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

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0
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1answer
50 views

Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is ...
0
votes
1answer
27 views

Deriving definition of the complex logarithm

Given that: $$z = Re^{i\theta} = R(a + bi) = R\left( \cos(\theta) + i\sin(\theta) \right)$$ In its polar form. $$\log(z) = \log(R) + i\theta$$ $$|z| = \sqrt{(Ra)^2 + (Rb)^2} = R\sqrt{a^2 + b^2} ...
0
votes
4answers
79 views

Prove $\ln(1+x^{2})=\arctan x$ has two solutions in $\Bbb{R}$. How do I finish my proof?

Attempt: Define a function $g(x)=\ln(1+x^{2})-\arctan x$. Then $g'(x)={2x\over 1+x^2}-{1\over 1+x^2}$ and $x={1\over2}$ is a potential extremum. $g''(x)={-2x^2+2x+2\over (1+x)^4} \Rightarrow ...
0
votes
2answers
33 views

How to interpret a vertical line at the end of something?

For example: The last three lines have a |t=ti, what does that mean?
0
votes
1answer
33 views

Construct a non-constant sequence

Let $$S^{n}_{r}=\bigl\{{\overline{x}\in\mathbb{R}^{n+1}\;:\; \|{\overline{x}}\|=r}\bigr\}$$ thn $n-$ sphere with radius $r$ where $\|{\cdot}\|$ is the usual norm in $\mathbb{R}^{n+1}$ and a point ...
5
votes
2answers
310 views

Prove a convergent sequence has either a minimum, a maximum or both.

Let $a_n$ be a convergent sequence. Prove $a_n$ has a minimum, a maximum or both. I am being prepared for a final exam, which is why it is important to me to know that $I$ am correct in $my$ ...
2
votes
1answer
69 views

Domain of a composition of log functions

Domain of $$\log_3(\log_{1/3}(\log_4(\log_{1/4} x)))$$ Please guide me to solve this problem. Since it is a composition of logs, I am confused how to start.
2
votes
6answers
109 views

Limit $\lim _{ \theta \to 0 }{ \frac { cos2\theta -cos\theta }{ \theta } } $

$$\lim _{ \theta \to 0 }{ \frac { cos2\theta -cos\theta }{ \theta } } $$ Steps I took: $$\lim _{ \theta \rightarrow 0 }{ \frac { 1-2sin^{ 2 }\theta -cos\theta }{ \theta } } =$$ $$\lim _{ ...
0
votes
1answer
468 views

Help me complete finding the Reduction formula of $J_n=\tan^{2n} x \sec^3 x dx$?

Please don't mark my question as a duplicate of Find the reduction formula for the following integral. This question, was asked by a user and I was trying to answer this, I couldn't complete my work ...
1
vote
1answer
176 views

Find the reduction formula for the following integral.

$$ \mathrm{(b)} \quad\quad J_n = \int \tan^{2n}(x) \sec^3(x) \;\mathrm{d}x $$ I have no idea how to start this question when it comes to the reduction formula. I know there are some cases for when ...
7
votes
2answers
419 views

$\int_0^\infty\int_0^\pi\frac{k^2(e^{-it\sqrt{k^2+m^2}}-e^{it\sqrt{k^2+m^2}})\sin(\theta)}{e^{-ikx\cos{\theta}}\sqrt{k^2+m^2}}d\theta dk$

$$\int_0^\infty\int_0^\pi\frac{k^2\left(e^{-it\sqrt{k^2+m^2}}-e^{it\sqrt{k^2+m^2}}\right)\sin(\theta)}{e^{-ikx\cos{\theta}}\sqrt{k^2+m^2}}d\theta dk$$ I saw this Integral at Quora, and I have not ...
0
votes
2answers
409 views

Let f be a convex differentiable function. Prove that if u is any continuous function, then … [closed]

Let $f$ be a convex differentiable function. Prove that if $u$ is any continuous function, then $$\frac1 a \int_0^a f(u(t))dt \geq f \bigg(\frac1a \int_0^a u(t) dt\bigg) $$ I need insight on this ...
0
votes
3answers
89 views

Let $f$ be a continuous function on $[a,b]$ such that $\int_{a}^{b}f=0$ Prove that there is a number $z$ in $[a, b]$ such that $f(z)=0$.

Let $f$ be a continuous function on $[a,b]$ such that $\int_{a}^{b}f=0$ Prove that there is a number $z$ in $[a,b]$ such that $f(z)=0$. Show by an example that the continuity assumption is necessary. ...
1
vote
2answers
56 views

Evaluating $\lim _{ \theta \rightarrow 0 }{ \frac { \sin { 7\theta } }{ sin3\theta } } $

$$\lim _{ \theta \rightarrow 0 }{ \frac { \sin { 7\theta } }{ sin3\theta } } $$ My impression: I don't see how I can change the variables in any way to get it in the form where I can simplify it ...
1
vote
2answers
20 views

indefinite integration the result is -1 instead of $1 \over 2$

Below is from a book, When 0 $\le$ x < 1, F(x) = $\int_0^x$ t dt = $x^2 \over 2 $; When 1 $\le$ x < 2, F(x) = $\int_0^1$ t dt + $\int_1^x$(2-t) dt = -$x^2 \over 2$ + $2x$ ...
0
votes
2answers
46 views

Limit $\lim _{ h\to0 }{ \frac { \sin { 9h } }{ h } } $

$$\lim _{ h\to0 }{ \dfrac { \sin { 9h } }{ h } } $$ Steps I took: $$let\quad \theta \quad =\quad 9h$$ $$\lim _{ h\rightarrow 0 }{ 9\left(\frac { \sin { 9h } }{ 9h } \right) } $$ $$\lim _{ ...
0
votes
2answers
96 views

Simplify ratio of integrals $\frac{\int f(x-t) t e^{-t^2/2} dt}{\int f(x-t)e^{-t^2/2} dt}$

I am trying to simplify the following expression: \begin{align*} \frac{\int_{-\infty}^\infty f(x-t) t e^{-t^2/2} dt}{\int_{-\infty}^\infty f(x-t)e^{-t^2/2} dt} \end{align*} by getting it in terms of ...
1
vote
1answer
59 views

Evaluating a limit using the Squeeze Theorem

$$\lim _{ x\rightarrow 1 }{ (x-1) } \sin { \frac { \pi }{ x-1 } } $$ Steps I took: $$-1\le \sin { \frac { \pi }{ x-1 } } \le 1$$ $$-\left| x-1 \right| \le \sin { \frac { \pi }{ x-1 } } \le ...
0
votes
1answer
106 views

Solving an Equation by going to the 3rd dervative

Given the equation $y'=5x^2+2y^2-7$, where $y(0)=-2$, find $y'(0), y''(0), y'''(0)$ using the above, I have to find $y'''(0)$, which is what I'm having trouble with. I solved $y'(0)=1$, and then ...
1
vote
1answer
27 views

Solving A Second Order Ordinary Differential Equation

Given the equation $y'=5x^2+2y^2-7$, where $y(0)=-2$, find $y'(0)$. So I'm sure you have to manipulate the equation to integrate both sides, solve for the constant, then use that to find what ...
0
votes
3answers
80 views

Evaluate $ \int_{a}^{b}(A - f(x))dx$ where $A = [1/(b-a)] \cdot \int_a^b f(x)\,dx$

My solution: Using the definition of the integral, rewrite $f(x)$ in the expression $A = [1/(b-a)] \cdot \int_a^b f(x) \, dx$ as: $$A = \frac1{b-a} \sum_{i = 0}^{n \to +\infty} f(x)\frac{b-a}n$$ ...
4
votes
2answers
174 views

Solutions of the functional equation $f(2x) = \frac{f(x)+x}{2}$

How can I solve the following functional equation? $$f(2x) = \frac{f(x)+x}{2},$$ for $x \in \mathbb{R}$ with $f$ being a continuous function.
3
votes
2answers
75 views

Limit $(1+\frac{1}{a_n})(1+\frac{1}{a_{n-1}} )\cdots(1+\frac{1}{a_1}) $ where $a_n=n(1+a_{n-1})$ and $a_1 =1$

Suppose $a_n=n(1+a_{n-1})$ and $a_1 =1$. Then the limit of $(1+\frac{1}{a_n})(1+\frac{1}{a_{n-1}} )\cdots(1+\frac{1}{a_1}) $ where $n$ tends to infinity is? I got $1/2$ for answer is it correct?
0
votes
1answer
57 views

Solutions of the functional equation $f(x) + f(qx) = 0$

How can I find the solutions of $$f(x) + f(qx) = 0,$$ where $q \in \mathbb{Q}, q\neq1, x \in \mathbb{R}$, with $f$ being a continuous function?
-1
votes
1answer
144 views

How do I find the area shared by the circles $r = 2\cos(\theta)$ and $r = 1$?

I figured out the intersection points: $r=2\cos(\theta)$, $r=1$ $2\cos(\theta) = 1$ $\cos(\theta) = \frac{1}{2}$ $\arccos(1/2) = π/3$ (I), $5π/3$ (IV)
0
votes
1answer
50 views

Shifting Velocity and Position functions

I'm given a function $A(t)$ that defines the acceleration of an object w.r.t. time $t$ and am tasked with finding the position function and velocity function for that object. Finding the functions ...
1
vote
1answer
23 views

$f:\mathbb{R}^2\rightarrow \mathbb{R}$, $C^0. f(x,0)=0 \Rightarrow \exists r>0$ such that $|f(x,y)|<\frac{1}{4}$ in $(x,y)\in[0,1]\times[-r,r]$?

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ continue and with $f(x,0)=0$ for each $x\in\mathbb{R}$. Show that there is $r>0$ such that $|f(x,y)|<\frac{1}{4}$ for each ...
2
votes
1answer
216 views

How to prove the limit of Thomae function?

Given the Thomae Function: $$t(x)= \begin{cases} 1 & \text{if }x=0, \\ 1/n & \text{if }x=m/n \in \mathbf Q\setminus\{0\}\text{ is in lowest terms with }n>0,\\\ 0 & ...
4
votes
2answers
141 views

Closed form for $ \prod_{k=1}^n (a+k^2) $

I have come across the following product: $$ \prod_{k=1}^n (a+k^2) $$ where $a$ is a positive constant. Could anyone suggest a closed form for this product? I need to approximate this for large $n$, ...
1
vote
4answers
502 views

Integral of $\cos^4(2t)\,dt$ with bounds from $0$ to $\pi$

$$\int_0^\pi\cos^4(2t)\,dt=?$$ I have attempted this problem two different ways and got two different answers that are nowhere near the correct answer. Could you please show me detailed steps on how ...
0
votes
3answers
57 views

Local Max of an Integral

I'm having trouble with the following problem. $f(x)=\int_0^x \frac{t^2-4}{1+cos^2(t)}dt$ At what value of $x$ does the local max of $f(x)$ occur? I've tried just taking the integral then ...
0
votes
2answers
31 views

Determining whether an inequality provides sufficient information to determine the limit

State whether the inequality provides sufficient information to determine the $\lim _{ x\rightarrow 1 }{ f(x) } $, and if so, find the limit. $$4x-5\le f(x)\le x^{ 2 }\\ 2x-1\le f(x)\le x^{ 2 }\\ ...
3
votes
2answers
123 views

How to show that $ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \ln\left(\tan(x)+\tan\left(\frac{\pi}{6}\right)\right)\tan(x)\space dx=\frac{\zeta(2)}{6} $

I was trying to prove the well known result: $$ \sum_{k=1}^\infty \frac{1}{\binom{2k}kk^2}=\frac{\zeta(2)}{3} $$ and it came down to prove the following equation: $$ ...
0
votes
0answers
55 views

Calculus - application of the intermediate value theorem

Let $f$ be a continuous function, $f:[0,1] \rightarrow [0,1]$. For which values of a$ \in \mathbb{R}$ does there necessarily exist $c \in [0,1]$ such that $f(c) = ac$? Is the following answer ...
0
votes
1answer
50 views

Finding the Volume of an Oddly Shaped Region

I am currently working on some homework for my calculus course and I have been stuck on a problem for quite some time. The problem sounds simple enough: Find the volume of the solid generated by ...
0
votes
1answer
28 views

Fourier series: $\lim_{n\to\pm\infty} n^p \hat{f}(n) = 0$

Let $f:\mathbb{R}\to\mathbb{C}$, $f\in C^\infty$ (differentiable infinitely many times) and periodic,$T=2\pi$. Prove that for every $p>0$: $$ \lim_{n\to\pm\infty} n^p \hat{f}(n) = 0$$ So I ...
0
votes
1answer
26 views

Find a polynomial $Q$ of degree $k$ and a remainder function $E$ for $f(x)=\frac{1}{1-x}$.

There is a theorem in our textbook saying that rather than calculating all the derivatives needed to compute the taylor polynomial, if one can find, by any means, a polynomial $Q$ of degree $k$ such ...
2
votes
2answers
91 views

What does $C^{\infty}_0$ stand for

In my course material I have the following notation: $$f\in C^{\infty}_0(\Omega, \mathbb{R}),$$ where $\Omega \subset\mathbb{R}^n$ is a bounded open set. I was wondering what does this notation ...
0
votes
2answers
218 views

Showing y≈x for small x if y=log(x+1)

Given: $y=\log(1+x)$ Show that $y≈x$ if $x$ gets small (less than 1). I don't think we're supposed to use Taylor series (because they were never formally introduced in class), but I do think we have ...
1
vote
1answer
44 views

Proofs from the book: in Praise of inequality's

I am reading a book with nice proofs, but i struggle at a few points. 1) why is $\sum_{i=1}^{k} p_i \int_{a_i}^{G} (\frac{1}{t} - \frac{1}{G}) dt + \sum_{i=k+1}^{n} p_i \int_{G}^{a_i} (\frac{1}{G} - ...
0
votes
1answer
203 views

Solids of Revolution Question (Method of Cylinders vs Disc/Washers)

Find the volume of the solid formed by revolving the region bounded by y=x^2+1, y=0, x=0, and x=1 about the y-axis. I was practicing this concept and I came across this problem. I did it using the ...
0
votes
1answer
50 views

Explicit form for $\left(e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)\right)^2$

Basically I have been working with polynomials of the form: $$P_n(x)=e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)$$ I do realize that an explicit form for $P_n(x)$ has been asked for on this site ...
3
votes
4answers
471 views

A calculus proof for the general term of the Fibonacci sequence [duplicate]

Let $a_0=1$,$a_1=1$ and $a_n=a_{n-1} + a_{n-2}$ for $n \geq 2$, I would like to prove: $$a_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n + 1}- \left(\frac{1-\sqrt{5}}{2}\right)^{n + ...
2
votes
1answer
92 views

Clarifying the elementary calculus used in this statistics problem

Let $X \sim N(\mu, \sigma^{2})$ and $Y = \alpha X + \beta$ for $\alpha > 0$. I'm looking at a demonstration that $Y = \alpha X + \beta \sim N(\alpha\mu + \beta, (\alpha\sigma)^{2})$, and find ...
0
votes
1answer
74 views

Why is the euclidean norm not differetiable at $0$?

I denote $N(x)$ as the norm-function, although in the denominator it stays $\|x\|$. $$\lim_{x\to 0} \frac{N(x)-N(0)}{\|x\|} = \lim_{x\to 0} \frac{N(x) - 0}{\|x\|} = \lim_{x\to 0} 1 = 1 \ne 0$$ 1) ...
2
votes
2answers
248 views

Differential equation type

How can I solve this differential equation $$(1 + x^2)(1+y^2)\mathrm dx +xy\mathrm dy = 0$$ It doesn't look like separable and I don't think it's neither homogenous. Maybe I need to use the ...
0
votes
3answers
38 views

A problem in calculus mean value theorem

Hi tried to solve this for hours, any idea how to approach this question: prove for every $x>0$ $$2x\times\arctan(x)>\ln(1+x^2)$$
23
votes
2answers
478 views

How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
0
votes
1answer
17 views

Determining at what points multiple variable functions are continuous

With a two variable function what is the procedure to figure out at what points it is continuous? Do I basically just look at what points it would be undefined and anywhere between those points it is ...
2
votes
1answer
23 views

Fourier series: Show that $f$ is a trigonometric polynomial

Let $N\in\mathbb{N}$ and $f_m:\mathbb{R}\to\mathbb{R}$, continuous functions and periodic, $T=2\pi$. Let's assume that $f_m \to f$ uniformly and for all $m\ge 1$: $$\left| \hat{f_m}(n)\right| \le ...