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

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3
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2answers
74 views

On the solution of constant coefficients PDEs (exponential method)

Having a look to my old PDE notes, I have come across with the following problem: Consider the 2nd order PDE: $$ \varphi_{xx} - \varphi_{xy} = 0, \quad (x,y)\in \mathbb{R}^2, \quad \varphi = ...
4
votes
3answers
91 views

Is it always true? $\left|A-B\right| \le \left|A\right| + \left|B\right|$

Is it always right to claim that: $$\left|A - B\right| \le \left|A\right| + \left|B\right|$$ where $A, B \in \mathbb{R}$ ?
0
votes
2answers
77 views

A Matrix Integral Equation

We have an integral equation on matrix. ${\Im(t)}=\Im(0)+\int_{0}^{t} \Im(s)[K(s)]_{ \times }ds \tag 1$ $[\hspace{.2cm} ]_{\times}$ is skew symmetric matrix with diagonals zero and is non ...
4
votes
2answers
221 views

Simple integration

I'm currently learning calc 2 and feel I'm making a very silly, obvious mistake with solving the integral $\int\frac{-5\sqrt[3]{x^2}}3 dx$ I guess I'm making an algebraic mistake somewhere, this is ...
1
vote
1answer
22 views

infinitisimal part and the directional integral

In the paper A circle detection approach based on Radon Transform by Erman Okman and Gozde B. Akar. I have a few questions on some basics. first of all what does $$ds^2 = dx^2 + dy^2$$ ...
0
votes
1answer
36 views

The motion of the particle satisfies $\textbf{v} = \textbf{c}\times \textbf{r}$

Why is the path is contained in a circle that lies in a plane perpendicular to $\textbf{c}$ with centre on a line through the origin in the direction of $\textbf{c}$
0
votes
1answer
50 views

Which technique of integration should use to solve the question?

Quote from the paper I read: Given $F=(1-\lambda)f$ + $\lambda zf'$, we find that $f(z)= \tfrac{1}{\lambda} z^{1- \tfrac{1}{\lambda}}\int_0^zF(t)t^{\tfrac{1}{\lambda}-2}dt.$ My questions is Which ...
0
votes
0answers
25 views

A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit

In one of my books there was an exercise to prove that: A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit (I know a sequence doesn't really converge to ...
0
votes
0answers
18 views

Where can I read about the techniques for computing areas and volumes before calculus?

I've read the following here: The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most ...
1
vote
2answers
44 views

Finding derivative

$\lim\limits_{x\to\ 2}\frac{f(x)-f(2)}{x^2-4}=4$ where $f(x)$ is defined on $\Bbb R$ and $g(X)=\frac{f(x)e^x}{1-x}$. What is $g'(2)$?
0
votes
1answer
27 views

Rearranging equation with algebra

I'm having a difficult time showing that the two are equivalent: $2(x_1-\theta)(1+(x_2-\theta)^2)+2(x_2-\theta)(1+(x_1-\theta)^2) = 2(\bar{x}-\theta)(1+(x_1-\theta)(x_2-\theta))$ I have multiplied ...
2
votes
3answers
76 views

any other method for evaluating $\int\limits \frac{ x^2+x+1 }{ \sqrt{x^2-x+1} } dx$?

I tried below and its getting tedious : $\begin{align}\\ \int\limits \frac{ x^2+x+1 }{ \sqrt{x^2-x+1} } dx &= \int\limits \frac{(2x-1)+ x^2-x+2 }{ \sqrt{x^2-x+1} } dx \\~\\ &= \int\limits ...
1
vote
1answer
157 views

Is it possible for a triangular matrix in echelon form to not have a unique solution and how?

I want to know if it is possible for a triangular matrix in echelon form to not have a unique solution and how? Isn't there something to do with the determinant that shows this? or am I wrong?
0
votes
1answer
42 views

Convergence of function

Suppose that $V(t)$ is nonnegative continuous function ($\forall t:V(t)\ge0$). $\dot{V}(t) = -|h(t)|^2 + f(t)g(t)$ $f(t)$ is a bounded and uniformly continuous function. $g(t)$ is a bounded and ...
0
votes
2answers
73 views

Integration by parts. integrate $\ln(x^2-x+2)$ [closed]

Integrate by parts $\ln(x^2-x+2)$
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3answers
37 views

How to find the arc length of this graph?

Could you please help me with this problem? Find the arc length of the graph of $y = \frac{x^{3}}{3} + \frac{1}{4x}$ between $x = 1$ and $x = 2$. Note: It may be helpful to use identities like $$x^{2} ...
1
vote
2answers
77 views

Find $ \int_0^2 \int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy\hspace{1mm}dx$

I need the approximation to four decimals Not sure how to start or if a closed form solution exists All Ideas are appreciated
1
vote
1answer
31 views

Area of a Self-intersecting Curve

I was doing some work finding the areas of rose curves. The rose curve is a polar curve given by the equation $$ r(\theta) = \cos{k\theta} $$ When $k$ is even, the area is $\pi/2$, and when $k$ is ...
2
votes
1answer
48 views

Equation of the tangent line to the curve $x^2y^2+y^2 = 2x^2$ at the point $(1,-1)$ [closed]

Find the equation of the tangent line to the curve $x^2y^2+y^2 = 2x^2$ at the point $(1,-1)$. I did implicit differentiation and my answer is $y = -\frac12x-\frac12$ however this is a take home test ...
2
votes
1answer
43 views

verifying extrema found by Lagrange multipliers

This question was inspired by reading this problem: Prove the inequality $\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74$ Suppose I have a function $f(x,y,z)$ with continuous ...
2
votes
1answer
73 views

Looking for an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$

I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$. But I don't remember details, and searching my books and the Internet ...
3
votes
3answers
85 views

What is $\frac{d^n}{dx^n} \frac{e^{\lambda x}}{x}$?

I was wondering whether there is an explicit way to say what the derivative of $\dfrac{d^n}{dx^n} \dfrac{e^{\lambda x}}{x}$ for $n \in \mathbb{N}_0$is, where we assume that $\lambda \neq 0$.
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votes
3answers
36 views

Solving the Riemann Sum $\sum_{i=1}^{n}(1+\frac{6i}{n})^3(\frac{2}{n})$?

So I have the Riemann sum. $\sum_{i=1}^{n}(1+\frac{6i}{n})^3(\frac{2}{n})$. From my understanding that turns into $(\frac{2}{n})\sum_{i=1}^{n}(1+\frac{6i}{n})^3$ and what is really perplexing me is ...
1
vote
1answer
30 views

Chapter five of Spivak's: the second lemma

In chapter five of Spivak's, the chapter on limits, Spivak lists a lemma (the second out of three total) that is the following: if $|x-x_0|< \min(1,\frac{\epsilon}{2(|y_0|+1)})$ and ...
2
votes
1answer
32 views

Strict local extremum without $f'$ “changing signs”

Let $f:\mathbb{R}\to \mathbb{R}$. Is it possible that $f$ has the following properties: $f$ is differentiable in a neighborhood of $a\in \mathbb{R}$ $a$ is a strict local minimum There is no ...
3
votes
5answers
173 views

Calculate $\ln(1+\sqrt3)$

I distributed the natural logarithm and got $(0 + 0.549)$ [placing the values in a calculator] However, the answer key states that the answer is $1.0051$? Where did I go wrong?
0
votes
1answer
67 views

Rigorous proof of the bounds of $f(x) = (x-2)/x(x-3)$

I need to prove that for the function $$f(x) = \frac{x-2}{x\,(x-3)}, \qquad x \in (0,3)$$ we have $-\infty \leq f(x) \leq +\infty$. This statement follows from the asymptotes at $x=0$ and $x=3$. I ...
0
votes
1answer
34 views

Trouble with integration using the definition of integral

I'm playing with integration for the first time and I can understand now why everyone tells me calculus II is the hardest calculus. I'm trying to solve this problem but I think I have the wrong ...
0
votes
1answer
47 views

Convergence of bound integral

Prove or disprove: Given $f(x)$ a continuous function in $(0,1]$, and exists $M$ such that $$\left|\int_x^1 f(t)~\mathrm{d}t \right| \le M$$ for every $0\lt x \lt 1$, then $$\int_0^1 ...
4
votes
0answers
87 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
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vote
2answers
47 views

Constants for anti-derivatives

Hey StackExchange I'm diving into integral calculus for the first time and I have a few questions about this problem. A steel ball bearing at rest is accelerated in a magnetic field in a line with ...
6
votes
4answers
78 views

Continuity of function where $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function which satisfies $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$ is continuous at $x=0$, then it is continuous at every point of $\mathbb{R}$. ...
0
votes
2answers
21 views

Volume of revolution x-axis

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line $y=6x+7$ and the parabola $y=x^2$ around the $x$-axis I'm getting the wrong answer. I read ...
6
votes
2answers
207 views

Evaluating a limit. What makes the equality right?

I'm reading a proof of a limit calculation. The limit is: $$\lim\limits_{x\to 0}\left(\frac{a^x+b^x}{2}\right)^\frac{1}{x}$$ where $a,b>0$. The aother claims that: $$\lim\limits_{x\to ...
0
votes
1answer
52 views

Evaluating $\int{(a^2-x^2)^n}dx$ using repeated integration by part

The problem is as follows: Prove that $$\int{(a^2-x^2)^n}dx=\frac{x(a^2-x^2)^n}{2n+1}+\frac{2a^2n}{2n+1}\int{(a^2-x^2)^{n-1}}dx+C$$ using integration by part. I can easily obtain partially ...
0
votes
3answers
107 views

Prove $\lim\limits_{x\to 0^+} \frac{\ln x}{x} = -\infty$

Prove $\lim\limits_{x\to 0^+} \frac{\ln x}{x} = -\infty$ I've seen the following proof but I think it's invalid: $$\lim\limits_{x\to 0^+} \frac{\ln x}{x} = \lim\limits_{x\to 0^+}\ln x \cdot ...
1
vote
1answer
37 views

Calculating the value of integral of $f''(x)$ when given values of $f(x)$ and $f'(x)$.

Here is the original question. Suppose that $f(1)=2$, $f(4)=7$, $f'(1)=5$, $f'(4)=3$, and $f''$ is continuous. Find the value of $\int_1^4 {xf''(x)\,dx}$.
2
votes
2answers
21 views

Convergence of series with root

Given $$\sum_{n=1}^\infty {((-1)^n + \alpha^3) (\sqrt{n+1} - \sqrt{n})}$$ find all values of $\alpha$ such that the series converges. My try: By multiplying the series with the expression $$\frac ...
4
votes
3answers
124 views

Determine the greatest interval where the function is invertible

The assingment is to determine the greatest interval around $x=0$ where the function: $$f(x)=x^5-5x+3$$ is invertible. After that, determine $(f^{-1})'(3)$ I have totally forgotten all about ...
2
votes
0answers
51 views

Simple proof of L'Hôpital's Rule

I would like to prove the following result (along with understanding its conditions) without using Cesaro-Stolz’s theorem and the one Wikipedia has got. Assume that $f$ and $g$ are differentiable ...
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vote
2answers
98 views

Integral Involving Dilogarithms

I came across the identity $$\int^x_0\frac{\ln(p+qt)}{r+st}{\rm ...
1
vote
1answer
26 views

Area of the surface generated by revolving curve around y-axis

So I did something wrong in my solution because I'm not seeming to get the right answer. $$\int_c^d 2\pi (4 \sqrt{9-y}\sqrt{1-\frac{4}{9-y}})~\mathrm{d}y$$ combine square roots and move out ...
1
vote
3answers
85 views

Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
13
votes
3answers
226 views

How to prove: $\left(\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt[4]{25}-\sqrt[4]{125}}}-1\right)^{4}=5$?

Question: show that: the beautiful ${\tt sqrt}$-identity: $$ \left({2 \over \sqrt{\vphantom{\Large A}\, 4\ -\ 3\,\sqrt[4]{\,5\,}\ +\ 2\,\sqrt[4]{\,25\,}\ - \,\sqrt[4]{\,125\,}\,}\,}\ -\ ...
2
votes
3answers
101 views

About integrating $\sin^2 x$ by parts

This is about that old chestnut, $\newcommand{\d}{\mathrm{d}} \int \sin^2 x\,\d x$. OK, I know that ordinarily you're supposed to use the identity $\sin^2 x = (1 - \cos 2x)/2$ and integrating that ...
1
vote
2answers
64 views

Convergence of infinite $ \sum (\frac{n}{n+1})^{n^2} $?

I have being trying to solve this convergence but with no success. Using the ratio test I have reached here: $$ a_n = \left( \frac{n}{n+1} \right)^{n^2} $$ And $$ \frac{1}{a_n} = \left( ...
0
votes
1answer
33 views

Unsure of definition of composite functions and integrals

Could someone explain to me what this function represents and how it is possible. Lets have $y :\mathbb {R} \rightarrow \mathbb {R}^2 $ and that $f: \mathbb{R}^2\rightarrow\mathbb{R}^2$, and lets ...
4
votes
2answers
107 views

Integral of $\int_0^\infty\frac{1}{e^{x}-x} dx$

I am curious as to whether a closed form exists for the following integral: $$\int_0^\infty\frac{1}{e^{x}-x} dx$$ I have tried a few elemetary methods on it, but I believe this integral (if it has ...
0
votes
2answers
32 views

Convexity of functions

I'm trying to read up on convex/concave functions (is that the same as concave up, concave down?) If I were asked to prove a convexity of a function, what are the general steps to follow? (So far I ...
0
votes
1answer
39 views

Specify the values of $p$ and $p'$ for a polynomial

Problem 10-26 from Spivak's Calculus, 4th edition: Let $a_1, \dotsc, a_n$ and $b_1, \dotsc, b_n$ be given numbers. If $x_1, \dotsc, x_n$ are distinct numbers, prove that there is a polynomial ...