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

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1
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2answers
43 views

If $\int_{\mathbb R} hf=\int_{\mathbb R} gf$ for every probability density $f$, does it follow that $h=f$ a.e.?

If $\int_{\mathbb R} hf=\int_{\mathbb R} gf$ for any arbitrary probability density $f$, does it follow that $h=f$ a.e.? It seems to be right for that taking f as an indicator function over ...
0
votes
1answer
25 views

polar equation related question and im really stuck

Show that the polar equation of the line x + (√3y) = 2 can be expressed as r= sec( θ - (π/3) )
1
vote
1answer
55 views

$Lu = -u'' + f(x) u$ has a strictly negative eigenvalue

Let $f(x)$ be a real valued continuous function such that $\int_{0}^{1}f(x)\, dx = 0$ and $f$ is not identically 0. The problem I am working on is to show that $Lu := -u'' + f(x)u$ with boundary ...
1
vote
1answer
87 views

Is changing variables the same as substitutions?

I have asked several on a similar matter. This time the question is tad different. $$\int_{\mathbb{R}} e^{-x^2} dx$$ We let $x=y \implies dx=dy$ $$\int_{\mathbb{R}} e^{-(x^2 + y^2)} dxdy$$ But ...
1
vote
2answers
87 views

Finding derivative $f(x)={2\over x^3}$

I have to find the derivative and the slope at $a=6$ The function is $f(x)={2\over x^3}$ I have to find the answer using the formula, $$f'(x)= \lim_{\Delta x \to 0} {f(x+ \Delta x) - f(x) \over ...
6
votes
4answers
203 views

Is the usual proof of $\lim_{x\rightarrow 0}\frac {\sin(x)}{x} = 1$ an honest proof?

In a lot of textbooks on Calculus a proof that $\lim_{x\rightarrow 0}\frac {\sin(x)}{x} = 1$ is the following: Comparing the areas of triangles $ABC, ABD$ and circular sector, you get: $$ ...
-3
votes
5answers
145 views

What allows change of variables? [duplicate]

In school, especscially, one is not taught "why" we can change variables, dummy variables in integration. $$\int_{a}^{b} f(x) dx$$ We can change the $x$ variable to $y$ for example. The idea is ...
0
votes
2answers
40 views

Question about continuous from right or left

I saw the claim that say: "Suppose that g is continuous at $x_0$; $g\left(x_0\right)$ is an interior point of $D_f$ ; and f is continuous at $g\left(x_0\right)$: Then $\left(f\:\circ \:g\right)$ is ...
1
vote
0answers
28 views

Prove that $f$ has a global maximum given a product of Taylor polinomials

Let $f:\mathbb{R}\to\mathbb{R}:f\in C^{\infty},f\left(1\right)=0$ and the product of its Taylor polinomials of order 2 in $x_{0}=0$ and $x_{0}=1$ be $P(x)=x^{4}-2x^{3}+2x^{2}-x$. Prove that $f$ has a ...
4
votes
1answer
69 views

Infinite Riemann sum of $x^3$

Write $$\int_{1}^{3} x^3 dx$$ as a riemann sum. Here is what I thought: $$\Delta(x) = \frac{2}{n}$$ $$f(x) = (\Delta(x)k)^3 = \left(\frac{2k}{n}\right)^3$$ $$I = \int_{1}^{3} x^3dx = \lim_{n ...
1
vote
1answer
54 views

Find the point where the slope changes drastically

I have a distribution for which I have to find the point where the slope changes drastically. In visual terms, I have to find this point: I though I could use derivatives, but for the following ...
5
votes
2answers
71 views

Prove that $ \sum_{k=1}^\infty {\ln(k) \over k^2} \le{2+3\ln2 \over 4} $

Prove that $$ \sum_{k=1}^\infty{\ln(k) \over k^2} \le{2+3\ln2 \over 4} $$ Start with $$ \sum_{k=1}^n {\ln(k) \over k^2} \le \int_1^n {\ln(x) \over x^2}\,dx + f(1) $$ where $$ f(x) = {\ln(x) ...
2
votes
0answers
40 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
3
votes
2answers
59 views

Question about a differentiable function at point $a$.

Let $f$ be differentiable at point $a$. Prove than if $\lim \limits_{n \to \infty}x_n =\ a^{+}$ and $\lim \limits_{n \to \infty}y_n = a^{-}$ then $$\lim \limits_{n \to \infty} \frac{ f(x_n) - ...
1
vote
1answer
244 views

Delta Epsilon Proof Limit as $x\to -\infty$, equals infinity

Prove using the formal definition of limit that, limit as x goes to negative infinity of $$\lim_{x\to-\infty}x^2+3\sin x =\infty$$ I know the definition is for every $\epsilon>0$, there exists a ...
2
votes
3answers
89 views

If $2^x=9$ and $4^y=27$ then what will $\frac {x+2y}{2y-4x}$ be?

If $2^x=9$ and $4^y=27$ then what will $\frac {x+2y}{2y-4x}$ be? I have seen such problems many times but no specific method to solve them. I hope I find the general rule to solve them here.
1
vote
3answers
287 views

Determining slope that cuts off least area

So here is the question: Determine the slope of the line that passes through the point $(1,2)$ and that cuts off the least area from the first quadrant. I've thought about this question, and I ...
5
votes
3answers
139 views

Is the limit a function?

We're aware of the existence of the limit in the context of calculus, e.g. where $f:\mathbb{R}\to\mathbb{R}$, we may have: $$\lim_{x\to z} f(x) = y$$ My question is whether it is valid to see the ...
2
votes
6answers
277 views

Math, how do we know if a substitution is true?

For instance, in calculus we often do u-substitutions. Quite often, we do trignometric substitutions to solve integrals. For instance, if we have the following relation $y=\sqrt{1-x^2}$ And we ...
1
vote
1answer
31 views

Basic FTC question

I was working through some Fundamental Theorem of Calc questions on Brilliant and I had some trouble with the following: "Given $$f(x)=\int_3^{x^2}\frac {\sqrt {1+t^6}} tdt$$ ...
5
votes
1answer
101 views

proof that function is constant

I'm annoyed by quite a simple problem in calculus (I apologize in advance if I'm not using adequate terms in English, I don't take the course in English nor am I a native speaker): Let $f:\mathbb ...
0
votes
2answers
73 views

Show that $V=\frac{Z_1}{\sqrt{(Z^2_1 + Z^2_2)/2}}$ has pdf $f(v) = 1 / (\pi \sqrt{2-v^2}),-\sqrt2<v<\sqrt2$

Let $Z_1, Z_2$ have independent standard normal distributions, $N(0,1)$. If the random variable in the numerator did not also appear in the denominator this would be a t distribution. Should ...
3
votes
3answers
120 views

Trouble with definite integral calculating probabilities

I cannot solve this: $$\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} \tan ^{-1}(a+\tan (x)) \, dx$$ it apeared when trying to find out the probability: $$P\{\tan a - \tan b \leq 2x\},\ \ 0 < x < ...
9
votes
3answers
254 views

$\lim_{x\to +\infty}\frac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$

Determine if the following limits exist $$\lim_{x\to +\infty}\dfrac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$$ note that $\lfloor x \rfloor \leq x < \lfloor x \rfloor + 1 \implies ...
2
votes
1answer
22 views

Is $\operatorname{Gr}(\log(x)) \subset \mathbb{R}^2$ a closed subset.

We know that graph of a continuous function (target space being $T_2$) is a closed set. But the usual logarithm function , $\log:(0,\infty) \to \mathbb{R}$ is continuous but cannot be extended ...
1
vote
1answer
67 views

Problem with the correction of series exercise

Well, I have a little problem with the correction of an exercise, I have to calculate : $$ S(x)=\sum_{n=2}^{\infty} \frac{n+(-1)^{n+1}}{n+(-1)^n}x^n, x \in \mathbb{R} $$ So I have : $$ ...
0
votes
2answers
130 views

Derive the DE that is satisfied by the family of curves $2y=k(x^{2} + y^{2}) $

$2y=k(x^{2} + y^{2})$ can be rewritten as $y = cx^{2} + cy^{2}$ then differentiate both sides wrt x: $dy/dx = 2cx + 2cy*dy/dx$ => $ dy/dx ( 1 - 2cy)= 2cx $ => dy/dx = 2cx/(1-2cy) I checked my ...
-1
votes
2answers
141 views

Definition of Bound/Free Variables

You may have already seen that: $$\int_0^1 x \, dx = \int_0^1 y \, dy$$ But the formal reason why this is done is because $x$ is a bound variable correct? QUESTION: We are allowed to change ...
1
vote
0answers
84 views

Borel-/Laplace-transform and $\psi$-function

I'm considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A (of course thought as of infinite size) $ \qquad $ The ...
6
votes
1answer
148 views

Evaluate $\int \frac{\mathrm dx}{1+\cos^2 x}$

$$\int \frac{1}{1+\cos ^2x} \,\mathrm dx$$ I have to integrate the expression above: I tried with substitutions $\cos x=t$ and $1+(\cos x)^2=t$, but those didn't work, and I couldn't find any useful ...
1
vote
1answer
141 views

Calculate the surface integral $\iint_S (\nabla \times F)\cdot dS$ over a part of a sphere

How can I calculate the integral $$\iint_S (\nabla \times F)\cdot dS$$ where $S$ is the part of the surface of the sphere $x^2+y^2+z^2=1$ and $x+y+z\ge 1$, $F=(y-z, z-x, x-y)$. I calculated that ...
6
votes
1answer
201 views

Sum $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{10^{mn}}$

Does this double series has closed form (i.e. can be computed) ? $$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{10^{mn}}$$
4
votes
2answers
161 views

Why do you reject negative base solution for Logs?

$log_x64=2$ translates to $x^2=64$ This solves to $x=\pm8$ Why do you reject the solution of $x=-8$ ? Doesn't it successfully check? $log_{-8}64=2$ means "The exponent for -8 to get 64 is 2" which ...
0
votes
1answer
78 views

A condition for mid-convex implies convex

Let I an open interval, and $ f: I \rightarrow \mathbb{R} $ such that: $\forall (x,y) \in I^2 ; f(\frac{x+y}{2}) \le \frac{f(x)+f(y)}{2}$ There exists an interval $[a,b]$ such that $a<b$ and ...
-1
votes
5answers
80 views

$\lim\limits_{x\to\infty} \frac{\sqrt{(4x +3) +2\sqrt{1+ x}}}{\sqrt x}$ without L'Hôpital's rule [closed]

How do you evaluate the following limit without using L'Hôpital's rule? $$\lim\limits_{x\to\infty} \frac{\sqrt{(4x +3) +2\sqrt{1+ x}}}{\sqrt x}$$
4
votes
7answers
119 views

Find $\lim_{x\to 0}(x+\sin x)^{\tan x}$

Find the limit $\lim_{x\to 0}(x+\sin x)^{\tan x}.$ I tried: $\lim_{x\to 0}{\tan x}.\ln(x+\sin x)=\lim_{x\to 0}\frac{\ln(x+\sin x)}{\cot x}=\lim_{x\to 0}\frac{(1+\cos x)(\cos^2 x)}{-(x+\sin x)}$ I ...
2
votes
2answers
123 views

Definite integration involving square root function

How to integrate this definite integral: $$\int_{0}^{\pi/2} \big(\sqrt{\cos x}+ \sqrt{\cot x}\,\big)\,\mathrm dx$$
7
votes
2answers
863 views

When is differentiating an equation valid?

I wonder that Is it true to differentiate an equation side by side. Under which conditions can I differentiate both sides. For example, for the simple equality $x=3$, Is ıt valid to differentiate both ...
1
vote
1answer
61 views

Proving that $\liminf_{n\to\infty} x_n = \sup\left\{z : \{n : x_n < z\} \text{ is finite}\right\}$; what does the answer mean?

Suppose $x_n$ is a bounded sequence. Prove that $$\liminf_{n\to\infty} x_n = \sup\left\{z : \{n : x_n < z\} \text{ is finite}\right\}$$ I'm having trouble even interpreting what this is suppose to ...
1
vote
2answers
103 views

For which values of $p \geq 0$ does the integral $\int_0^{\infty} \frac{dx}{x^{p}+x^{-p}}$ converge?

For which values of $p \geq 0$ does the integral $$\int_0^{\infty} \frac{dx}{x^{p}+x^{-p}}$$ converge? I tried applying the p-test, but I could not get the integrand into a suitable form. So what ...
1
vote
1answer
250 views

How to take the definite integral on both sides of a differential equation?

For instance, $$a \cdot ds=dt$$ I know that one can take the indefinite integral on both sides to get $$\int a ds = \int 1 dt$$ But how do I take the definite integral of both sides, and exactly ...
0
votes
2answers
88 views

Fundamental Theorem of Calculus of a definite integral

$$\frac{\mathrm{d}}{\mathrm{d}x} \int_{2x}^{3x+1}\! \sin\left(t^4\right)\, \mathrm{d}t$$ could you just use the Fundamental Theorem of Calculus to get $$\sin\left(t^4\right) \bigg|_{2x}^{3x+1}$$ ...
2
votes
3answers
103 views

find the limit of the following expression

Let $f\left(x\right)=\left(x^{2}+1\right)e^{x}$. Find the following:$$\lim_{n\to+\infty}n\int_{0}^{1}\left(f\left(\frac{x^{2}}{n}\right)-1\right)\,dx$$ I dont know if L'Hopital's Rule may be used ...
0
votes
2answers
61 views

Trigonometric inequality $\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$

How can I solve the following inequality? $$\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$$
0
votes
1answer
24 views

Max value of a expression

given $M\subset\mathbb{R}$ where $M=\{|z^2+az-1|:z\in\mathbb{C}\wedge|z|=1\},a\in\mathbb{R}$ find max value of $M$ in function of $a$ i tried to make $z=e^{\theta i}$ $$\begin{align} ...
1
vote
3answers
84 views

Preimage of non-invertible matrix

I am given the matrix $$\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{-1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$ Apparently this one is not invertible. ...
1
vote
1answer
63 views

Can $\pi(x)-\operatorname{Li}(x)=O(\sqrt x\log(x))$ be proven without using $\psi(x)$?

Assuming the truth of the Riemann hypothesis, $\pi(x)-\operatorname{Li}(x)=O(\sqrt x\log(x))$. Apparently the proof that $\psi(x)$ approaches $1$ for sufficiently large $x$ proves the error term. ...
3
votes
8answers
167 views

Find $\lim_{x \to 0}\frac{x-\sin(x)\cos(x)}{\sin(x)-\sin(x)\cos(x)}$

Find $$\lim_{x \to 0}\frac{x-\sin(x)\cos(x)}{\sin(x)-\sin(x)\cos(x)}\;.$$ Applying L'Hopital's rule directly does not seem to get me anywhere. I also tried dividing the numerator and denominator ...
4
votes
2answers
156 views

a way to integrate: $\int (\sqrt{x} +3)/(2+ x^ \frac{1}{3}) dx$

Im looking for a way to integrate: $$ \int \frac{ \sqrt{x} +3}{2+ x^ \frac{1}{3}} dx $$ that would make it efficient and not too difficult... Any suggestions?
0
votes
1answer
26 views

$\{u_n^2\}$ is increasing

how he does Know that $\{u_n^2\}$ is increasing this question is related to that one solution verification here is his solution Note first that $$ 1-\frac{1}{k}+\frac{1}{4k^2} ...