Questions tagged [calculus]

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

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10 views

Not using Stokes theorem for line integral

I have a question for the integral of $f(x,y,z)=(y-1)dx+z^2dy+ydz$ over the curve of intersection between the surface $x^2+y^2=z^2/2$ and the plane $z=y+1$. I know that I can do this using Stokes ...
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4answers
84 views

Range of $f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$

I am trying to find the range of this function: $$f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$$ So I think that means I have to find minima and maxima. Using partial derivatives gets messy, so I was ...
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2answers
23 views

How to evaluate $\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$, where t is constant

I need to evaluate the following one. Can't understand the method in my textbook. $$\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$$ My textbook is to let $\alpha=1-\sqrt{1-t^2}$, $\beta=1+\...
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2answers
23 views

Help with algebra in evaluating a limit

Good evening folks. I'm doing some self-study from the eighth edition of Stewart Calculus (metric version), and I ran into this problem on page 103: Evalute $$\lim_{x\to 1} \frac{x^{1/3} - 1}{x^{1/2}-...
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3answers
39 views

what did i do wrong when trying to prove the derivative of ln(x)

when trying to prove the derivative i ended up with $$\lim_{h \to 0}\frac{\ln(x+h)-\ln(x)}{h}=\lim_{h \to 0}\frac{\ln(\frac{x+h}{h})}{h}=\lim_{h \to 0}\ln((1+\frac{h}{x})^{(\frac{1}{h})}$$ and $$\...
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4answers
121 views

At what point of mathematical education can you start inventing new math?

I am a 2nd year student doing an honors program in math and statistics. Everything that I have been learning has been formulas, theorems, and mathematical concepts that other people have discovered/...
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1answer
28 views

About double map function $D(x) = 2x\pmod 1$ iterations

Well, I have a basic question about double map function, let's go to it... The double map $D:[0,1) \to [0,1)$ is define by $$ D(x) = 2x\pmod 1 = \begin{cases} 2x & \quad \text{if } 0 \...
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3answers
1k views

Methods to evaluate $ \int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx} $

Today I saw a question with an answer that made me rethink of the following question, since it's not the first time I try to find an answer to it. If you look at the answer of Mhenni Benghorbal here ...
3
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1answer
43 views

How to prove $\sum_{n=-\infty}^ \infty {\rm sinc}\bigl( \pi(t-n)\bigr) = 1$?

Thank you by avance for your help. So, I found on this website, that $\sum_{n=-\infty}^{\infty} {\rm sinc}( \pi n)= 1$. But I could not find any way to prove it. I know it’s about fourrier, but I don’...
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1answer
33 views

Calculate $\int\sqrt{(x^2+1)}dx$

Calculate $I$ =$\int\sqrt{(x^2+1)}dx$ I have tried calculating it using integration by parts: $$f'(x) = 1, f(x) = x$$ $$g(x) = \sqrt{x^2+1}, g'(x) = \frac{x}{\sqrt{x^2+1}}$$ $$\int\sqrt{x^2+1}dx = x\...
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2answers
52 views

Limit involving roots

$$\lim_{x\to\infty}(x^3+x^2)^{1/3}-(x^4-x^3)^{1/4}.$$ The solution is 7/12. I dont know who to get to that solution. My thoughts were that the second root is bigger that the first one so the solution ...
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1answer
64 views

Can you get the function equations just based on the graph ( without knowing whether it is linear,quadratic…)

For example, I tell a kid draw a line(it doesn't have to be straight, anything you like) in the coordinate system. Am i able to find the equation for f(x)? Also, if i have a program to which i give an ...
2
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1answer
191 views

Generalized polar coordinates, how to switch form cartesian to polar

I was solving a problem: $$\left(\frac xa + \frac yb\right)^ 4=4xy,\quad a>0 , b>0 $$ Find the are bounded by curve. It is not hard to see that x and y must have the same sign, that function ...
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0answers
27 views

Formation of differential equation of Circle two cases

Why is differential equation of circle at origin $x^{2}+y^{2}=c^{2}$ is $2x+2y \dfrac {dy}{dx}=0$ and the differential equation of circle at (h,k) , $\left( x-h\right) ^{2}+\left( y-k\right) ^{2}=c^{...
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1answer
29 views

Proving discontinuity of a function defined by a double integral

Show that $$f(\alpha)=\int_{\alpha}^1 dx \int_{\alpha}^1\dfrac{(x-y)}{(x+y)^3} dy$$ is not continuous at $\alpha=0$. I have found out the iterated integral at $\alpha=0$ which is $1/2$. But after ...
3
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3answers
41 views

How to check if a function is convex

According to a calculus book I have been reading, we call a function $g(x)$ a convex function if $$g(\lambda x +(1-\lambda)y) \leq \lambda g(x) +(1-\lambda)g(y)$$, for all $x,y$ and $0<\lambda&...
2
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4answers
84 views

Is my analysis of the series $\sum_{n=1}^\infty{\frac{\ln n}{n}}$ correct?

I came across the following series and I'm supposed to analyse whether it converges or not. $$\sum_{n=1}^\infty{\frac{\ln n}{n}}$$ My attempt: At first sight, the thought of using the integral test ...
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2answers
118 views

How to find volume from Washer method?

So, I want to use Washer method to find $$\pi \int_{y=0}^2 (\sqrt{4y})^2-(\sqrt{4y}-0.1)^2 dy.$$ I've literally spent a day trying to do so, and when I treat each part separately and then subtract ...
13
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4answers
307 views

Integration with $\ln(x)$ in the denominator

Find$$\displaystyle\int_1^\infty\frac{(x^2-1)(x^4-1)(x^6-1)}{\ln(x)(x^{14}-1)} dx$$ I tried simplifying the terms without logarithm $x^2-1=(x-1)(x+1)\\x^{14}-1=(x^7-1)(x^7+1)$ to see if any ...
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1answer
43 views

Derivative of trace of a product containing an inverse matrix

What's the derivative of $$f(X)=\text{Tr}(YX^{-1})$$ with respect to $X$, where $X$ and $Y$ are square matrices of the same dimension? My first attempt is to apply the chain rule as: Let $h(X)=X^{-1}$...
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1answer
30 views

Finding a formula for $[(1+f(t))^{b}]^{(k)}$ with $b$ constant.

I am trying to find a formula for the kth derived from the function $(1 + f(t))^b$ where $f(t)$ is an infinitely differentiable function and b a constant number. With the help of Wolfram, see many ...
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2answers
68 views

Essence of Weierstrass approximation theorem.

Weierstrass approximation theorem is a quite strong theorem,even stronger than the Taylor's theorem because: Statement:Suppose $f:[a,b]\to \mathbb R$ is a continuous function then $\exists$ a ...
8
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3answers
390 views

Seemingly impossible double integral reduction

Show that $\int_{0}^{1}\int_{0}^{1}(xy)^{xy}dxdy = \int_{0}^{1}y^{y}dy$ I have tried the method used in the Gaussian integral, polar coordinates, exp(.) and ln(.) of both sides of the integrand... ...
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70 views

Related rates problem: How rapidly is the area of an equilateral triangle increasing at the instant when each side is $69.28$ inches?

A metal plate in the shape of an equilateral triangle is being heated in such a way that each of the sides is increasing at the rate of ten inches per hour. How rapidly is the area increasing at the ...
8
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3answers
231 views

Is this closed-form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is $0....
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1answer
34 views

Calculus for the Practical Man: Chapter 4, Problem 7

A man standing on a wharf is hauling in a rope attached to a boat at the rate of four feet a second. If his hands are nine feet above the point of attachment, how fast is the boast approaching the ...
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15 views

How do we calculate $\int_{i}^{f} (L_1 - L_2) dt$ where $L_1 = f(t)$ and $L_2 = f(t+dt)$? Thanks. [on hold]

I have two functions f(u) and g(u). I have to use these two functions to calculate a function $h(u) = \int_{i}^{f} (L_1 - L_2) du$ where $L_1 = func(f(u),g(u))$ and $L_2 = func(f(u+du),g(u+du))$.
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1answer
16 views

Proving limit of bounded convergent sequence is bounded

Question: Show that if $a \leq x_n \leq b$ for every $n$ and $x_n \rightarrow x$, then $a \leq x \leq b$. Proof: Let $\epsilon>0$. By assumption $a_n \leq x_n \leq b$ for all $n$. By definition ...
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0answers
52 views

Solve: $2y'' = e^y$ [on hold]

I wish to find the solution of the differential equation: $$ 2 y^{\prime \prime}=\mathrm{e}^{y} $$
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4answers
88 views

Strange! Right solution of equation is partially wrong.

Let $0<a<1$ be arbitrary but fixed. The equation $$ \frac{x^2 (1+y)^2}{\left(a y + \sqrt{1+(1-a^2)y}\right)^2} = 1$$ in $y$ has according to straight-forward calculus and Mathematica two ...
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2answers
31 views

find total surface area of solid using formula for surface area using integration

find the total surface area of solid which is formed when the region enclosed by $x^2+y^2=4$ in the first quadrant is rotated about $y=-1$. Edit: I'm getting that on rotating this solid I'm getting ...
6
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4answers
210 views

Double Integral $\int\limits_0^a\int\limits_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$

How to solve this integral? $$\int_0^a\!\!\!\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$$ my attempt $$ \int_0^a\!\!\!\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}= \int_0^a\!\!\!\int_0^a\...
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0answers
37 views

Behaviour of Laplace transform as $s \to \infty$

I have shown that if $f$ is a continuous function with scaling symmetry given by $f(mt)=m^kf(t)$ for any $m>0$, than $(\mathcal{Lf})(s) = m^{-k-1}(\mathcal{Lf})(\frac{s}{m})$. For this $f$, what ...
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3answers
32 views

Find inflection points of $e^{\cos x}$

I got a function $$ f(x) = e^{\cos x}$$ I would like to find inflection points of the function above within range $[-2\pi, 2\pi]$ Find first derivative $$f'(x) = e^{\cos x}(-\sin x)$$ Find second ...
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1answer
38 views

“every regular functions is differentiable infinitely” [on hold]

Is there a theorem that states something like "every regular functions is differentiable infinitely"? I want to prove that $$\frac{1}{1+x^2}$$ (or similar "simple" functions) has a second derivative ...
36
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3answers
2k views

Integral $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful (yes beautiful) and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\...
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0answers
22 views

What, if anything, is fundamentally special about plugging certain functions into certain ODEs?

While studying differential equations, I have noticed that several solution techniques are nearly identical. Reduction of order, variation of parameters (first-order version), and transforming ...
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2answers
1k views

Is there a basis which spans the real numbers?

Is there a finite set of real numbers $S=\{a_1, a_2, ..., a_n \}$ such that every real number can be written as a linear combination (with integer coefficients) of the elements of $S$? If no, is there ...
2
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1answer
19 views

If $f: R \to R$ is one one and differentiable function and graph of $y = f(x)$ is symmetrical about the point $(4,0)$ then…

If $f: R \to R$ is one one and differentiable function and graph of $y = f(x)$ is symmetrical about the point $(4,0)$ then, If $f'(-100) >0$ then roots of $x^{2} - f'(10)x - f'(10) = 0$ may be non ...
6
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5answers
81 views

Proving the Continuity of $e^x$

Can one say that $e^x$ is the sum of an infinite number of terms (Taylor expansion), every term being a continuous polynomial in itself, the sum of all the terms is continuous and so $e^x$ is ...
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2answers
56 views

find largest $x$ based on $f(x,y)=c$

Consider the following equation: $$ \frac{p}{\left(a-x\right)^2+y^2}+\frac{1-p}{\left(b-x\right)^2+y^2}=1 $$ where $0<p<1$ and $a,b\in\mathbb{R}$. I want to look for the largest $x$ satisfying ...
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1answer
26 views

Can the sum property of integrals be stated for indefinite integrals?

Some time ago I learned about the following property of integrals: If $ f $ and $ g $ are bounded, integrable functions on $\color{red}{[a, b]}$, then so is $f + g$ and $$ \displaystyle \int_\...
2
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1answer
49 views

Find the range of $P= \frac{\sum_{1}^n x_i}{\prod _{1}^n (x_i^2+1)}$

When I solved the following problem. Find the range of $B\equiv\dfrac{x+y}{(x^2+1)(y^2+1)}$ where $x,y \in\mathbb{R}$. Solution. Setting $x=\tan u$ and $y=\tan v$, expression is $\frac{\sin 2u+\sin ...
1
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1answer
54 views

Continuity of partial derivatives.

I am trying to find out if the partial derivatives of my function $$f(x,y)=\begin{cases} 2\frac{x^3y}{x^2+y^2}-xy & \text{if } (x,y)\neq 0 \space \\ 0 & \text{if } (x,y)= 0 \space\\ \end{...
0
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1answer
26 views

Spherical Cap Area

Find the Area of the upper Cap cut from the sphere $x^2+y^2+z^2=2$ by the cylinder $x^2+y^2=1$. I got how to solve it after seeing solution using $dS=\iint\sqrt{\frac{dz}{dx}^2 +\frac{dz}{dy}^2-1 } ...
4
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2answers
173 views

Continuity of a strange function

Let $f: [0,1)\to\mathbb{R}$ such that $f(x)=0.a_1a_3a_5\ldots$ where $x=0.a_1a_2a_3a_4\ldots$, i.e, $f(x)$ skips the even digits of $x$. Prove $f$ is continuous at $0$, and find a point where $f$ is ...
0
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1answer
27 views

Taylor-Remainder divided by $(x-x_0)^n$ goes to zero as $x$ approaches $x_0$.

In a script about the Taylor-Theorem I'm reading I have stumbled across the property that for remainder $R_n(x,x_0): \lim_{x \to x_0} \frac{R_n(x,x_0)}{(x-x_0)^n} = 0$. Since there are no further ...
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1answer
63 views

Calculating $\int_{\pi/4}^{\pi/2} \frac{x \cos x-\sin x}{x} dx$ [on hold]

How to calculate this trigonometric integral ? $$\int_{\pi/4}^{\pi/2} \frac{x \cos(x)-\sin(x)}{x} dx$$
6
votes
2answers
349 views

For which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}\ln^{\beta}x}$ converge?

I'm trying to find out for which values of $\alpha$ and $\beta$ the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}\ln^{\beta}x}$ does converge. I know that when $\alpha=1$ then $\beta$ must be ...
2
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0answers
21 views

How to estimate $\sum_{2 \leq n \leq X} \frac{1}{(\log n)^A} $?

I was wondering how does one estimate $$ \sum_{2 \leq n \leq X} \frac{1}{(\log n)^A}? $$ where $A>0$. I feel like it should be $\ll \frac{X}{(\log X)^A}$... Any comments would be appreciated. ...