Questions tagged [calculus]

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

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8
votes
6answers
5k views

How to evaluate $\lim\limits_{h \to 0} \frac {3^h-1} {h}=\ln3$?

How is $$\lim_{h \to 0} \frac {3^h-1} {h}=\ln3$$ evaluated?
4
votes
1answer
1k views

Trig substitution for a triple integral

This problem involves calculating the triple integral of the following fraction, first with respect to $p$: $$ \int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^{2} \frac{p^2\sin(\phi)}{\sqrt{p^...
14
votes
2answers
765 views

Integrating $\frac{x^k }{1+\cosh(x)}$

In the course of solving a certain problem, I've had to evaluate integrals of the form: $$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$ for several values of k. I've noticed that that, for k a ...
20
votes
3answers
63k views

How to calculate the derivative of this integral?

Here it is : $$ \frac{\mathrm d}{\mathrm dx}\left( \int_{\cos x}^{\sin x}{\sin \left( t^3 \right)\mathrm dt} \right) $$ I've got the answer but I don't know how to start , what to do ? Here is ...
4
votes
1answer
90 views

Is $\int_0^t \left[f(x)-g(x) + l\right]\mathrm dx > \int_t^1 \left[f(x-t)-f(x)\right] \mathrm dx$?

Consider $f,g$ where $f(x) > g(x) \ge 0 \ \forall x \in (0,1)$ and $f(0) = g(0)$, $f(1) = g(1)$ . Is the following inequality true? $$\int_0^t \left[f(x)-g(x) + l\right]\mathrm dx > \int_t^1 \...
1
vote
1answer
5k views

How do I find the minimum distance from a point to a graph?

Assume that I have a graph $$ (C): y=f(x)$$ and a point $$ A(x_0, y_0)$$ How do I find the minimum distance from point A to graph (C)?
6
votes
1answer
549 views

Coordinate-free differentiation techniques in Riemannian geometry

I encountered the following identities while reading this article on global calculus (p. 10): $$ d(\|df\|^2)=2\mathop{\iota_{\mathop{\mathrm{grad}} f}} \mathop{\mathrm{Hess}} f, $$ $$ \mathop{\...
8
votes
1answer
5k views

What is the formal definition of a one sided limit?

I'm looking for the formal definition of $\displaystyle \lim_{x \to a^+}f(x) = L$ and $\displaystyle\lim_{x \to a^-}g(x) = M$ I took a guess at it intuitively, but I need to make sure this is correct:...
2
votes
1answer
529 views

Bessel integral solution or simplification

I am trying to verify the following formula involving Bessel functions of the first kind and am having no luck. The formula is $$ \int{\omega} J_n(\rho \omega)\mathrm d\omega = \frac {1} {\rho} \left\...
6
votes
3answers
331 views

How to calculate $\lim_{x \to 0}\left(\frac1{x} + \frac{\ln(1-x)}{x^2}\right)$?

How to calculate the following limit? $$\lim_{x \to 0}\left(\frac1{x} + \frac{\ln(1-x)}{x^2}\right)$$
11
votes
2answers
2k views

When a change of variable results in equal limits of integration

Here is something that's been bugging me for a while. Say we want to find $$I = \displaystyle\int\nolimits_{a}^{b}f(g(t))g'(t)\,\mathrm dt.$$ If we substitute $x = g(t)$, then $$I = \displaystyle\...
1
vote
2answers
226 views

What is $\lim\limits_{x \to 0^{-}}x^{\frac{1}{2x}}$?

What is $\lim\limits_{x \to 0^{-}}x^{\frac{1}{2x}}$?
2
votes
2answers
411 views

For what real values of $k$ does the series $\sum\frac{k^n}{\sqrt{n}}$ converge?

I think it's $|k| < 1$, but I don't know how to prove it. It's either that or it never converges. $\sum\frac{1}{\sqrt{n}}$ obviously diverges, but can't an exponential beat it and make the sum ...
1
vote
2answers
297 views

integration using partial fraction

can any body please tell me how to integrate the following expression: $$\int\frac{x}{1+x^4}\,\mathrm {d}x$$ please help...
10
votes
0answers
776 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f{} = 2\pi+2\tan^{-1}(y,x)$ $y = -...
5
votes
3answers
4k views

Proving a function is infinitely differentiable

I haven't done too many proofs, but I'd like to attempt more and thought I'd take a shot at this one, which is from Penrose's book "The Road To Reality." The problem is to prove that the function ...
7
votes
3answers
2k views

Proving the odd Bernoulli numbers are zero

suppose we define the Bernoulli numbers $b_n, n = 1, 2, 3, \ldots$ by the Faulhaber's fomula $$\begin{eqnarray*}1 + 2^k + 3^k + \ldots n^k &=& \frac{1}{k+1}[n^{k+1} + b_1 c(k+1,2) n^k + b_2 ...
2
votes
1answer
118 views

Is there a name for this function?

this should be simple A polynomial could be defined as \begin{equation} P_n (x) = \sum_{i=1}^{n} a_i x^{i-1} \end{equation} Would the infinite-dimensional version of that \begin{equation} F_l (x) = ...
9
votes
4answers
591 views

How to find $\int\frac{\sin x}{x}dx$

How do I integrate $$\int\frac{\sin(x)}xdx$$? I tried using integration by parts, but it led me to nowhere. Please help.
2
votes
2answers
148 views

convex inequality

why does it follow from convexity that $\sum^n_{i=1} 2^{k_i} \geq n2^{k/n}$, where $k_1 +...+k_n = k$ holds?
45
votes
4answers
7k views

Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$

It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why? I got: $2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\...
6
votes
2answers
3k views

Integration of $\cos(2x)\cos(nx)$

I'm struggling to integrate $\int \cos(2x)\cos(nx)\,\mathrm{ d}x$ I seem to be going round in circles and would be grateful if someone could help? I think I need to use a trig expansion or identity ...
8
votes
6answers
21k views

Don't understand why this binomial expansion is not valid for x > 1

today I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...
0
votes
1answer
986 views

Verify Stokes's Theorem for the given surface and vector field

$S$ is parametrized by $X(s,t) = (s\cos(t), s\sin(t), t)$, $0 \leq s \leq 1$ and $0 \leq t \leq \frac{\pi}{2}$ $$\mathbf{F} = z \mathbf{i} + x \mathbf{j} + z \mathbf{k}$$ I have two things ...
1
vote
1answer
285 views

Where is my (algebra) mistake? Converting parametric to Cartesian equation

I'm having a problem with my solution to a textbook exercise: Find the Cartesian equation of the curve given by this parametric equation: $$x = \frac{t}{2t-1}, y = \frac{t}{t+1}$$ The textbook's ...
0
votes
1answer
2k views

You can't find an equation for the cross-sectional shape when given only the formula for volume

Why not? I'm trying to design a calculus I problem. I want them to eventually prove that the volume of a cone (oblique or even with irregular base) is $\frac{\pi r^2 h}{3}$, I also want them to know ...
2
votes
2answers
1k views

Calculus: Second Order Derivative with Sin and Cos

$D^2 [\sin(\theta)+\cos^3(\theta)]$ The answer should be $-\sin(\theta)+6\sin^2 (\theta)\cos(\theta)-3\cos^3 (\theta)$ I understand it's applying $D$ twice, but I can't tell which rules to use. ...
5
votes
4answers
2k views

Is the sum of two functions, one of which is non differentiable, necessarily non-differentiable?

Suppose $f(x) = g(x) + h(x)$ where $g(x)$ is known to be non-differentiable. When $h(x) \ne -g(x)$ is some other function (differentiable or not), can $f(x)$ ever be differentiable? We can assume $f, ...
2
votes
3answers
390 views

Evaluating the series $\sum\limits_{n=0}^{\infty}\frac{n}{3^n}$ [duplicate]

Possible Duplicate: How to find the sum of this infinite series Hello all, I have one last major question, where would I get started on the following question: $$\sum_{n=0}^{\infty}\frac{n}{3^n}$...
4
votes
1answer
4k views

Arrow in limit operator

In this Wikipedia article, I see a limit operator such as in: $$\lim_{x \searrow 0} \frac{e^{-1/x}}{x^m}=0\,\,;\,\,\,\, m\in \mathbb{N}$$ I am assuming that the downward pointing arrow indicate the ...
7
votes
3answers
7k views

Finding the limit of $(1-\cos(x))/x$ as $x\to 0$ with squeeze theorem

How do I find: $$ \lim_{x\to0}\frac{1-\cos(x)}{x} $$ Using the squeeze theorem. Particularly, how would I arrive at its bounding functions? If possible, please try not to use derivatives.
3
votes
1answer
170 views

Is there a closed-form expression for $\int (a-b\ln(cx))^{-1} \mathrm{d}x$?

Is there a closed form expression for $$\int\frac{1}{a-b\ln(cx)}\,\mathrm dx\ ?$$ I was wondering how to integrate the above function. I have spent a lot of time on it. First i did an integration by ...
4
votes
1answer
720 views

How we get the area by subtracting two end points of a function in Integration?

For example consider the following integration: f(x) = x^3 [from 1 to 3] $$\int_{1}^{3}x^{3}dx$$ when we subtract: {(3^4)/4}-{(1^4)/4} why we get the result? I meant to say, how we get the area ...
3
votes
1answer
2k views

Algorithm for joining two polygons based on set of 2D points

[Thanks for votes, I fixed the images now] I am working on system, that does some fuzzy logic. I don't have any special types, everything is done with simple math. The last problem I am facing is ...
2
votes
1answer
166 views

Calculus: Limit at infinity

Suppose I have two continuous functions on $\mathbb R$, $f(x)$ and $g(x)$, such that $f(x)\leq g(x)$ for all $x\in \mathbb R$ and $\lim _{x \to\infty }g(x)=0$. Is the following true: $$ \lim_{x \to\...
14
votes
5answers
650 views

prove that $\lim\limits_{x\to 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}$

I'm trying to find a way to prove this: EDIT: without using LHopital theorem. $$\lim_{x\rightarrow 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}.$$ Honestly, I didn't come with any good idea. We know ...
3
votes
3answers
13k views

How to prove a function is positive or negative in $x \in \mathbb{R}$

A homework question: I know the solution but I don't know how to prove that the function is negative XOR positive for $x \in \mathbb{R}$ f is continuous in $\mathbb{R}$. $$\text{ prove that if } |f(...
5
votes
5answers
6k views

Distance traveled by a bouncing ball with exponentially diminishing rebounds

This is kind of an odd question, but can somebody please tell me that I am crazy with the following question, I did the math, and what I am told to prove is simply wrong: Question: Show that a ball ...
28
votes
8answers
44k views

Integral of $\frac{1}{(1+x^2)^2}$

I am in the middle of a problem and having trouble integrating the following integral: $$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$ I tried doing partial fractions and got: $$1=A(1+x^2)+B(1+x^2)$$ I ...
1
vote
2answers
638 views

expectation of incomplete gamma

Is the expectation of the (upper/lower) incomplete gamma function known? $$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx$$
2
votes
1answer
2k views

Finding Constants for a probability density functions given a finite interval

Given the probability density function f(x) with constant K on the interval [a,b], how do you solve the value of K?
5
votes
2answers
21k views

Find the arc length of the cardioid: r = 3-3cos θ

This is what I have so far: Using the formula $\mathrm ds = \sqrt{r^2 + \left(\frac{\mathrm dr}{\mathrm dθ}\right)^2}$ $$\frac{\mathrm dr}{\mathrm d\theta} = 3\sin\;\theta $$ $$r^2 = 9 - 18\cos\;\...
5
votes
3answers
182 views

More Computing Integrals

This particular problem has been giving me trouble, and while the math dept tutors did help a great deal, the resulting answer hasn't been accepted by the online homework submission website. Find the ...
5
votes
4answers
219 views

Calculus one problem about substitution and area

I'm looking for a fun (not too many tedious calculations) calculus one problem that uses the concept that, after subsitution, you have two integrals of diffent functions with different limits, but ...
1
vote
2answers
209 views

Is it allowed and if so, how to differentiate this integral?

I have the following expression (everything is $\in \mathbb R$): $$f(a,b,c)=c\cdot\int_a^b g(t) \cdot h(t,c) \,dt,\quad0\leq a<b$$ I now want to differentiate this function with respect to c: $$\...
8
votes
3answers
736 views

Integrals $ \int_0^1 \log x \mathrm dx $,$\int_2^\infty \frac{\log x}{x} \mathrm dx $,$\int_0^\infty \frac{1}{1+x^2} \mathrm dx$

I don't get how we're supposed to use analysis to calculate things like: a) $$ \int_0^1 \log x \mathrm dx $$ b) $$\int_2^\infty \frac{\log x}{x} \mathrm dx $$ c) $$\int_0^\infty \frac{1}{1+x^2} \...
3
votes
1answer
17k views

Parameterizing the upper hemisphere of a sphere with an upward pointing normal

Can someone explain how to do this? area we're dealing with: $x^2 + y^2 + z^2 = a^2, z \geq 0$ I'm aware that the answer is: $x = a \sin(\phi) \cos(\theta)$ $y = a \sin(\phi) \sin(\theta)$ $z = ...
5
votes
4answers
1k views

Proof that this limit equals $e^a$

Can someone please explain to me why the following identity is true? $$\lim_{x \to \infty}\left(1 + \frac{a}{x} \right)^x = e^a$$ (I'll make a notation $L$ that is equal to the limit above.) One '...
0
votes
2answers
325 views

Is there a difference between these integral notations?

I've come across these two notations for calculating an indefinite integral but I'm not sure whether or not they are equal: $f(x)dx$ $\int f(x)dx$ When calculating the indefinite integral, the first ...
3
votes
1answer
446 views

How to evaluate $\int \frac{\cos(x) - 1}{x^2}\mathrm dx$?

would like a hint with the integral $$\int \frac{\cos(x) - 1}{x^2}\mathrm dx$$Thanks