{b(a-x)}\right]$ if $x$ is constant using l'Hopital's rule

if x is a constant what do I differentiate with respect to? My best guess would be $b$. However, is this correct? Also how do you differentiate that function with respect to b? Do you have to use the ...
0
votes
0answers
57 views

How to prove $f(c) = f(d) = 0$ [duplicate]

I ONLY NEED HELP WITH PROVING: $f(c) = f(d) = 0$ Robert Green's answer here : Is the first part of the answer, but I cannot problem that $f(c) = 0$ and $f(d) = 0$? How should I do this? Here was ...
1
vote
0answers
178 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
2
votes
3answers
85 views

Help me find the following limit : $\lim_{{n}\to{\infty}} (\frac{2^x+3^x+\cdots+n^x}{n-1})^\frac{1}{x} = ?$

I have no idea where to start.$$\begin{align}\lim_{{n}\to{\infty}} \left(\dfrac{2^x+3^x+\cdots+n^x}{n-1}\right)^{1/x} = ?, n >1\\\end{align}$$
0
votes
1answer
21 views

Finding the general solution of a 2nd order ODE?

SO here's a problem that I'm not having much progress with: Using substitution $u=cosx$, how can I find the general solution of $sinx(d^2y/dx^2)-cosx(dy/dx)+2ysin^3x=0$ Thank you so much for ...
0
votes
1answer
48 views

$f: \mathbb R \to \mathbb R$, define $f^2(x)=f(f(x))$

Given $f: \mathbb R \to \mathbb R$, define $f^2(x)=f(f(x))$, then which of the following statements are true: $1.$ $f$ is strictly monotonic then $f^2$ is strictly increasing. $2.$ ...
0
votes
1answer
22 views

Find the stationary points of the curve $z(x,y)=xy(12-4x-3y)$

$$z(x,y)=xy(12-4x-3y)$$ First, I expanded the brackets: $$z(x,y)=12xy-4x^2y-3xy^2$$ Then I found the partial derivatives with respect to $y$ and $x$: $$(\frac{\partial z}{\partial ...
3
votes
1answer
63 views

Evaluating a line integral

What's the quickest way of evaluating $$ \int_{|y| = r} \frac{1}{|x - y|^2} d \sigma_y $$ in real plane, where $x \in B(0,r)$. Could complex contour integration help us here?
2
votes
1answer
43 views

Given $x,y,z>0$: $\frac{2}{3x+2y+z+1}+\frac{2}{3x+2z+y+1}=(x+y)(x+z)$. Find Minimum Value Of: $P=\frac{2(x+3)^2+y^2+z^2-16}{2x^2+y^2+z^2}$

Given $x,y,z>0$: $\frac{2}{3x+2y+z+1}+\frac{2}{3x+2z+y+1}=(x+y)(x+z)$ $(1)$ Find Minimum Value Of: $P=\frac{2(x+3)^2+y^2+z^2-16}{2x^2+y^2+z^2}$ I found $2x+y+z\geq 2$ from (1) but it not work ...
2
votes
4answers
78 views

Integral $\int \frac{x+2}{x^3-x} dx$

I need to solve this integral but I get stuck, let me show what I did: $$\int \frac{x+2}{x^3-x} dx$$ then: $$\int \frac{x}{x^3-x} + \int \frac{2}{x^3-x}$$ $$\int \frac{x}{x(x^2-1)} + 2\int ...
-1
votes
1answer
45 views

Integral $\int(\frac{1}{n+\cos x})^{\frac{3}{2}}dx$

Find an expression for this indefinite integral.I tried using some online calculators, but it is not coming. $$\int\left(\dfrac{1}{n+\cos x}\right)^{\frac{3}{2}}dx$$
0
votes
1answer
228 views

Chain rule in a Sobolev space

(Chain rule) Assume $F : \mathbb{R} \to \mathbb{R}$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u \in W^{1,p}(U)$ for some $1 \le p \le \infty$. Show $$v :=F(u) \in W^{1,p}(U) \quad ...
2
votes
1answer
54 views

Find Distance Function from Acceleration Function

The (non-constant) acceleration as a function of time, $a(t)$, is defined and known over $[t_0, t_2]$. It is also known that $a(t)$ is integrable. Also, $a(t)=\frac{dv(t)}{dt}$ and ...
0
votes
1answer
42 views

strong convexity of loss function in multi-dimensional (high-dimensional) space

My question is based on this paper (see the last 10 rows in page 7). It seems this is a general claim: In machine learning or statistic, the loss function $l(W^TX, y)$ (a linear predictor) can never ...
11
votes
4answers
236 views

Prove $\int_{-\pi}^{\pi}\sin \sin x \,dx=0$ without using the fact that $\sin(x)$ is odd.

Prove $$\large\int_{-\pi}^{\pi}\sin (\sin x) \,dx =0$$ without using the fact that $\sin(x)$ is odd. Computing this in wolfram says that it is uncomputable, which leads me to believe that the only ...
1
vote
3answers
54 views

find total number of point at which f attains extremum is , $F = | x^{2} - 25 | $

To find total numberof point at which F attains extremum is $F = | x^{2} - 25 | $ I have made its graph and and from it i have got answer , but how do i use calculus to see ? I am confused with ...
0
votes
2answers
30 views

How to find the limit $\lim_{x \rightarrow 1^+}\left (1 - \frac{1}{x}\right)^x \left( \log\left(1 - \frac{1}{x}\right) + \frac{1}{x - 1}\right)$

How can I find the limit $\lim_{x \rightarrow 1^+} \left (1 - \frac{1}{x}\right)^x \left( \log\left(1 - \frac{1}{x}\right) + \frac{1}{x - 1}\right)$? I tried turning into a fraction so that I could ...
7
votes
3answers
504 views

Why $\lim_{x\to -\infty} \frac{x^3}{\sqrt{x^6+4}} =-1 $ and not $1$?

$$\lim_{x\to -\infty} \frac{x^3}{\sqrt{x^6+4}} $$ When we deal with infinities then 4 is negligible and so the above limit is equal to $$\lim_{x\to -\infty} \frac{x^3}{\sqrt{x^6}} = \frac{x^3}{x^3} = ...
0
votes
1answer
39 views

Finding the position vector from the velocity vector?

A block moves outward along the slot in the platform with a speed of $\frac{dr}{dt}=4t$ m/s, where t is in seconds. If the block starts from rest from half the distance of the platform, determine the ...
1
vote
0answers
46 views

Proof of Riemann Integration involving convex differentiable function

Define the convex function by: A function $f$ is convex on an interval if for $a$, $x$ and $b$ in the interval with $a<x<b$, we have $$\frac{f(x)-f(a)}{x-a} < ...
2
votes
1answer
33 views

A Sequence of Functions Converging to the Derivative at a Point

I'm reading Neal Carothers' Real Analysis and while in the process of constructing an everywhere continuous but nowhere differentiable function, he claims that $$\dfrac{f(v_n)-f(u_n)}{(v_n-u_n)} \to ...
2
votes
0answers
40 views

Positivity of the Fourier transform of a certain function

I am trying to show that the Fourier transform of $\cosh(x)^{-\nu}$ is positive for every $\nu>1$. I know that such a function has even Fourier transform and... that's about it. Could you suggets ...
1
vote
2answers
21 views

On adding terms to limits

Is it always possible to add terms into limits, like in the following example? (Or must certain conditions be fulfilled first, such as for example the numerator by itself must converge etc) $\lim_{h ...
1
vote
3answers
137 views

How to understand this derivation.

I don't understand how the first equality comes about. I read that L'Hopital's Rule was used. $ \lim_{h \to 0} \frac{f(x + h) + f(x - h) - 2f(x)}{h^2} = \lim_{h \to 0} \frac{f'(x + h) - f'(x - ...
0
votes
2answers
51 views

Different answer for integral for two different methods

I am trying to integrate $\frac{1-x}{(x+1)^2}$, but I get to answers for two different methods: First, $\frac{1-x}{(x+1)^2} = \frac{1-x+1-1}{(x+1)^2} = \frac{2}{(x+1)^2} - \frac{x+1}{(x+1)^2} = ...
4
votes
1answer
60 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
0
votes
1answer
42 views

Volume using Triple Integral

I would like to find the volume of the solid with $$2-z \geq x^2+y^2$$ $$z^2\geq x^2+y^2$$ $$x^2+y^2 \geq 1/4$$ and $x,y,z\geq 0$, using triple integrals. The problem I have is that I don't know ...
-2
votes
1answer
54 views

Analysing the derivatives of a sketched curve

I have the following graph and want to know: Are these statements correct and if not why? question A: $f'(-2) > 0$ question B: $f(0) = 0$ and $f'(0) = 0$ and $f''(0) \neq 0 $ question C: there ...
0
votes
1answer
48 views

A particle which travels a unit distance in a unit time, and starts and ends with velocity 0, has at some time an acceleration $\ge 4$.

(a) Prove that if $f$ is a twice differentiable function with $f(0) = 0$ and $f(1) = 1$ and $f'(0)=f'(1)=0$, then $|f''(x)| \ge 4$ for some $x \in (0,1)$. Hint: Prove that either $f''(x) \ge 4$ for ...
1
vote
3answers
177 views

What Does $y=A\exp(6x)$ mean?

So my professor used this and I don't really know what this equation means. $A$ is a positive constant, different $A$'s give different curves and these curves form a family $\mathcal{F}$. Given a ...
3
votes
1answer
101 views

Exterior derivative of a 2-form

I want to prove that the exterior derivative of a 2-form in $\mathbb{R}^n$: $${\alpha = \sum a_{ij} dx_i \wedge dx_j}$$ is: $${d \alpha = \sum (\frac{\partial a_{ij}}{\partial x_k} + \frac{\partial ...
1
vote
1answer
23 views

Show the $i$-th row of $D_f$ is $\nabla f_i$

Let $f:\mathbb{R}^m\to\mathbb{R}^n$. Show that the $i$-th row of the differential, $D_f$ is the gradient of $i$-th function, $\nabla f_i$ I understand it intuitively, because I know that ...
-1
votes
2answers
65 views

Is there a specific rule or theorem to do differentiation for integration?

I have seen many problems while doing my homework asking me to do a differentiation for an integral. How could I solve such problems? For example, how would I solve the following definite integral $$ ...