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

learn more… | top users | synonyms

3
votes
5answers
538 views

Deriving the summation formula for $x^2, x^3,\ldots,x^n$

How is the summation formula's for $x,x^2,x^3,x^4,\ldots$ derived? I know how to do it for $x$ which is $n^2/2 + n/2$ but I am having hard time deriving the summation formula for $x^n$ on my own. I ...
1
vote
0answers
65 views

Question about proof for $f\in C^1\Rightarrow f$ is local Lipschitz continuous

I have found the following proof in this very forum: If $f:\Omega\to{\mathbb R}^m$ is continuously differentiable on the open set $\Omega\subset{\mathbb R}^d$, then for each point $p\in\Omega$ ...
0
votes
3answers
832 views

Calculate double integral of function over triangle

Find the limis $\int\int dy \, dx$ and $\int\int dx\,dy$ and computer the area over the region, based on the function $f(x,y) = 3x^2y$ Region = triangle inside the lines $x=0$, $y=1$, $y=2x$. ...
0
votes
2answers
455 views

Given this piecewise-defined function $f$, prove that $f$ is differentiable at $0$, but $f'$ isn't

Prove $f$ is differentiable at $0$, but $f'$ isn't. $$f(x)=\begin{cases} x^2 & \text{if }x>0,\\ x^3 & \text{if }x\leq 0. \end{cases}$$ The derivative at a point is equal to the limit as it ...
5
votes
3answers
222 views

Integration and area - why is integrating over a single point zero?

This is a naive question, but I'm only starting to learn calculus so please cut me some slack. So we all know the integral from $a$ to $b$ of a function over an interval measures the area under the ...
2
votes
2answers
116 views

Solve system of Charpit equations

I have a partial differential equation $$u_xu_y = xy \mbox{ with } u(0,y)=y+1$$ Calling $u_x = p, u_y = q$ gives the following Charpit equations $$\frac{dx}{dt} = q, \frac{dy}{dt} = p, \frac{dp}{dt} ...
1
vote
0answers
74 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
3
votes
2answers
75 views

Deducing Euler Equation

From Sydsaeter / Hammond (Further Mathematics for Economic Analysis, 2008, 2nd ed., p. 293): $$ \max \int\limits_{0}^T [N(\dot{x}(t)) + \dot{x}(t)f(x(t))] e^{-rt} dt $$ where N and f are $C^1$ ...
8
votes
5answers
319 views

Are derivatives defined at boundaries?

Given a differentiable function $f : [-5,5] \rightarrow \mathbb{R},$ I was under the impression that the derivative $f'$ has domain $(-5,5).$ However, according to Wikipedia ...a differentiable ...
5
votes
1answer
133 views

Calculus on surfaces and chain rule

Define the surface gradient operator on any surface $S$ as $$\nabla_S f = \nabla f - \nabla f \cdot \nu_S \nu_S$$ where $\nu_S$ is the outward unit normal on $S$. Let $T:S_1 \to S_2$ be a $C^2$ ...
0
votes
0answers
78 views

Request for exercises in calculus II

I took this semester calculus II and I want to practice towards the final test some questions. The subject of the course were: function investigation integrals: anti derivatives Definite ...
0
votes
1answer
319 views

Find gradient of this implicit function

How to find a gradient of this implicit function? $$ xz+yz^2-3xy-3=0 $$
1
vote
1answer
119 views

How to find a directional derivative of an implicit function?

it's not my homework, I just want to find out how to find a directional derivative of an implicit function. I know what is a directional derivative and how to find it when I have a function in normal ...
6
votes
4answers
250 views

Integrating $\ln x$ by parts

I am asked to integrate by parts $\int \ln(x) dx$. But I'm at a loss isn't there supposed to be two functions in the integral for you to be able to integrate by parts?
3
votes
1answer
53 views

On convergence of $\prod (1 - \alpha_n)$

Suppose $\{ \alpha_n \}$ is a decreasing sequence of real numbers such that $0 < \alpha_n < 1$ and $\alpha_n$ goes to $0$ as $n$ goes to infinity. I was wondering if there is a known condition ...
3
votes
1answer
71 views

Continuity of the function $f=1/x$

How do I show that the function $f(x)=\frac{1}{x}$ is continuous using the $\epsilon - \delta$ definition? I have been trying for quite a while now without success. My attempts Suppose that $\left ...
10
votes
1answer
217 views

Evaluate $\int\limits_0^1\frac{(1-x)e^x}{x+e^x}\,dx$

I`m trying to evaluate this integral $$\int\limits_0^1\frac{(1-x)e^x}{x+e^x}\,dx.$$ Would you please give me any idea?
0
votes
1answer
75 views

Not able to solve $\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $

If $p=\frac{7}{8}$ then what should be the value of $\displaystyle\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $ when $$g(x) = x \log x \quad \text{or} \quad g(x) = \frac{x}{\log x}? $$ ...
0
votes
1answer
30 views

For what values ​​of $m$ the function $y=x^m\sin(x)$ have horizontal asymptote

I want to figure for what values ​​of $m$ the function have horizontal asymptote.$$y=x^m\sin(x)$$ so what I understand from that is this that the function dont have a vertical one, so I will find ...
2
votes
1answer
81 views

Calculate a $\infty^0$ limit using `de l'hopital` rule.

I have to calculate the following limit: $$\lim_{x\to \infty} 2x^{1/\ln x}$$ So I tried to start: $$\lim_{x\to \infty} 2x^{1/ \ln x} = \infty^0 $$ From here on I noticed that I have to use ...
2
votes
2answers
220 views

Weak / Classical derivative

I know the definitions of both weak and classical derivative. But I am trying to see the classical derivative as a weak derivative. When we have $\int f' \varphi = -\int f\varphi'$ for all $\varphi\in ...
1
vote
3answers
1k views

What is wrong with treating $\dfrac {dy}{dx}$ as a fraction? [duplicate]

If you think about the limit definition of the derivative, $dy$ represents $$\lim_{h\rightarrow 0}\dfrac {f(x+h)-f(x)}{h}$$, and $dx$ represents $$\lim_{h\rightarrow 0}$$ . So you have a ...
0
votes
4answers
270 views

Why is $e^x$ the only nontrivial function with a repeating derivative? [duplicate]

Why is $e^x$ the only nontrivial function with a repeating derivative, i.e. is its own derivative? It says so in the Wikipedia article about $e$. Is there a proof of this that I (a calculus AB ...
2
votes
0answers
120 views

Proving that a set is measurable and has a zero area

I want to prove that each of the following sets is measurable and has zero area. a) A set consisting of a single point. b) A set consisting of a finite number of points in a plane. c) The union of a ...
5
votes
4answers
205 views

Finding another way of doing this integral $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$

Problem : Integrate : $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$ I have the solution : We can substitute $\sqrt{x}= \cos^2t$ and proceeding further, I got the the answer which is ...
0
votes
1answer
64 views

Is there a quick way (a way) to do this integral

$f(x,y) = 2xy-y$ over the square $0-2$ $\int _0 ^2 \int _0 ^2 (\frac{2xy-y}{\sqrt{1+2x^2+2y^2-2x}} )dxdy$ My fallback is to write the square-root as (something to do with x)^2 + (some 'constant' ...
2
votes
1answer
64 views

Is constructing a function that DNE a sufficient counterexample to show the function does not diverge to $\infty$?

Prove or disprove: If $f(x)\to 0$ as $x\to a^+$ and $g(x)\geq 1$ for all $x\in \mathbb{R}$, then $g(x)/f(x)\to\infty$ as $x\to a^+$. Counterexample: Let $f(x)=0$ and $g(x)=1$ for all ...
0
votes
1answer
624 views

Show that the limit does not exist.

Using deltas and epsilons... $f(x)=\sqrt{|x-1|}$. Show that $\lim_{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ does not exist. Well, I tried to get the fraction into x-a form, and I got to ...
1
vote
1answer
72 views

Polynomials $p$ of degree $ \le 2$

Find all polynomials $p$ of degree $ \le 2$ which satisfy the given condition: a) $p(0)=p(1)=p(2)=1$ b) $p(0)=p(1)$ IN each case, find all polynomials $p$ of degree $ \le 2$ which satisfy the given ...
0
votes
2answers
75 views

Definite integral doubt

I'm starting to study Calculus now, I've got the following problem: What's the minimum value of $\displaystyle F(a) = \int_{0}^{2} |x^2 - a^2|\, \mathrm{d}x$ When $0 < a < 2$, I managed to ...
17
votes
6answers
815 views

If $f$ is continuous at $a$, is it continuous in some open interval around $a$?

If $f: \mathbb{R} \to \mathbb{R}$ is continuous at $a$, is it continuous in some open interval around $a$?
3
votes
5answers
100 views

Did I derive this correctly?

I derived this $$(2x+1)^2 \sqrt{4x+1}$$ and got $(8x+4)(\sqrt{4x+1})$+$\frac{2}{\sqrt{4x+1}}(2x+1)^2$ Is this correct? I ask because Wofram Alpha gave me a different answer. Thanks in advance.
5
votes
1answer
649 views

Prove or disprove: if $f$ and $fg$ are continuous then $g$ is continuous.

Prove of provide a counterexample: Suppose that $f$ and $g$ are defined and finite valued on an open interval $I$ which contains $a$, that $f$ is continuous at $a$, and that $f(a)\neq 0$. Then $g$ is ...
1
vote
2answers
175 views

Struggling to interpret exam past paper question, calculus and vectors involved.

I hate to do this but there are no solutions and I am really struggling to interpret what it means. $f:\mathbb{R}^3\to\mathbb{R}$, $f(X)=\|X \|^2$ ($X=$ the vector $(x,y,z)$) and ...
1
vote
4answers
71 views

Regarding u-substitution

1) $\displaystyle \int_{1}^{4} \frac{(\ln x)^3}{2x}dx$ 2) $\displaystyle \int_{}^{} \frac{\ln(\ln x)}{x \ln x}dx$ 3) $\displaystyle \int \frac{e^{\sqrt{r}}}{\sqrt{r}}dr$ 4) $\displaystyle \int ...
3
votes
6answers
2k views

integrating by parts $ \int (x^2+2x)\cos(x)\,dx$

I seems to be stumped in this integral by parts problem. I have $$ \int (x^2+2x) \cos(x)\,dx $$ step 1- pick my $u , dv, du, v$ $$u=x^2+2x$$ $$du=(2x+2) \,dx$$ $$dv = \cos(x)$$ $$v= \sin(x) $$ step ...
1
vote
2answers
56 views

Integration of $\int(2-x/2)^2dx$

Got an exam tomorrow and my head is no longer working. Could someone walk through the integration of this function $$\int\left(2-\frac x2\right)^2dx$$ I understand integration by parts and stuff ...
3
votes
4answers
107 views

Find $ \int {dt\over 2t+1}$

a simple question, but I'm stuck anyway: How to integrate this: $$ \int {dt\over 2t+1} = ? $$ Is it simply: $\ln|2t +1| $ or do I need Chain rule like: $$\ln|2t +1| \cdot \frac{d}{dt}(2t + 1) $$
0
votes
1answer
428 views

Calculus - Indefinite integration Find $\int \sqrt{\cot x} +\sqrt{\tan x}dx$ [duplicate]

Problem : Find $\int \sqrt{\cot x} +\sqrt{ \tan x}dx$ My Working : Let $I_1 = \sqrt{\cot x}dx$ and $I_2 = \sqrt{\tan x}dx$ By using integration by parts: Therefore , $I_1 = \sqrt{\cot x}.\int1 ...
3
votes
2answers
162 views

How to integrate $\iiint\limits_\Omega \frac{1}{(1+z)^2} \, dx \, dy \, dz$

I know how to integrate functions of the form $$ \iiint_\Omega f(r,\theta, \phi) \,dV,$$ where $dV = r^2 \sin \phi \,dr \,d \theta \,d \phi$ and $x=r \sin \phi \cos \theta$, $y=r \sin \phi \sin ...
5
votes
1answer
108 views

How can I integrate $\int\frac{\ln(1+xy)}{1+x^2} dx$?

$$\int\frac{\ln(1+xy)}{1+x^2} dx$$ Give me some idea how to solve to solve this. Is it possible to use some complex analysis to solve this. Or some nice substitution will work?
2
votes
0answers
43 views

Integrating a function of itself?

In physics, the radial velocity of a particle around a black hole is given by this equation: $$ \left(\frac{dr}{d\tau}\right)^2 = \left({E\over m}\right)^2 - \left(1-{2M\over r}\right)\left(1+ ...
0
votes
1answer
140 views

proof of unique solution of particular differential equation

If $k$ is a given non zero constant show that the functions $y = c\operatorname{exp}(kx)$ are the only solutions of the differential equation $\dfrac{dy}{dx} = ky$. Hint: assume that $f(x)$ is a ...
2
votes
4answers
350 views

Prove that $x+e^{2x}=1$ have only one solution

I`m trying to prove that this equation have only one solution. $$x+e^{2x}=1$$ so what I did is to set $\ln$ on this equation and get: $$\ln(x)+2x=0$$ I need some hint how to continue from here. ...
5
votes
2answers
468 views

How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$

For $\alpha \in \mathbb{R}$, define $\displaystyle I(\alpha):=\int_{0}^{2\pi}e^{\alpha \cos \theta}\cos(\alpha \sin \theta) d\theta$. Calculate $I(0)$. Hence evaluate ...
1
vote
1answer
29 views

Find the tangent line of $\frac{x^2}{y+1}+xy^2=4$ at $y=1$ and where $y<x$

I want to find the tangent line for the function $\frac{x^2}{y+1}+xy^2=4$ at the point $y=1$ and where $y<x$. First step: finding the point so I inserted y=1 and get : $$x^2+2x-8 \rightarrow x_1=2 ...
4
votes
3answers
597 views

Closest point in $y = \sqrt{x}$ to the origin is at $x=-1/2$?

When I solve for the point in $y = \sqrt{x}$ closest to the origin using calculus, I get $x = -1/2$. And this is the case for ALL functions $y = \sqrt{x + c}$ using the distance formula $d^2 = x^2 + ...
4
votes
3answers
86 views

Limits and continuous functions

I have always been told that if $f(x)$ is a continuous function at $a$ so that $f(a) = L$, then $\lim_{x\to a}f(x) = L$. Please, could someone explain in detail why this is true?
3
votes
1answer
228 views

integrating logarithm or x raised to a power?

$\int\frac{15}{x}dx$ would be 15$\int\frac{1}{x}dx$ = $15\ln|x|+c$. This seems like a silly question but I'm feeling exceptionally dense today. Why would you apply the logarithm rule, why wouldn't ...
1
vote
1answer
68 views

How find $\int(x^7/8+x^5/4+x^3/2+x)\big((1-x^2/2)^2-x^2\big)^{-\frac{3}{2}}dx$

How can I compute the following integral: $$\int \dfrac{\frac{x^7}{8}+\frac{x^5}{4}+\frac{x^3}{2}+x}{\left(\left(1-\frac{x^2}{2}\right)^2-x^2\right)^{\frac{3}{2}}}dx$$ According to Wolfram Alpha, the ...