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

learn more… | top users | synonyms

0
votes
0answers
6 views

Area between two functions

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 94. Exercise 16. Let $f(x)=x-x^2$, $g(x)=ax$. Determine $a$ so that the region above ...
0
votes
0answers
15 views

Number of zeros of $ f^n $

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $ f^{n} $ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following ...
2
votes
2answers
21 views

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$ Why is it wrong to... $$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$
0
votes
0answers
7 views

Using Intermediate value theorem and Rolle's theorem

Find how many solutions $2\ln x+2x^2+7=0$ has. Define: $f(x)=2\ln x+2x^2+7$, derive it and equate to $0$: $f'(x)=0 \\ 2+4x^2=0$ The discriminant is negative so there are no solutions, so from ...
0
votes
1answer
7 views

Transformation of an equation

How do you get from the left side to the right side in this equation? $$\frac{1+\sqrt{5}}{2} + 1 =\left(\frac{1+\sqrt{5}}{2}\right)^2$$
0
votes
1answer
11 views

Which one of given set is connected…

Which of the following are connected? (Notation: $c(a, r) =\{(x, y) \in\mathbb R^2: (x-a)^2+(y-b)^2=r^2\}$) $c(0,1) \cup c(0,2)$ $c(0,1) \cup c(1,3)$ $c(0,1) \cup c(1,1)$ $c(0,1) \cup c(2,1)$
1
vote
3answers
25 views

Image of (0,1] under continous function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Which of the following sets cannot be image of $(0,1]$ under $f$. {$0$} $(0,1)$ $[0,1)$ $[0,1]$ My initial guess was using intermediate ...
0
votes
1answer
25 views

Find a Taylor series around $x=0$

I don't know how to find the Taylor series around $x=0$ for: $$f(x)=\frac{\tan(2x)-\arctan(4\sinh(x))}{\sin(x^{2})}$$ Thank you in advance.
2
votes
3answers
26 views

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$.

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$. What I have: Since $\{a_n\}\rightarrow \alpha$ we know that ...
0
votes
0answers
32 views

Riemann sum/integral of $\sin(x)$ from $0$ to $A$ [duplicate]

Hello I keep getting stuck on calculating the Riemann sum/integral of $\sin x$ from $0$ to $A$ I know this has been looked at before but I just don't understand it and was hoping someone could ...
0
votes
2answers
59 views

how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$?

would someone give me a hint or a solution ? how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$? Thanks a lot.
0
votes
5answers
61 views

Limit $\lim \limits_{x\to0}{\frac{\ln(x+1)}{2^x-1}}$ wihout LHospital

I want to find this limit but without using L'Hospital, with which is one-liner. $$\lim \limits_{x\to0}{\frac{\ln(x+1)}{2^x-1}}$$
0
votes
2answers
89 views

Find the antiderivative of $(x^2+x+1)^{20}$ [on hold]

How do I find the antiderivative for that? The online calculators say that there's no solution. **NVM, the little scratch on my paper turned out to be a faintly copied handwritten $'2'$, turning it ...
-4
votes
0answers
17 views

question from Radius of curvature [on hold]

prove that for the ellipse x^2/a^2 + y^2/b^2 =1, Radius of curvature =a^2b^2/p^3, p being the perpendicular from the centre upon the tangent at (x,y)
-2
votes
0answers
20 views

Definite integrals and piecewise defined functions [on hold]

Consider the function $G(x) = \int_0^x g(u)\, du$ , where: $ g(u) = \begin{cases} 4 - \frac 43 u, & \text{for $0 \leq u < 6$} \\ u - 10, & \text{for $6 \leq u \leq 12$}. \end{cases} $ i. ...
0
votes
0answers
26 views

Prove that exists $\delta>0$ such that, if $(x,y)\in S$ satisfies $\lVert(x,y) \rVert < \delta$, then $f(x,y) \leq f(0,0)$.

This exercise appeared on my Calculus II exam, and I didn't know even how to start doing it. Any hint is appreciated. Let $\ f, \ g : \mathbb{R^2}\to \mathbb{R}$ two $C^2$functions over the plane. ...
4
votes
2answers
94 views

Who first found the value of $\int_{-\infty}^{+\infty}e^{-x^2}dx$?

A fairly pretty technique of showing that $$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt{\pi}$$ is to square the integral, writing that square as the product of two integrals with integration variables ...
0
votes
1answer
42 views

Prove statement related to dot product

$a, b$ and $x$ are vectors in $\mathbb R^3$ and satisfy $$a\cdot x=b\cdot x$$ Prove $$ a=b $$ By using the definition of dot product, I come up with something like ...
1
vote
2answers
46 views

How do I evaluate $\int \cot^2x$? [duplicate]

I have an integral with $\frac{1}{\tan^2x}$ needed to be evaluated. But instead of searching online for the antiderivative of $\cot^2x$, how would i find it from first principles?
6
votes
3answers
220 views

Trig substitution; why can we ignore the absolute value?

If we have to integrate $$f(x)=\frac{x}{\sqrt{1-x^2}}$$ and we substitute $x=\sin \theta$ then we eventually have to take the square root of $\cos^2x$ which is equal to $|\cos x|$. But in my textbook ...
0
votes
2answers
22 views

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$.

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$. What I have: Assume that $\beta>B$, so $\beta-B>0$. Since $\{b_n\}$ ...
5
votes
1answer
175 views

Why do we put absolute brackets for ln?

When writing out the final answer in $\ln$ form, why is it necessary to put absolute brackets? How does it affect the answer? I have this answer of $-3\ln|\frac{3+\sqrt{9-x^2}}{x}|$, but why does it ...
0
votes
0answers
62 views

Prove the limit as $x$ approaches $0$, $\frac{\sin(x)}{x}$ approaches $1$ using the epsilon delta definition [duplicate]

Prove that $\lim_{x\to0}\frac{\sin(x)}{x} = 1$. So far i have things such as $|\sin(x)|\leq|x|$ for small $x$ and $|\sin(x)|\leq1$ so it is bounded but I'm rather stuck, Also I am not looking for a ...
0
votes
2answers
49 views

How to prove that $f$ is differentiable at every point beside $x=-1$? [duplicate]

Consider $f(x) = |x+1|$. I want to show that for every $x_0\neq-1$ , $\lim_{h \rightarrow 0}\frac{f(x_0+h)-f(x_0)}{h}$ is exist. So far i just wrote the definition, and looked on 2 options:When ...
0
votes
0answers
14 views

Limit of sequence (limit of Bilateral sequence)

I have a question related sequence and limits of sequence. From definition we know that sequence is a function whose domain is natural number.Then we called a sequence (a_n) converges if for every ...
1
vote
1answer
24 views

If a continuous function is nonzero at a point $a$, there is a ball around $a$ in which it has the same sign as $f(a)$

Let $f$ be a scalar field continuous at an interior point a of a set $S\in \mathbb{R}$. If $f(a)\ne 0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The ...
1
vote
1answer
18 views

Trick to finding length of parametric curve

I was giving the parameters of the curve: $x = 2cos(2t)$ $y = 2sin(2t)$ and $z = 1$, where $ 0 \leq t \leq 10 \pi$ This curve describes a cylinder in the $z$ direction, and seems very straight ...
1
vote
1answer
38 views

get a integral from another

if $\int\limits_{0}^{+\infty}x^3e^{-\alpha x^2} dx=\frac{1}{2A}$ then $\int\limits_{0}^{+\infty}x^4e^{-\alpha x^2} dx=$ i tried to use integration by parts $$\begin{align} ...
1
vote
2answers
76 views

How to integrate $((x^2-1)(x+1))^{-2/3}$ using the substitution $u=(x-1)/(x+1)$?

I was asked to find the indefinite integral $$\int \frac{1}{((x^2-1)(x+1))^{2/3}} dx$$ using the substitution of $u=(x-1)/(x+1)$. How do I make this substitution? I attempted to solve this ...
2
votes
2answers
44 views

what is the integral $\int_{-\pi/3}^{\pi/3} \tan (\theta)$?

I tried evaluating the integral $\int_{-\pi/3}^{\pi/3} \tan (\theta)$. I keep getting $\ln(2) - \ln(2) = 0$, but my textbook says its $\ln(4)$. I'm not sure what I am doing wrong because when I ...
2
votes
5answers
100 views

What is $\lim_{n\to\infty}\left(\frac{2}{5}\right)^{1/n}$? [on hold]

What is $$\lim\limits_{n\to\infty}\left(\frac{2}{5n}\right)^{\frac{1}{n}}$$ This is clearly equal to ...
0
votes
1answer
20 views

Linear Transformations of Functions

$\textbf{Problem}$ Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = mx + b$. $\textbf{a.}$ Show that $f$ is a linear transformation when $b = 0$. $\textbf{b.}$ Find a property of linear ...
0
votes
0answers
9 views

Continuous scalar field at an interior point of S and same sign proof.

Let $f$ be a scalar field continuous at an interior point $a$ of a set $S \in R$. If $f(a)$ is not $0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The above ...
1
vote
1answer
26 views

Double integral of $e^{3+y^2}$ over a triangle

Evaluate $\iint_{A}^{} e^{3+y^2}dxdy$ where $A$ is a triangle with vertices $(0,0)$, $(0,-1)$ and $(1,-1)$. I don't know how to bite that. I tried multiplying it by $e^{x^2}$ and then changing the ...
0
votes
2answers
22 views

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$ I've started like this: $X= B - \mathop{\bigcup}_{A\in F} A$ $Y= \mathop{\bigcap}_{A\in ...
0
votes
1answer
12 views

Normal vectors of planes using partial derivatives

I have a question concerning two planes and a given point of intersection in which I am asked to show that they do intersect at that point. The sample method for answering the question goes as ...
0
votes
2answers
25 views

Series with floor function - convergent?

I am trying to figure out where does the following series converge to as $n$ goes to infinity (if it doest at all) $$\frac{1}{n} \sum^{n}_{t=\lfloor \rho n \rfloor +1} \frac{n}{t}$$ where $\rho$ is ...
0
votes
0answers
9 views

Associated Laguerre polynomial

As you know the Rodrigues formula for Associated Laguerre polynomial is $$L_n^{\beta }(x)=\frac{\left(e^x x^{-\beta }\right) \frac{\partial ^n\left(e^{-x} x^{\beta +n}\right)}{\partial x^n}}{n!}$$ i ...
0
votes
1answer
32 views

Linear approximation $y= \ln(1+x)$ for small x

How can I show with linear approximation that $y \approx x$ for small x? I know the rule $$f(x) \approx f(a) + f^{\prime}(a) (x-a),$$ but I don't know how to put it to use in this case.
0
votes
2answers
25 views

Mathematical Puzzle: A Drag Race of Who Wins

I'm having a real difficult time understanding how this problem is solved: "Two drivers, Alison and Kevin, are participating in a drag race. Beginning from a standing start, they each proceed with a ...
0
votes
3answers
30 views

How does integrating over absolute values work with definite integrals?

I have $ \int_0^\pi | \sin(x/2) | \, dx $, and according to Wolfram Alpha, the indefinite integral is: $$ -2\cos(x/2)\operatorname{sgn}(\sin(x/2)) + C $$ but the definite integral above evaluates to ...
0
votes
0answers
18 views

estimates Gaussian moments

Let $X_i \sim N(0,\sigma_i^2)$. Let $k\geq0$ be a fixed integer. I would like to compute $$A:=E[|X_1-X_2|^k|X_2|^k]$$ My idea was \begin{align*} A=&\int_{\mathbb{R}^2}|x_1-x_2|^k |x_2|^k ...
0
votes
1answer
42 views

Calculate this infinite sum [duplicate]

$$s= \sum_{n=1}^\infty \frac{n+3}{(2^n)(n+1)(n+2)}$$ Any method to calculate this type of infinite sums?
0
votes
1answer
34 views

$\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ [on hold]

$f$ an integratable function defined in $[a;b] \rightarrow \mathbb{R}$: prove that exists $c \in [a;b]$ that: $\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ and then give an example that might not ...
0
votes
2answers
55 views

How to evaluate $\lim_{x \to +0} \frac{\sqrt{x}}{x}$

Looking on each function by itself we get $\lim_{x \to 0}\sqrt{x}=0$ and $\lim_{x \to 0}\frac{1}{x}=\infty$ so it is an expression $\frac{0}{\infty}=0$ but when looking at $x=\frac{1}{100}$ we get ...
0
votes
0answers
34 views

$f$ is differentiable twice but not in $0$?

Let $f$ be differentiable twice in any punctured neighborhood of 0. Is $f$ necessarily differentiable at 0? How can I show or prove such a thing? Does it mean that the derivative is subtracted by $x$ ...
2
votes
1answer
48 views

If $f,g$ are unifromly continuous, then $\alpha f+\beta g$ is uniformly continuous?

If $f,g$ are uniformly continuous, then is $\alpha f+\beta g$ uniformly continuous? So far, I looked at here If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous, but I didn't ...
0
votes
2answers
20 views

Hyperbola question

the graph $ y^2=16x $ is a hyperbola; it can be rewritten as $ y= \pm 4\sqrt{x}$ when I draw it down however It is clearly not a function..question is whether it has to be one in order to perform ...
1
vote
1answer
55 views

Calculus of Variations: Understanding functional derivative

I am trying to understand the basics of the Calculus of Variations and the first thing to understand is the functional derivative. I failed to find a good introductory material, so I am trying to make ...
1
vote
1answer
42 views

Given a solution of a differential equation, determine the differential eqution itself

Sorry if my layout is bad, I'm new. So this question was asked a couple of years ago on an exam about differential-equations. Suppose you have a third order differential-equation with the following ...