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

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3
votes
3answers
153 views

How find this value $\frac{a^2+b^2-c^2}{2ab}+\frac{a^2+c^2-b^2}{2ac}+\frac{b^2+c^2-a^2}{2bc}$

let $a,b,c$ such that $$\left(\dfrac{a^2+b^2-c^2}{2ab}\right)^2+\left(\dfrac{b^2+c^2-a^2}{2bc}\right)^2+\left(\dfrac{a^2+c^2-b^2}{2ac}\right)^2=3,$$ find the value ...
2
votes
1answer
361 views

Definite integral involving modified bessel function of the first kind

I would like to solve the following integral that is a variation of this one (Integral involving Modified Bessel Function of the First Kind). Namely, I have: $$\frac{1}{\sqrt{2\pi ...
2
votes
2answers
82 views

Trigonometry - Finding the range of the function

Problem : $$f(\theta)=(2\sqrt{3}+4)\sin\theta +4\cos \theta $$ I have studied if the function is in the form : $f(\theta)=a\cos\theta + b\sin\theta$ then the range of this function can be given as ...
3
votes
2answers
191 views

Proving The Average Value of a Function with Infinite Length

This is the given: One can extend the definition of the average value of a continuous function $f(x)$ to the interval $[a,\infty)$ of infinite length as follows: ...
1
vote
1answer
145 views

Change of Variables via Line Element Differential vs. Jacobian

If I had a double integral $\int \int_R f(x,y) dxdy $ I would change variables to polar coordinates by expressing the position vector $\vec{r} \ = \ \vec{r}(x,y) \ = \ x \vec{e}_x \ + \ y\vec{e}_y$ ...
2
votes
2answers
58 views

Comparing Areas under Curves

I remembered back in high school AP Calculus class, we're taught that for a series: $$\int^\infty_1\frac{1}{x^n}dx:n\in\mathbb{R}_{\geq2}\implies\text{The integral converges.}$$ Now, let's compare ...
1
vote
1answer
73 views

Showing $\max\limits_{|z|=r}|p(z)| \ge |a_n|r^n$, without Cauchy integral formula.

Let $p(z) = a_n z^n + a_{n-1}z^{n-1} + \cdots + a_0$. My question is: Is there an elementary way to show that for all $r > 0$ $$ \max \limits _{|z| = r} |p(z)| \ge |a_n|r^n$$ without using ...
1
vote
1answer
35 views

Differentiability of projection

Where $\pi_i:\Bbb R^n\rightarrow\Bbb R$ is projection onto the $i$th coordinate, the differentiability of $\pi_i$ at $X$ is given by: $$\pi_i(X+H)-\pi_i(X)=\textrm{grad}\ \pi_i(X)\cdot H+||H||g(H)$$ ...
4
votes
3answers
131 views

$f(x)=x^{x}$ what happens when $x$ is a negative irrational number?

Just looking at negative numbers, $x^{x}$ is defined for all rational numbers (on the real plane) in all instances except whenever $x=\large \frac {2a+1}{2b}$ where $(a, b)$ are integers . However, ...
2
votes
2answers
742 views

Lipschitz continuous

Let $\delta$ be an interval in $\mathbb{R}$. Recall that a function $f$ is called Lipschitz continuous on $\delta$ with Lipschitz constant $L$ if there holds $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$ ...
3
votes
1answer
46 views

How can I transform this equation in a conical?

In this equation $$2x²+y²-4x-6y+11=0$$ I got the result $(1,3)$ completing squares $2(x - 1)² + (y - 3)² = 0$   But on my list exercises, demanded that determine the foci, straight guideline ...
0
votes
0answers
74 views

When is the “inequality” approach to limits valid?

For example, let's say $\lim_{x\to \infty} [g(x)]^{f(x)}=1$ . If we know that as $x \to \infty$, $h(x)> g(x)$ , we can say that $\lim_{x\to \infty} [h(x)]^{f(x)}$ equals $\infty$ . However, I ...
2
votes
1answer
106 views

Inequality involving definite integral

Just wondering, what may be the best way to show that $$\int_0^1 xf(x)dx \leq \frac{1}{2}\int_0^1 f(x)dx,$$ provided that $f(x) \geq 0$ over the interval $[0,1]$ and that $f(x)$ is monotonically ...
8
votes
4answers
237 views

How to find the limit $\lim\limits_{m\to\infty}\frac{m^{m-2}}{(m-1)^{m-2}}$?

I am trying to evaluate the limit of this: $$\lim_{m\rightarrow \infty} \frac{m^{m-2}}{(m-1)^{m-2}}$$ That is just basic calculus I think but I forget those methods for finding the limit. I think ...
7
votes
2answers
169 views

Fundamental Theorem of Calculus in Multivariate Case

From the FTC we have, for continuously differentiable $f: \mathbb{R} \to \mathbb{R}$, $$ f(a) - f(b) = \int_b^a \frac{d}{dx} f(x) dx $$ I'm trying to write the difference between a vector function ...
2
votes
2answers
1k views

Intersection between a plane and a curve

Find the point(s) at which the following plane and curve intersect. The plane: $$3x+4y-12z=0.$$ The curve $$r(t)=\langle 3\cos t,3\sin > t,\cos{t} \rangle; 0≤t≤2\pi.$$ I started out by ...
3
votes
3answers
68 views

Limit of sum with parameter [duplicate]

$$\lim_{n \rightarrow \infty} \sum^n_{k=1} \frac {1} {k+n} = \ln 2$$ I'm supposed to find the limit and get the right part as an answer. I don't know what to do with it. Could someone explain?
2
votes
2answers
182 views

Why do we include deleted neighborhoods when defining limits?

Very often, we define the limit of a function as $0 < |x -a|< \delta \implies |f(x) - L|< \epsilon$. A lot of times we don't let $x \neq a$, for the case of discontinuity, it is clear. ...
1
vote
5answers
87 views

solution of recurrence relation $x_{n+1} = \frac {1}{x_n + 1}$

$$x_{n+1} = \frac {1}{x_n + 1}; x_1 > 0$$ How to transform it into the form $x_n = $? I need the solution in order to check if it converges at any $x_1 > 0$.
1
vote
2answers
90 views

Evaluation of limits: $\lim n \cdot \exp( f(nx) )$

I have two similar (from my point of view) limits: $$\lim_{n \rightarrow \infty} x \sqrt n \exp(-\sqrt {nx})$$ $$\lim_{n \rightarrow \infty} n \exp(-n \cdot |x|))$$ How to deal with such limits? Is ...
2
votes
1answer
443 views

Derivation of Riemann Stieltjes integral with floor function

I come back again only to confirm (or not) a generalization. In my post on yesterday the integral $$ \int_{0}^{6}(x^2+[x])d(|3-x|) $$ was worked out based on a change of variable. I tried to get ...
32
votes
12answers
2k views

explaining the derivative of $x^x$

You set the following exercise to your calculus class: Q1. Differentiate $y(x) = x^x$. A student submits the following solution: Let $g(a)=a^x$ and $f(x)=x$. Then $y(x) = g(f(x))$, so by ...
3
votes
2answers
45 views

Relations between radius of convergence

Let $R$ denote the radius of convergence. Then $\sum_{n=0} a_{2n} x^{2n}$ has $R = 2$, and $\sum_{n=0} a_{3n} x^{3n}$ has $R = 3$. How to prove that $\sum_{n=0} a_{n} x^{n}$ has $R \leq 2$? For ...
5
votes
1answer
139 views

Zeroes of a Particular Function

I'm looking for the zeroes of $f(k) = e^{\sqrt{k}}[\frac{s}{k} - \frac{d}{\sqrt{k}}] - 1$ on the set $k > 0$. Is there a nice way to describe the set of solutions for given $s$ and $d$? Thanks!
3
votes
3answers
43 views

Finding parameter of equation

$$x^4 + 1 = kx; k>0$$ The question is at what k the equation has 1 solution. I understand that it could rephrased as follows: at what k $f(x) = x^4 - kx + 1$ has 1 $x:f(x) = 0$. but I've no idea ...
1
vote
1answer
33 views

How can I calculate the expected change of my function?

I am attempting to model the following equation with 3 variables (H, M, and S) and 1 constant (C): D = .05*H*M + .05*S*(M+C) or, when factored D = .05*M (H+S) + .05*C(S) It'd be a simple ...
0
votes
1answer
132 views

Directional derivative, gradient and a differential function

Let there be some function $f$, some point $(x_0,y_0)$ and some vector $u$. Is $D_{u}f(x_0,y_0)=∇f(x_0,y_0)⋅u$ always correct? Even if the the function is not differentiable at the point? Or in more ...
1
vote
2answers
68 views

How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent?

How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent? I applied integral test and found the series is divergent. I wonder if there exist easier ...
6
votes
5answers
854 views

Integration trig substitution $\int \frac{dx}{x\sqrt{x^2 + 16}}$

$$\int \frac{dx}{x\sqrt{x^2 + 16}}$$ With some magic I get down to $$\frac{1}{4} \int\frac{1}{\sin\theta} d\theta$$ Now is where I am lost. How do I do this? I tried integration by parts but it ...
2
votes
1answer
70 views

Proof including continuous function

Question Let $f: \Bbb R \to \Bbb R$ a continuous function. Prove that $$\forall x \in \Bbb R\,\, \forall A >0\,\, \exists B>0\,\, \forall y \in \Bbb R\,\, |y-x|\le A : |f(x)-f(y)|\le B $$ ...
1
vote
2answers
63 views

differentaibility of a piecewise continuous function

For each $n\in \mathbb{N}$ let $$ f_n = \begin{cases} x^{n+1}, & x \in \mathbb{Q} \cap (-1,1), \\ x^{2n}, & x \in \mathbb{Q}^c\cap(-1,1). \\ \end{cases}$$ Prove that for each $n\in ...
0
votes
1answer
469 views

Filling a conical tank

I have been working on this problem for about 2 hours and I can't seem to get it, here is exactly what the question reads. "Water is poured into the top of a conical tank at the constant rate of 1 ...
0
votes
1answer
79 views

Fundamental theorem of calculus integration

Suppose that $g$ is continuous on $[a,b]$ and that $f$ is differentiable on $[a,b]$ and its derivative is continuous on $[a,b]$ with $f'(t)\geq 0$ for each $t\in [a,b]$. Prove that there exists $c \in ...
0
votes
2answers
171 views

Conditional/Absolute convergence of $\int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$

I need to check conditional/absolute convergence of the integral: $$f(x) = \int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$$ I tried for a long time and I can't understand what I should do. I know that ...
0
votes
1answer
88 views

proving continuity

For each n belongs to natural numbers let, $$f_n(x) = \begin{cases} 0, & x \in \mathbb{Q} \cap[-1,-1/n] \cup [1/n,1] \\ 1, & x \in \mathbb{Q}^c\cap[-1,-1/n] \cup [1/n,1] \\ 0, & x ...
6
votes
2answers
280 views

Show that $\lim\limits_{n\to\infty} x_n$ exists for $0 \le x_{n+1} \le x_n + \frac1{n^2}$

Let $x_1, x_2,\ldots$ be a sequence of non-negative real numbers such that $$ x_{n+1} ≤ x_n + \frac 1{n^2}\text{ for }1≤n. $$ Show that $\lim\limits_{n\to\infty} x_n$ exists. Help please...
10
votes
3answers
274 views

the value of $\lim\limits_{n\rightarrow\infty}n^2\left(\int_0^1\left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$

This is exercise from my lecturer, for IMC preparation. I haven't found any idea. Find the value of $$\lim_{n\rightarrow\infty}n^2\left(\int_0^1 \left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$$ ...
1
vote
2answers
63 views

Double Integral over Solids

I know how to do double integral in polar coordinates but I just dont know how to visualize the 3D solid in order to find volume,$V=f(x,y$). Question : Find the volume of the solid bounded by the ...
8
votes
6answers
1k views

Finding $\lim\limits_{x\to0}x^2\ln (x)$ without L'Hospital

I am preparing a resit for calculus and I encountered a limit problem. The problem is the following: $\lim\limits_{x\to0}x^2\ln (x)$ I am not allowed to use L'Hospital. Please help me, I am stuck ...
2
votes
4answers
113 views

Evaluating $\int \frac{\mathrm dz}{z^3 \sqrt{z^2 - 4}}$

$$\int \frac{dz}{z^3 \sqrt{z^2 - 4}}$$ $z = 4\sec\theta$ $dz = 4\sec\theta \tan d\theta$ $$\int \frac{\sec\theta \tan\theta}{4^3 \sec^3 \theta \tan \theta}$$ $$ \frac{1}{4^3} \int \frac{d ...
2
votes
2answers
50 views

The difference with sup and without sup

The difference with sup and without sup,how to judge and choose the use For example, here is rudin's "Root" Test: Given $\sum a_n$, put $\color{Green}{\{\alpha =\lim_{n\to \infty}\sup ...
2
votes
1answer
519 views

Finding the interval for increase of the function $y =x^2e^{-x}$

Problem : Find the interval in which the function $y =x^2e^{-x}$ is increasing . My approach : We can take first derivative to the find the increase or decrease of function ie. ...
2
votes
4answers
128 views

Evaluation of $\int \frac{dx}{\sqrt{x^2 - 9}}$

$$\int \frac{dx}{\sqrt{x^2 - 9}}$$ $x = 3\sec\theta$ $dx = 3\tan\theta \sec\theta\,d\theta$ $$\frac{1}{3} \int \frac{3\tan\theta\sec\theta}{\sqrt{\sec^2 + 1}} d\theta$$ $$ \int \sec\theta ...
2
votes
0answers
68 views

Try the exercise using green

Let $\phi$ and $\psi$ be functions $\left [a,b \right ] \to \mathbb{R}$ of class $C^1$, such that $\phi(a) = \psi(a)$, $\phi(b)= \psi(b)$, $\phi(x) \leq \psi(x)$, $\phi$ is convex, and $\psi$ is ...
3
votes
2answers
50 views

Differential equation two solutions, how so?

I tried to solve $7x^3y'=4*\sqrt{y}$ with $y(1)=1$ now I thought that Picard Lindelöf would tell me that there is a (at least in a local area for x=1) unique solution unfortunately I found two: ...
1
vote
0answers
110 views

Isolated points

Every point $x \in S \subset\Bbb R$ is isolated. 1) $S$ is closed? 2) $S$ doesn't have any limit point? My attempt: by definition any isolated point is boundary point, and cannot be limit point. ...
0
votes
1answer
26 views

proving convergence of a power series with the use of another power series

Let <$a_n$> be a sequence of nonnegative numbers such that $\sum_{n\ge1} a_{{_2}{^n}} $ converges.Prove that $\sum_{n\ge1} a_n $ converges. I tried but couldn't find a way to prove this.Any help ...
1
vote
2answers
56 views

Inequality involving Maclaurin series of $\sin x$

Question: If $T_n$ is $\sin x$'s $n$th Maclaurin polynomial. Prove that $\forall 0<x<2, \forall m \in \Bbb N,~ T_{4m+3}(x)<\sin(x)<T_{4m+1}(x)$ Thoughts I think I managed to prove the ...
3
votes
2answers
168 views

non empty set with empty interior is countable at most

A is non empty set of R and set of interior points of A is empty. Then A is countable at most. How to (dis)prove it? Empty interior for non-empty set implies that A consist of isolated points. I ...
4
votes
1answer
87 views

What's $\sum_{n \ge 0} q^{n^2}$?

Is there a relatively simple way of calculating the sum of the series $$\sum_{n=0}^\infty q^{n^2}, \quad |q|<1 ?$$