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

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2
votes
2answers
35 views

Express $\arctan(2x)$ as power series

I am trying to express $\arctan(2x)$ as power series. Let $f(x) = \arctan(2x)$, then $f'(x) = \frac{2}{1+4x^2}$ $$\arctan(2x)=\int \frac{2}{1+4x^2} \, dx$$ $$\int\frac{2}{1+4x^2} \, dx= \int2\frac{...
3
votes
1answer
43 views

Proving that $\lim_{x\to 2}\frac{x^2-5x}{x^2+2}=-1$ using the $\epsilon$-$\delta$ definition of a limit

My attempt: $$ \left|\frac{x^2-5x}{x^2+2}+1\right|<\left|\frac{x^2-5x}{x^2+2}\right|< \left|\frac{x^2-5x}{x^2}\right|<\frac{1}{x^2}|x^2-5x|,$$ using the restriction $|x-2|<2$, so $0<x&...
2
votes
1answer
24 views

If set $A_n$ converges to $A$ in $L^1$, then is it also true for the $r$-neighborhood?

Let $A$ be a bounded simply connected set in $\mathbb{R}^2$, and $A^r$ denote the $r$-neighborhood of $A$, that is, $A^r:=\{x: d(x,A) \le r\}$. Suppose there exists $A_n \rightarrow A$ in $L^1$, that ...
4
votes
2answers
102 views

What is the derivative of $x^i$?

What would the derivative be of $x^i$? Would it simply be $ix^{(1-i)} $? I tried running the Power rule, and I got that is that right?
0
votes
1answer
34 views

Recursive formula in term of original value

$$P_1=P_0G_{0,1}A_1\\ P_m=P_{m-1}G_{m-1,m}A_m+A_0\sum_{i=0}^{m-2}P_i G_{i,m}~~\text{for}~~m\geq 2$$ Is it possible to write $P_m$ in terms of only $P_0$, i.e., without other $P_j$ terms?
1
vote
4answers
42 views

Why must the determinant of the hessian of a scalar function be positive for there to be a local min/max? Intuition needed

Is there any intuition behind having the determinant of the Hessian matrix being negative corresponding to a saddle point, and positive corresponding to a max/min depending on the sign of $f_{xx}$ for ...
0
votes
0answers
13 views

Reparametrization of a function [on hold]

Given \begin{equation} S(t)= \left( 1- \left( 1-{{\rm e}^{- \left( \rho\,t \right) ^{k}\theta}} \right) ^{\gamma} \right) ^{\frac{1}{\theta}} \end{equation} where $\rho,k,\theta$ and $\gamma$ are ...
2
votes
2answers
30 views

Calculate $P'(x)$ for $x \in (-1,1)$

$P$ is a power series with $P(x)=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ Calculate $P'(x)$ for $x \in (-1,1)$ When I read this task (not homework), I got some questions: 1) ...
6
votes
2answers
63 views

Proving $\lim_{x\to1}(x^3+5x^2-2)=4$ using the $\epsilon$-$\delta$ definition of a limit

I want to prove that the limit of $f(x)=x^3+5x^2-2$ when $x\to 1$ is $4$. So, I want to show that for any $\epsilon >0$ $\exists \delta_{\epsilon}$ such that for all $x$ that satisfies $|x-1|<\...
-1
votes
0answers
34 views

FourerIntegrals

Trying to understand a proof about Fourer Intergrals in my book. I can't understand how they got formula (3). As you can see delta W goes to $0$ then $L$ goes to infinty. Doesn't that mean that we get ...
2
votes
0answers
31 views

integration by parts of multiple functions

Can anyone help me solve this problem? $I_0$ represents the modified Bessel function.Thanks. $$\int_{a-b}^{a+b}\frac{\exp\left(-\frac{c\,t^2}{2ab}\right)\,t\,I_0(t)}{ab\sqrt{1-\frac{(t^2-a^2-b^2)^2}{...
2
votes
2answers
53 views

Proof that $P(x)=x-\frac{1}3 x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$

Proof that $P(x)=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$ First of all, I need to convert this to a series: $$\sum_{k=1}^\infty \frac{x^{2k-1}(-1)^k}{...
4
votes
2answers
48 views

Prove: the function $g$ has a global minimum in $\mathbb{R}$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a polynomial of a even $n$ degree, such that $0\leq f(x)$ let $g=f+f'+f''+\cdots+f^{(k)}$, prove $g$ has a global minimum in $\mathbb{R}$ when $k$ is the $...
0
votes
3answers
30 views

Distance travelled on moving walkway.

I am reading Manga guide to calculus. I am kind of stuck at one place. The statement says; The moving walkway moves f(x) meters in x minutes. When measured on walkway, Futoshi travels g(x) meters in ...
1
vote
2answers
41 views

Proof of sum convergence?

$$\sum_{n=1}^{\infty} \frac{2\sqrt{n} + 1}{n^2 + n + 1}$$ This seems like a problem that could be handled by the comparison test. So we need an $f(n) > \displaystyle \frac{2\sqrt{n} + 1}{n^2 + n ...
0
votes
0answers
20 views

When can you “plug in” a function $g(x)$ directly into a taylor expansion of a function $f(x)$ to get the expansion of $f(g(x))$, specifics below

I have asked a couple of questions Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to ...
1
vote
1answer
34 views

Best Differential Equation, Partial Differential Equation and Calculus of Variations books?

Electrical Engineer here thinking of switching to physics. What are the best Differential Equation, Partial Differential Equation and Calculus of Variations books? Ideally they explain the topic ...
0
votes
3answers
37 views

Derivative $f(x)={{x \ln (x)} \over 1+ \ln (x)}$ and $f(x)=\ln( \ln x)$

If $f(x)={{x \ln (x)} \over 1+ \ln (x)}$ Find $df\over dx$ My answer : $${(\ln(x)+1).(\ln(x)+1)-ln(x)}\over {(1+\ln(x))^2}$$ $${(\ln(x)+1)^2-\ln(x)}\over {(1+\ln(x))^2}$$ True or false ? And if ...
0
votes
3answers
129 views

Integral that makes square root of $\frac{\pi}{2}$ [duplicate]

My question is regarding a integral that´s giving me a huge headache. I want to show $$\int_{0}^{\infty}y^2e^{-\frac{y^2}{2}}dy=\sqrt{\frac{\pi}{2}}$$ I'm studying for an exam. I'm suppose to find ...
3
votes
1answer
57 views

Numerical Method Sample Question via Truncation error Methods?

I have one multiple choice question: Approximation of integration $\int_0^{0.1} e^{x^2}dx $ by using simple formula of following options has lower Truncation error: Choice Part: $a)$ ...
6
votes
3answers
102 views

Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\large\frac { 1 }{ 1+\frac { 1 }{ 2+\frac { 1 }{ 3+\frac { 1 }{ 4+\dots } } } } $$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I ...
3
votes
1answer
29 views

Clarification on what this author means by “logging does not change the maximum” of a function

I am self-studying for an actuarial exam and I encountered the following: The author seems to suggest that if we want to find the maximum of a function $f(x)$ with respect to $x$: We can drop any ...
8
votes
5answers
125 views

Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + … + 1/(x+a_n) = 1/x$ are all real

Prove that the roots of the equation: $$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$ are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.
0
votes
1answer
19 views

Definition of locally integrable function

I was given two definitions: Let the function $f(x)$ be defined in a interval $[a,\infty)$ we will say that $f$ is locally integrable in $[a,\infty)$ if for all $a<b$ $f$ is integrable in $[...
1
vote
1answer
26 views

Is the mathematical syntax correct here (Taylor-polynomial)?

Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$ I'd like to know if the syntax is correct, how I solved this little task. We have defined the ...
0
votes
2answers
53 views

Absolutely Convergent, Conditionally Convergent or Divergent?

$$\sum_{k=0}^{\infty} (-1)^{k+1}\frac{\sqrt{k}}{k+1}$$ This problem is asking me to prove if this series is absolutely convergent, conditionally convergent or divergent, but I don't know how to start ...
0
votes
0answers
39 views

Can this integral equality hold for any nonzero real $b$?

Please may you kindly assist me on this integration exercise: Suppose for some fixed real $a, b$ with $a \neq 0$, we have: $$\int_1^N f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^N f(x)\sin(a\log \...
1
vote
1answer
73 views

Inner Product Examples, what is the points?

Example: For $ -\pi<x<\pi$, $$x =-2 \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx)$$ and $$x^3 =-2 \sum_{n=1}^{\infty} \left( \frac{\pi^2}{n}-\frac{6}{n^3} \right)(-1)^n \sin(nx)$$ by ...
-1
votes
3answers
24 views

Does this series converge or diverge? application of root test

Suppose $\sum a_n $ is convergent and $a_n > 0$ for all $n$, does it follow that $\sum \left( \frac{ 1 + \sin (a_n) }{2} \right)^n $ is convergent?? yes Since $\sum a_n$ is convergent, then $\sum ...
1
vote
2answers
35 views

Why is the gradient always perpendicular to level curves?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. A level set is a set of points: $$L(c) = \{x \in \mathbb{R}^n | f(x) = c\}$$ Two vectors $a, b \in \mathbb{R}^n$ are perpendicular, when ...
0
votes
0answers
34 views

Does Taylor series around point zero (maclaurin series) always exist for differentiable function?

For every differentiable function $f(x)$, is $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ always true and can be written? The only thing we have to be concerned is just whether ...
0
votes
2answers
30 views

When given a function of multiple variables, is it possible to differentiate all the variables simultaneously?

I understand that when given a function of two variables, for example $f(x,y)=x^2y+xy^2$, you can perform partial differentiation on it to get $(2xy+y^2,y+2xy)$. However, to my understanding, this is ...
0
votes
1answer
31 views

Difference between little o and big O notation in taylor expansion

I know I can say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+o(\Delta t) $$ But can I say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+O((\Delta t)^2) $$ In both cases, do I have to add that $\Delta t \...
0
votes
0answers
30 views

Recommended Textbook for Integral Equation [on hold]

I am doing a self reading in preparation for the courses I have next semester of which Integral Equation is part of it. I keep on seeing very strange notations in the materials given to me by my ...
0
votes
1answer
32 views

Integration by substitution $\int (x - b) dx$

Probably a simple one. So, integrating something like: $$\int (x-b) \space dx $$ Integrating by substitution seems to be the way to go, like so: $$\frac{d}{dx} (x-b) = \frac{d}{dx}x - \frac{d}{dx} ...
2
votes
2answers
75 views

If f is a function defined for all real x and satisfies f(x+y) = f(x) + f(y) + xy(x+y). Does f' and f'' exist?

Let $f$ be a function defined for all real $x$ and satisfies $f(x+y) = f(x) + f(y) + xy(x+y).$ If $f'(0) =-1,$ then prove that $f$ and $f'$ are differentiable for all $x.$ Also find $f'(3)$ and $f'(4)$...
1
vote
0answers
21 views

Maximum point of a modulus function

For our project on Maxima and Minima of functions, we have to do functions of type $\frac{k}{|x-a|+|x-b|}$. So, I chose $f(x)=\frac{2}{|x-1|+|x-2|}$ I noticed that the derivative is positive for $x&...
1
vote
1answer
61 views

“Of the order of” notation

I have a function where all terms have the same coefficient $x^3$ in it: For example $f(x) = ax^3 - bx^3$ Can I say that in big $O$ notation: $f(x) = O(x^3)$ $f(x) = O(x^3)$ as $x \rightarrow 0$ ...
0
votes
2answers
41 views

Is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative?

Assume a function $f(x)$ is differentiable, is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative? $$\max\left(\frac{...
0
votes
3answers
50 views

Find $|\boldsymbol{a \times b}|$ (graph included) if $|\boldsymbol{a}| = 2$, $|\boldsymbol{b}| = 5$

I don't see how I have enough information to figure this one out. Here's what I'm thinking: \begin{align*} |\boldsymbol{a \times b}| & = |\boldsymbol{a}|~|\boldsymbol{b}|\sin(\theta)\\ ...
-1
votes
1answer
47 views

As I can find the critical points of the function $\int_a^b (y^{2}+2(y')^{2}+(y'')^{2}) dx.$ [on hold]

Hi I'm stuck with a problem, greatly appreciate a suggestion to solve: As I can find the critical points of the function $$\int_a^b (y^{2}+2(y')^{2}+(y'')^{2}) dx.$$
1
vote
0answers
19 views

Maximum of derivative hierarchy

Given a function $a(x)$, is $\displaystyle\max\left(\frac{1}{2}\frac{da(x)}{dx}\right) > \max\left(\frac{1}{3!}\frac{d^2a(x)}{dx^2}\right) > \max\left( \frac{1}{4!}\frac{d^3a(x)}{dx^3}\right) &...
1
vote
3answers
44 views

Finding intersection between functions containing logarithms

At what point does $8x^2$ and $64x\log(x)$ intersect? I'm trying to catch up on my math, but this stumped me. Something awry in my understanding of logarithms. I figured that I could equate both ...
3
votes
2answers
91 views

Prove that $\lim a_{n}=e$?

Given that $a_{1}=0$, $a_{2}=1$ and $$a_{n+2}=\frac{(n+2)a_{n+1}-a_{n}}{n+1}$$ prove that $\lim\limits_{n\to\infty} a_n=e$ What I did: It was hinted to prove that $a_{n+1}-a_{n}=\frac{1}{n!}$ ...
0
votes
1answer
31 views

Finding the velocity of a position vector

Let $\{\tilde{i}, \tilde{j}\}$ be the standard basis vectors for IR2. Define two paths in IR2 by $\tilde{v1}$(θ) = cosθ$\tilde{i}$ + sinθ$\tilde{j}$ $\tilde{v2}$(θ) = −sinθ$\tilde{i}$ + cosθ$\tilde{j}...
1
vote
1answer
49 views

Be $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ a continuous function and $g(x)=\int_0^1 \! f(x,y) \, \mathrm{d}y.$ Proves that g is continuous.

I don't see how to solve the following problem, I think that it's like a generalization of the fundamental theorem of calculus. Be $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ a continuous function and $g(...
1
vote
2answers
36 views

Representing $f(x,y)$ as a Sum of Partial Derivatives

I was attempting an exam question which looked like this: Given the expression: $P(x, y)\text{d}x + Q(x,y)\text{d}y = 0$ Where: $P(x,y) = 6x +9y + 11 \\ Q(x, y) = 9x - 4y +3$ Find a function $f(...
1
vote
0answers
23 views

Evaluating a limit involving summation [duplicate]

Evaluate : $$ \lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right) $$ Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried ...
0
votes
0answers
20 views

Proving the Hilbert space filling curve is nowhere differentiable. [duplicate]

I am trying to understand the proof the Hilbert curve is nowhere differentiable. The Hilbert curve is defined as the mapping $f_h: I \to \mathcal{Q}$ where I in the unit interval in $\mathbb{R}$ and ...
0
votes
1answer
26 views

Cylindrical Coordinates

In the following example i am looking to find the volume of the solid bounded above by the plane $ z = y$ and below by the paraboloid $ z = x^2 +y^2 $ by the method of cylindridical coordinates. ...