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

learn more… | top users | synonyms

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
501 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 ...
2
votes
0answers
68 views

Prove that $\underline{\int_{a}^b} f \leq 0 \leq \overline{\int_{a}^{b} f}$

My question is just to make sure my proof is on the right track. Problem: Suppose that the bounded function $f\colon [a,b]\rightarrow \mathbb{R}$ has the property that for each $x\in \mathbb{Q}$, ...
2
votes
2answers
66 views

[Proof Verification]Prove that if f is differentiable at $c \in I$ and $f'(c) = 0$, then g is not differentiable at $d:=f(c)$.

Proposition. Let I be an interval, and let $f: I \to \mathbb{R}$ be a strictly monotone and continuous on I. Let $J := f(I)$ and let $g:J \to \mathbb{R}$ be the inverse function of f. Prove that if f ...
3
votes
2answers
115 views

Is $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ is rational number?

Can anyone help with this: Is $\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}$ is rational number?
9
votes
5answers
808 views

Issue with Spivak's Solution

Here was the problem: Here is the solution from his solutions book: This is barely a proof. How can he just say let $f(c) = 0$? How do you prove that $f(c) =0$ and how do you prove that $f(d) ...
1
vote
1answer
37 views

Volume using triple Integrals, cylindrical coordinates

I want to calculate the volume of a solid with $z+1\ge x^2+y^2$ and $3\left(z-1\right)\le -\left(x^2+y^2\right)$. After cylindrical coordinates x=rcosϕ, y=rsinϕ I got $r^2-1\le z$, and $z\le ...
0
votes
1answer
34 views

Minimum value of $F(a,b)$.

Let $$F(a,b) = \sum_{i=1}^n \left[ y_i - (ax_i+b) \right]^2$$ Find the minimum of $F$. Evaluating the dirctional derivatives: $$\frac{dF}{da} = \sum_{n=1}^n 2\cdot (y_i - (ax_i+b))(-x_i) \\ ...
1
vote
4answers
79 views

$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$

$$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$$ I tried to write it as $$I=\int \sqrt{\frac{\cos^6(x)}{\sin^7(x)}}\,dx$$ And $$I=\int \sqrt{\frac{1}{\tan^6(x)\sin(x)}}\,dx$$ but it seems to go ...
0
votes
2answers
29 views

Find $\lim_{b\to a}\frac {1}{b-a}\ln\left[\frac{a(b-x)}{b(a-x)}\right]$ if $x$ is constant using l'Hopital's rule

if x is a constant what do I differentiate with respect to? My best guess would be $b$. However, is this correct? Also how do you differentiate that function with respect to b? Do you have to use the ...
1
vote
0answers
190 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
3
votes
3answers
86 views

Help me find the following limit : $\lim_{{n}\to{\infty}} (\frac{2^x+3^x+\cdots+n^x}{n-1})^\frac{1}{x} = ?$

I have no idea where to start.$$\begin{align}\lim_{{n}\to{\infty}} \left(\dfrac{2^x+3^x+\cdots+n^x}{n-1}\right)^{1/x} = ?, n >1\\\end{align}$$
0
votes
1answer
22 views

Finding the general solution of a 2nd order ODE?

SO here's a problem that I'm not having much progress with: Using substitution $u=cosx$, how can I find the general solution of $sinx(d^2y/dx^2)-cosx(dy/dx)+2ysin^3x=0$ Thank you so much for ...
0
votes
1answer
49 views

$f: \mathbb R \to \mathbb R$, define $f^2(x)=f(f(x))$

Given $f: \mathbb R \to \mathbb R$, define $f^2(x)=f(f(x))$, then which of the following statements are true: $1.$ $f$ is strictly monotonic then $f^2$ is strictly increasing. $2.$ ...
0
votes
1answer
28 views

Find the stationary points of the curve $z(x,y)=xy(12-4x-3y)$

$$z(x,y)=xy(12-4x-3y)$$ First, I expanded the brackets: $$z(x,y)=12xy-4x^2y-3xy^2$$ Then I found the partial derivatives with respect to $y$ and $x$: $$(\frac{\partial z}{\partial ...
3
votes
1answer
67 views

Evaluating a line integral

What's the quickest way of evaluating $$ \int_{|y| = r} \frac{1}{|x - y|^2} d \sigma_y $$ in real plane, where $x \in B(0,r)$. Could complex contour integration help us here?
2
votes
1answer
44 views

Given $x,y,z>0$: $\frac{2}{3x+2y+z+1}+\frac{2}{3x+2z+y+1}=(x+y)(x+z)$. Find Minimum Value Of: $P=\frac{2(x+3)^2+y^2+z^2-16}{2x^2+y^2+z^2}$

Given $x,y,z>0$: $\frac{2}{3x+2y+z+1}+\frac{2}{3x+2z+y+1}=(x+y)(x+z)$ $(1)$ Find Minimum Value Of: $P=\frac{2(x+3)^2+y^2+z^2-16}{2x^2+y^2+z^2}$ I found $2x+y+z\geq 2$ from (1) but it not work ...