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

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1
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1answer
18 views

signature of the quadratic form: $f(x,y,z) = xy+yz+xz$

I am asked to find the signature of the following quadratic form: $f(x, y, z) = xy+yz+xz$ I have found that matrix wise, $f(x,y,z)= \begin{bmatrix}x&y&z\end{bmatrix}. ...
1
vote
1answer
32 views

When do I use the 'plus-minus' sign when square rooting both sides of an equation? (example in main body).

Above is the image of an integration by substitution question that I was doing; the answer can only have a plus sign in front (if you were to differentiate the answer to check if it's correct). ...
0
votes
1answer
60 views

Solve equation $\lim_{n \to \infty}\cos (nx)=1$

Solve equation $\lim_{n \to \infty}\cos(nx)=1$ Ok, $ \cos0 = 1$, but $0\times \infty \neq 0$ So, i think $x = \frac{0}{n}$ will work: $$\lim_{n \to \infty}\cos(n \times \frac{0}{n})=\lim_{n \to ...
1
vote
1answer
48 views

Is this estimate true or not true?

Let $\varepsilon>0$. Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ the standard normal density function. Then $$\lim_{\varepsilon\to 0}\int_0^1 \frac{1}{\sqrt{x}}\left[ ...
0
votes
1answer
37 views

find the values of $a$ for which function is invertible

I had a question in which i was told to find the range of values of $a$ for which the function is invertible (for which inverse exist) and function is $f(x)=ax+3\sin x+4\cos x$ what i tried for ...
0
votes
0answers
22 views

Truncation Error of 2-stage Runge-Kutta Method

I'm trying to derive the truncation error for the 2-step Runge-Kutta Method given by $$k_1 = f(x_n,t_n)$$ $$k_2 = f \left(x_n+\frac{2\Delta t}{3}k_1,t+\frac{2\Delta t}{3} \right)$$ $$x_{n+1}=x_n + ...
0
votes
0answers
28 views

A question on composite functions

Let $f:[a,b]\to\mathbb{R}$ and $g:[a,b]\to\mathbb{R}$ be two functions such that $f′(x)=g′(x)$ for all $x \in [a,b]$, where $f′$ and $g′$ denote the first derivative of $f$ and $g$, respectively. ...
2
votes
1answer
75 views

TIFR GS 2015 computer science: $G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$

Following expression was asked to be evaluated in TIFR GS 2015 exam, $$G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$$ where $x \in [0, 1]$ and $f(x)$ be any real valued continuous function. ...
3
votes
1answer
44 views

Find the interval of convergence to $\sum_{n=2}^\infty\frac{(-1)^nx^{n}}{n(n-1)}.$

My task is to find the interval of convergence to:$$\sum_{n=2}^\infty\frac{(-1)^nx^n}{n(n-1)}.$$ My work so far: Taking $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=|x|<1\implies ...
1
vote
2answers
51 views

Local extrema and minima of the multivariable function $f(x,y) = x^2y+y^2+xy$

Let $f(x,y) = x^2y+y^2+xy$ be a function, I want to find its local extrema an minima. I easily find that $f$ has 2 critical points: $(x,y)=(0,0)$ and $(x,y) = (-1,0)$. In order to find its local ...
0
votes
1answer
36 views

Level curve of the function $f(x,y)=\min\{x^2+y^2,xy\}$ [closed]

How I can find the level curve of the function $$f(x,y)=\min\{x^2+y^2,xy\}$$ From where I need to start to solve this problem? Thank you!
-2
votes
0answers
29 views

Application of dirac delta function [closed]

Please help me to find the $$\int e^{2\pi i a } \space \space \space \space -\infty<a<\infty$$ I am thinking of applying the dirac delta function.
-1
votes
0answers
19 views

Ratio version of mean value theorem [closed]

Let $f$ be a continuously second differentiable function. Under which conditions, we can say there exists M such that for every $a,b \in [0,1]$, we have $ \left |\frac{f(a)}{f(b)} - \left ...
1
vote
1answer
40 views

If $g(x) = xf(x^2)$ and $f(x)=\sum_{n=0}^\infty \sin(\frac{\pi}{n+2})x^n$, what is $f^{(20)}(0)$ and $g^{(35)}(0)$?

My task is this: (i)Let $$f(x)=\sum_{n=0}^\infty \sin\left(\frac{\pi}{n+2}\right)x^n.$$ Find $f^{(20)}(0)$ and $g^{(35)}(0)$ when $g(x) = xf(x^2)$. (ii)Find ...
-1
votes
0answers
19 views

Dirac delta function pplication [closed]

How do I find the int(1/(1-exp(-2*piai))),a, -inf, inf). I think of applying dirac delta function but I'm stucked. Can anyone please help me?
2
votes
3answers
59 views

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$ for $a, b > 0$. There are 2 problems. $|\sin(x)|^b = 0$ for $x = k \pi$ and $x^a = 0$ for $x = 0$. We can write ...
-3
votes
2answers
47 views

Find the Taylor's Series for $f(x)=x^3-10x^2+6$ about $x_0=3$ [closed]

Please help me. I want a solution for this question Find the Taylor's Series for $$f(x)=x^3-10x^2+6$$ about $x_0=3$.
1
vote
2answers
24 views

How to simplify from this thing to this (double derivative, stuck in the alegba part)

In my maths class I am doing double derivative to find concavity of the equation if I graph it, and getting these big functions. Plugging in even on online calculators skips from this big thing to ...
0
votes
2answers
24 views

How to determine the convergence radius and intervale of convergence from this sum

I have to find the converge radius and interval of convergence for the serie, I've tried the hHadamard criteria but I had no succes. I hope you can help me. $\sum_{n=1}^{\infty}(2+(-1)^n)(1+x)^{n-1}$ ...
1
vote
0answers
37 views

Stokes' theorem without the smoothness condition

I think I have proved the following version of Stokes' theorem: Teorem 1: Let $\beta: [0,4] \to \mathbb{R}^2$ the curve given by \begin{equation} \beta(t) = \begin{cases} (t,0) & \mbox{if } ...
-4
votes
0answers
36 views

Can someone tell me if constitutes enough proof to solve this infinite product?

I have a project do for my Calc II class where we must prove that $\lim_{n\to\infty}\prod_{k=1}^n(1-a_k)=0$ where $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
3
votes
3answers
39 views

Least value of $b$ for which inequality is true.

find the least natural number $b$ for which $x+bx^{-2}>2,\forall \space x\in (0,\infty)$ What I got confused with In my book they have done this: rearranged the function as ...
0
votes
1answer
19 views

Trying to find the square with minimum area inscribed in a Square of side L

A square has side length of L. Using the lagrange's multipliers, show that all squares inscribed in the square of side length of L, the square with minimum area has a side length of (sqrt(2)/2)L. I ...
0
votes
0answers
17 views

Are there any $k$-valued functions with first $k$ integrals all $0$? [closed]

Let $f(x)$ be a step function on the domain $[0, 1]$, which changes value at most $k-1$ times (and thus takes at most $k$ different values). Suppose that the first $k$ integrals are all $0$ (i.e. the ...
2
votes
2answers
58 views

Is continuity at a point only defined for points in the domain?

I'm using Michael Spivak's Calculus, 3rd edition textbook. Without ado, I'll state the definition given for continuity at a point: DEFINITION$\;\;\;\;$The function $f$ is continuous at $a$ if: ...
1
vote
1answer
31 views

Logarithmic Taylor series question [closed]

Consider the transformation of variables, x = $\frac{y+1}{y-1}$ How would you develop log(x) as a Taylor Series in y about zero?
1
vote
2answers
32 views

Limit of the Trapezoidal Rule

Let $f$ be Riemann-integrable and $\zeta=\{x_0,...x_n\}$ the most equidistant pratition. Show that $$\lim_{n\to\infty} \frac{b-a}{n} (\sum_{k=1}^n \frac{f(x_{k-1})+f(x_k)}{2})=\int_a^b f(x)dx$$ ...
1
vote
2answers
32 views

Prove that $\sup (-A) = -\inf A$

Prove that $\sup (-A) = -\inf A$. Note: Assume $A$ is a nonempty subset of $\mathbb{R}$ and $\alpha \in \mathbb{R}$ and define $\alpha A = \{\alpha A \mid a \in A\}$. I said let $x \in A$. Then ...
1
vote
2answers
31 views

$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$

I need to prove the following: $$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$$ It looks pretty intuitive since I can make $\frac{1}{n}$ as small as I want, thusk making $a$ as close as to ...
5
votes
2answers
87 views

Is the series convergent

Is series $\sum_1^\infty \frac{\ln(1+1/2) \ln(1+1/4) \cdots \ln(1+1/(2n))}{\ln(1+1/1) \ln(1+1/3) \cdots \ln(1+1/(2n-1))} = \sum_{n=1}^\infty \prod_{m=1}^n \ln(1+1/(2m))/(\ln(1+1/(2m-1))$ convergent ?
6
votes
3answers
97 views

How to see $\cos x \leq \exp(-x^2/2)$ on $x \in [0,\pi/2]$?

Can anyone help me with the above inequality? I tried looking at the series expansion and I guess the answer indeed lies there, but I fail to see it. Thanks
1
vote
1answer
36 views

prove a finite limit exists

Let $f$ be differentiable for any $x$. Given that $$\lim_{x\to \infty} f'(x) = 0,$$ prove there exists a finite $L$ such that $$\lim_{x\to \infty} f(x)=L.$$ By definition: ...
2
votes
1answer
27 views

Mean-value Theorem $f(x)=\sqrt{x+2}; [4,6]$

Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for $c$ that satisfies the conclusion of the ...
1
vote
1answer
32 views

Integrating using polar co-ordinates

Hey I've just finished an exam paper and just am stuck with one question. It's something that usually makes sense to me but for some reason I can't get this one: Let $R = \{(x,y) : x,y ≥ 0, x^2 + ...
0
votes
1answer
29 views

Showing $\mathbb{B}_{\mathbb{Q}}$ is a bases for $\mathbb{R}_{\text{usual}}$

Show that the collection $\mathbb{B}_{\mathbb{Q}} := \{(p, q) \subseteq \mathbb{R} : p, q \in \mathbb{Q}, p < q \}$ is a basis for the usual topology on $\mathbb{R}$. Solution: We know that ...
5
votes
7answers
507 views

Square root of both sides [closed]

If you have the equation: $x^2=2$ You get: $x=\pm \sqrt{2}$ But what do you do actually do? What do you multiply both sides with to get this answer? You take the square root of both sides, but the ...
0
votes
0answers
15 views

Kullback-Leibler Divergence (KL) and Approximation Symmetry Property

The Kullback-Leibler Divergence doesn't satisfy the symmetric property. But, it can be approximated (bounded) to such a value. in this paper: Compressing Interactive Communication under product ...
2
votes
1answer
54 views

$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{x^2}{b^2}}e^{-\frac{x^2}{a^2}+\frac{y^2}{b^2}}dxdy$

$$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}\,e^{-\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)}\,dx\,dy$$ Basically I have done problems similar to this, using the theorem that if ...
6
votes
2answers
77 views

Showing that $\prod_{n=1}^{\infty}\left(1+\frac{1}{F_{2^n+1}L_{2^n+1}}\right)=\frac{3}{\phi^2}$

Infinite product $F_{n}:=[1,1,2,3,5,8,\cdots]$ and $L_{n}:=[1,3,4,7,\cdots]$ for $n=1,2,3,\cdots$ respectively. $\frac{1+\sqrt5}{2}=\phi$ Show that, ...
3
votes
2answers
37 views

Evaluating $ \dfrac{1}{2 \pi} \int_{-\infty}^\infty \int_{-\infty}^\infty e^{tuv} e^{-u^2/2} e^{-v^2/2}\ du\ dv $

Solving a probability problem I came across this integral: $$ \dfrac{1}{2 \pi} \int_{-\infty}^\infty \int_{-\infty}^\infty e^{tuv} e^{-u^2/2} e^{-v^2/2}\ du\ dv $$ Can you explain how to ...
0
votes
3answers
24 views

Rearrangement of alternating harmonic series that does not converge

From Riemann's series theorem, we know that, given a conditionally convergent series, we can permute the elements of the series in order to basically do whatever we want. I have seen a rearrangement ...
0
votes
0answers
29 views

Proving absolute or conditionally convergence

I have the following partial series - $$ \sum_{n = 0}^\infty {a_n} $$ when $$a_n=\begin{cases} \frac{1}{n} &; \quad n \ \text{is even}\\ \frac{-1}{n^2} &; \quad n \ \text{is odd}\ . ...
0
votes
0answers
27 views

Show that $\{f_n(x) \}_{n \in \mathbb{N}}$ doesnt converge in M.

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
1
vote
2answers
84 views

Am I using sandwich theorem incorrectly?

I saw this question and wondered how OP of that question was able to do : $$0<\sin x+1<2$$ this $$\frac 0{|x|}<\frac{\sin x+1}{|x|}<\frac 2{|x|}$$ and when $x\to \infty$ he got the limit ...
0
votes
1answer
10 views

Finding the Components of a Hessian Matrix of a Quadratic Form

I'm trying to find the Hessian form of the following quadratic form: $f(x,y) = x^2y+y^2+xy$. I know that it's in the form of a matrix and that the elements of $H_f(a)_{i,j}=\dfrac{\delta^2f}{\delta ...
0
votes
2answers
33 views

Difficult projectiles question

A particle $P$ of mass $m$ lies on a plane inclined at an angle $\alpha$ to the direction vector $\mathbf{i}$. At $t=0$ the particle is projected from the origin of the coordinate system with speed ...
0
votes
2answers
41 views

Differentiate a Function (Help me Solve?!)

Find $\dfrac{d}{dx}$ for: $C(1+Ae^{-bt})^{-1}$ I have tried and arrived at: $-C(1+Ae^{-bt})^{-2}$ however that is not the correct answer.
1
vote
1answer
29 views

show that $ \left | y^{2} f(y)-x^{2} f(x) \right | \leq \left | y^{3} -x^{3} \right | (\forall x,y\in \mathbb{R})$

Q: If $f(x)$ differentiable function and $ \left | f '(x) \right | \leq 1 (\forall x\in \mathbb{R})$, $f(0)=0$ then show that $$ \left | y^{2} f(y)-x^{2} f(x) \right | \leq \left | y^{3} -x^{3} ...
2
votes
4answers
129 views

Extended $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ limit law?

So I've learned that $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ is true and the following picture really helped me get an intuitive feel for why that is I have been told that this limit is ...
0
votes
1answer
49 views

How do I evaluate $\displaystyle\prod_{r=1}^{\infty }\left (1-\frac{1}{\sqrt {r+1}}\right)$?

I am not being able to find the specific product $\prod_{r=1}^{k} \left(1-\frac{1}{\sqrt {r+1}}\right)$ so to evaluate the given problem when $k \to \infty $.