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

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0
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1answer
72 views

About odd functions and improper integrals e.g. $\int^{\infty}_{-\infty}\sin x \; dx$

Does $\displaystyle \int^{\infty}_{-\infty}\sin x \; dx$ converge? Since $\sin x$ is an odd function, and we know that in definite integrals $\displaystyle \int^{a}_{-a}\sin x \; dx=0$ then does ...
-2
votes
1answer
38 views

Evaluate lim through progression

Let $a_n$ be the $n$-th term of an arithmetic progression with the initial term $a_1=2$ and with the common difference $5$. That is, $a_n=2+5(n-1)$. Evaluate $\lim_{n\to\infty}n\left(\sqrt{a_n^2+3}-\...
3
votes
2answers
183 views

Evaluate an Integral

Evaluate: $$\int_{0}^{\infty}\dfrac{\sin^3(x-\frac{1}{x} )^5}{x^3} dx$$ I've been stumped by this Integral and cannot think of how to evaluate it. I substituted $\dfrac{1}{x^2}=t \Rightarrow \...
0
votes
3answers
59 views

Proof $\{(x,y,z)|4x^2+9y^2+16z^2<1\}$ is an open set

In order to prove that the points $(x,y,z)$ such that $$4x^2+9y^2+16z^2<1$$ form an open set, I tried this: Pick a generic point of the ellipsoid, lets say $$4x^2+9y^2+16z^2$$ Now, I'll form ...
1
vote
1answer
168 views

What is the ratio of the intensities of the two sounds?

1. Suppose that a jet engine at 50 meters has a decibel level of 130, and a normal conversation at 1 meter has a decibel level of 60. What is the ratio of the intensities of the two sounds? we ...
2
votes
4answers
180 views

Find $\int_a^b \sin |x| \, \mathrm{d}x $

How to find the integral $$\int_a^b \sin |x| \, \mathrm{d}x \,?$$ I'm able to obtain definite integral of form $ \int_a^b \lvert\sin x \rvert \, \mathrm{d}x$ but not when the modulus operator is ...
22
votes
4answers
4k views

Why doesn't L'Hopital's rule work in this case?

I have a very simple question. Suppose I want to evaluate this limit: $$\lim_{x\to \infty} \frac{x}{x-\sin x}$$ It is easy to evaluate this limit using the Squeeze theorem (the answer is $1$). But ...
1
vote
1answer
34 views

Help in understanding a limit involving an integral.

$$ I_n = \int^1_0\frac{x^n}{ax+b}dx $$ Where: $n \in N$ ; $a,b \in (0,\infty)$" Find $\Xi$, where: $$\Xi=\lim_{n \to \infty}nI_n$$
1
vote
1answer
74 views

Evaluate $\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$

Evaluate $$\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$$ Here's my attempt: $$u=1-tan \theta \implies -du=\sec^2 \theta d \theta$$ Substituting back in, I get this: ...
0
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1answer
36 views

Question about negative value using the ratio convergence test for integrals

Find for what $p$, $\displaystyle \int ^{\infty}_0 x^p \arctan x dx$ converges. By parts, it's equal to: $\displaystyle \lim_{b\to \infty}\frac 1 {p+1}x^{p+1}\arctan x |^b_0- \int ^b _0\frac {x^{p+1}...
0
votes
2answers
215 views

The meaning of differentiation of $x$ with respect to $y$

The physical meaning of the differentiation of $x$ with respect to $y$ is the rate of change of $x$ with respect to $y$. But, I am finding it difficult in understanding the geometrical interpretation ...
0
votes
0answers
74 views

Second Order Differentials: Using $y = A + Bxe^x$

I've went over some of my math work which I'm currently doing at Uni and came across a rather confusing example. The example I went over is based on Second Order Differentials. So basically what I ...
0
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2answers
119 views

Integration problem: $\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $

I need help in solving the following problem: $$\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $$ I really don't know how to start solving this problem; any tips or solutions will be greatly ...
3
votes
1answer
418 views

Reference Book for Calculus

I've made this post as detailed as I can so as to give you a fair idea of my level. Calculus (particularly Integration) is my passion and frankly I spend all my free time learning as much of it as I ...
1
vote
1answer
58 views

Existence of $x_0$ such that $f(|x_0 + a|) = f(|x_0|)$ given $f \colon \mathbb R \to \mathbb R$ and $a$

So I have this function $f : \mathbb{R} \to \mathbb{R}$ that is continuous and I have $a\in\mathbb{R}$. I have to prove that exists an $x_{0}\in\mathbb{R}$ such that this works: $$f(|x_{0}+a|) = f(|...
0
votes
1answer
40 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...
1
vote
2answers
129 views

Evaluate the indefinite integral $\int \frac{\cos \theta}{ \sqrt{2 - 9 \sin^2 \!\theta}} \mathrm{d}\theta$

I want to evaluate $$\int \dfrac{\cos \theta \, \mathrm{d}\theta}{ \sqrt{2 - 9 \sin^2 \theta}}$$ but I can't seem to get the answer, my working is as below:
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1answer
100 views

integration by part and a limit- Evans PDE Chapt2 problem 13

1) I am having a hard time in seeing how the integration by part done in this problem (page 11) enter link description here Could anyone help explaining? I cannot see how he got 3 terms instead of 2....
8
votes
5answers
10k views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
1
vote
1answer
87 views

Need hint to solve a nasty integral.

Let $f(x)=\frac{x+2}{2x+3}$, $x>0$. If $$\int \left( \frac{f(x)}{x^2} \right)^{1/2}dx=\frac{1}{\sqrt{2}}g \left(\frac{1+\sqrt{2f(x)}}{ 1-\sqrt{2f(x)}} \right) -\sqrt{\frac{2}{3}}h \left(\frac{...
1
vote
2answers
42 views

Function with 2 variables $ f(x,y)=(x-y)e^{xy} $

Find minimum and maximum of the next function: $$ f(x,y)=(x-y)e^{xy} $$ In the square: $$ \left\{\begin{matrix} -1\leq x\leq 1 \\ -1\leq y\leq 1 \end{matrix}\right. $$ I have found that $f(x,y)$ ...
4
votes
2answers
84 views

Proving that $\lim_{x {\to} \infty}f(x)=\infty $

Given $\ f(x)$ that is differential in $\ (x_0,\infty), f'(x)\ge a,$ for every $\ x> x_0$ and $\ a>0$, trying to show that $\lim_{x\to\infty}f(x)=\infty$. So far I've tried using Mean Value ...
6
votes
2answers
63 views

Convergence and divergence depending on whether $n$ is odd or even

It is part of one problem I am working on: I want to prove the following conjecture ($x\ne q\pi$ where $q\in\Bbb Q$) $$\sum_{n=1}^{\infty}\frac{\sin^{2k-1}(nx)}{n^{\alpha}}\quad\text{converges if}\...
1
vote
1answer
58 views

An inequality involving Sobolev embedding with epsilon

Let $\Omega$ be a nice bounded domain in $\mathbb{R}^n$ and let $2^*=2n/(n-2)$ for $n>2$ or anything finite for $n\leq2$, i.e., the Sobolev exponent. Then, for any $p\leq 2^*$, one has $$ \|u\|_p\...
1
vote
1answer
93 views

Integration of $\frac{1}{\sqrt{x^8+1}} $ [closed]

How can we integrate $\frac{1}{\sqrt{x^8+1}}$ thanks for help. I couldn't find any way. I tried to factor, substituted with $\tan(a)$ but it's not working :'(
2
votes
1answer
58 views

Help a beginner form an equation for a basic counting program.

I have programmed a counter. It counts at one point per second until it reaches ten points, at which time it resets back to zero and begins to count at two points per second. Again, upon reaching ten ...
0
votes
1answer
49 views

Why $\int_c^{x+h}f(t) dt-\int_c^x f(t) dt = A(x+h)-A(x)$?

I'm reading Apostol's: Calculus and trying to understand the fundamental theorem of calculus. I don't understand why $\int_c^{x+h}f(t) dt-\int_c^x f(t) dt = A(x+h)-A(x)$. I guess the ...
0
votes
2answers
298 views

How do I find the equations of line tangent to a unit circle that have a slope of 1?

I'm supposed to find the equations of the lines tangent to a circle of radius $1$ and centered at the origin that have a slope of $1$. I know these things: this is a unit circle, the equation for a ...
0
votes
0answers
48 views

Independence of Path for Line Integral of Vector Field Perpendicular to Curve

Let $C$ be a simple, piecewise-smooth curve between points $A$ and $B$, as shown below: Let $\vec{f}$ be a vector field that is defined on the curve (shown above in blue). The magnitude of $\vec{f}$ ...
1
vote
3answers
119 views

Prove that the following function is $C^{\infty}$ [duplicate]

Prove that the following function: $$r:x \mapsto \begin{cases} e^{-{1\over (1-x^2)}}, & \text{if $|x|<1$} \\ 0, & \text{if $|x| \ge 1$} \end{cases}$$ is $C^{\infty}$ I found this problem ...
-1
votes
3answers
130 views

Calculus: Find the limit of $(f(3+h)-f(3))/h$ as $h\to 0$ [closed]

I feel like I'm missing something because $f(x)$ is not given anywhere. Any help?
0
votes
2answers
51 views

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$.

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$. I tried to graph this using Wolfram Alpha, but it didn't help. I don't know how to start or ...
0
votes
1answer
472 views

How do I find the Laplace Transform of $ \delta(t-2\pi)\cos(t) $?

How do I find the Laplace Transform of $$ \delta(t-2\pi)\cos(t) $$ where $\delta(t) $ is the Dirac Delta Function. I know that it boils down to the following integral $$ \int_{0}^\infty e^{-st}\...
0
votes
1answer
82 views

How can I find $f'(0)$ of this function?

I need to find $f'(0)$ if $$f(x)={x^2\sin x-\cos\left(3x\right)\over e^{-3x}+1}$$ How do I do this? When I tried using the quotient rule it became messy very quickly so I thought that there must be ...
0
votes
0answers
157 views

A limit of an integral I

Let $f(x)$ and $g(x)$ be continuous and differentiable on $[0,1]$. Also let $0 \leq b \leq 1$. What is the resulting value of the limit \begin{align} \lim_{n \to 0} \frac{1}{n} \, \int_{0}^{1} f(b x^{...
0
votes
0answers
36 views

Proving multi-variable differentiability using the limit definition

I'm doing advanced calculus and I find it challenging to solve multi-variable limits while proving differentiability, more specifically 2 variable limits. could you show me how do I solve this limit?: ...
1
vote
3answers
61 views

In proving the product rule, how do we know to add and subtract f(x+h)g(x) from the numerator in the derivative definition?

I watched two YouTube videos to try to get a proof that makes sense, but in both videos, the authors said something to the effect of "add and subtract f(x+h)g(x)" without a good explanation as to how ...
3
votes
1answer
127 views

Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
1
vote
1answer
61 views

Convergence of $\int _{-\infty}^{+\infty}\sin(cx)dx$

At this forum there is an abundance of questions regarding the convergence of integrals and sums of infinite series. The mathematicians who answer these questions emphasize that only under strict ...
1
vote
6answers
227 views

What are the limits of these sine functions as x approaches infinity?

I have two different limit sine functions: $$\lim_{x\to \infty} {\sin x\over x}$$ $$\lim_{x\to \infty}x^2\sin\left({3\over x^2}\right)$$ My thoughts are that as $x$ becomes infinitely large, then ...
2
votes
0answers
60 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
11
votes
3answers
236 views

About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$

I came across the following Integral and have been completely stumped by it. $$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$ I'm extremely sorry, but the only thing I noticed was that the ...
-2
votes
2answers
123 views

Limit without l'hospital,derivatives,taylor series…just using limit properties and some basic limits [closed]

As the title says i have to solve this two limits without the help of these,i only know some basic limits from 1st semester calculus: 1st limit: $$\lim_{x\to 1}\frac{\sin{\pi x}}{\ln(2x^2-1)}$$ And ...
4
votes
2answers
86 views

Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$.

Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is ...
1
vote
0answers
55 views

A geometric interpretation of a geometric series [duplicate]

The following code is compiled by TikZ to draw a right triangle so that the enclosed area is partitioned by infinitely many triangles similar to itself. I saw on ...
1
vote
2answers
103 views

On proving the total differential.

I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from this lecture. In the lecture Denis Arnoux gives a sketch proof ...
1
vote
0answers
28 views

Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot B}){...
1
vote
1answer
44 views

Equation of plane

Find the equation of the plane through the point $(1,−1,2)$ which is perpendicular to the curve of intersection of the two surfaces $x^2+y^2−z=0$ and $2x^2+3y^2+z^2−9=0$. i've gotten as far as ...
14
votes
6answers
2k views

How is the area of a circle calculated using basic mathematics?

Area of a circle is addition of circumference of layers of a onion. If n is radius of a onion then area is $$ A = 2 \pi \cdot 1 + 2 \pi \cdot 2 + 2\pi \cdot 3 + \ldots + 2 \pi \cdot n $$ which $$ =...
1
vote
2answers
64 views

Domain of derivative on open interval is open

Let $f : (a, b) \to \mathbb{R}$. Suppose that the derivative $f'$ exists at every point of a set $E \subseteq (a,b)$. Is it true that the domain $E$ of $f'$ is open? And if it is not true, is it true ...