Questions tagged [calculus]

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

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76 views

If $f(x)$ is integrable then $f(e^{x})$ is also integrable on $\Bbb R^{+}$ [on hold]

If $f(x)$ is integrable then is it true that $f(e^{x})$ is also integrable on $\Bbb R^{+}$?
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0answers
64 views

Proof that $\lim_{s\to 1}\ln \zeta (s)$ diverges

I'm trying to prove that $$\displaystyle\lim_{s\to 1}\ln\zeta (s) \,\text{diverges}. \tag*{(*)}$$ Initially, I thought of $$\displaystyle\lim_{s\to 1}\ln\zeta (s)=\ln\displaystyle\lim_{s\to 1}\zeta (s)...
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0answers
31 views

A derivation operation which produces a definite integral

We are all familiar with the derivative, and the anti-derivative: $$ f(x)=\frac{d}{dx}\int f(x) dx $$ I am interested in a similar relationship but for definite integrals (with all the limitations ...
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1answer
67 views

Fake proof that $\zeta (1)=\gamma$

The following proof is fake, but I have no idea why it's not right (I know it's paradoxical, since $\zeta (1)$ diverges): $$\begin{align*}\zeta (1)&=\dfrac{2\zeta (1)}{2}\\&=\dfrac{\...
3
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2answers
47 views

Find maxima and minima of $f(x) = x \cot x$

So I got a function $$f(x)= x \cot x $$ I would like to find values of $x$ where $f'(x) = 0$ Applying product rule, we get: $$f'(x) = \cot x - x \cdot csc^2 x $$ Setting equation to zero $$\cot ...
2
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2answers
59 views

Spivak's proof that $\pi$ is irrational

I'm reading chapter 16 of Spivak's Calculus, 4th edition, specifically proof that $\pi$ is irrational. Last part is unclear to me. He states that because \begin{equation} 0 < \pi a^n f_n(x) \sin \...
0
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1answer
19 views

Complex Critical points of a Real Valued function

Assume that the function is $f:\mathbb{R}²\to\mathbb{R}³$ Exercise: Find and classify the crititcal points of the function \begin{equation*} f(x,y)=\frac{xy}{2+x^4+y^4} \end{equation*} Attempt Find $...
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4answers
49 views

Irrational inequality. Symbolab and Wolfram Alpha have different answers

Question : Find the solution for x! $\dfrac{\sqrt{x^2-4}}{x-2}\ge 0$ For my own work, i found : $x<-2 \quad \text{or}\quad x>2$ On Wolfram Alpha app, i type this command (sqrt(x^2 - 4))/...
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3answers
91 views

Local maxima and minima of $f(x,y,z)=(y+z)^2+(x+z)^2+xyz$

I saw this exam problem but I'm having trouble how to determine the local maxima and the minima of this function. This is what I did. I found $f_x=2x+2z+yz \\ f_y=2y+2z+xz \\ f_z=4z+2y+2x+xy$ One ...
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1answer
25 views

L'hopitals rule applied on probability density functions

I'm going through a set of solutions for the following question Let $Z_1,Z_2,\ldots$ be IID random variables with density $f$. Suppose that $P(Z_i>0)=1$ and that $\lambda = \mathrm{lim}_{x \...
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3answers
70 views

If $f$ is differentiable at $a$, then $\lim_{x\rightarrow a}f(x)$ exists. [duplicate]

I am looking at the usual proof that differentiability implies continuity (see e.g. here), which uses only the Difference and Product Rules for Limits (and in particular, doesn't use any results or ...
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0answers
35 views

Verify the Solution of an Integration

I have following interrogation to be solved: $$I=\int_{0}^{\infty}\frac{\Gamma \left(a,b \sqrt{x}\right)}{x+1}\,dx; a,b>0$$ Since this is a complicated one I first try in Mathematica which gives ...
4
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0answers
68 views

Evaluate $\lim_{n\to \infty}\frac{n+n^2+n^3+…+n^n}{1^n+2^n+3^n+…+n^n}$ [duplicate]

Evaluate $$\lim_{n\to \infty}\frac{n+n^2+n^3+....+n^n}{1^n+2^n+3^n+....+n^n}$$ My Attempt Was able to write as $$\lim_{n\to \infty}\frac{\frac{n^n-1}{n-1}}{n^n(\frac{1}{n})\sum_{r=1}^n\left(\frac{r}{...
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0answers
16 views

How to take integral of $\frac{pdf}{cdf}$ of two normal distributions with different means?

Is it possible at all to find the closed-form solution? If $x=0$ it's obvious. But what to do for $x\neq 0$? $\int\limits_y^{\infty } \frac{e^{-\frac{(\xi -x)^2}{2 \sigma ^2}}}{\sigma \left(1+ \text{...
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5answers
85 views

Is $f(a)-f(b) \approx f^{\prime}(b)(a-b)$ for small differences and if so, why?

As in the question, I asked myself whether $f(a)-f(b) \approx f^{\prime}(b)(a-b)$ holds true for small differences between a and b and if so, why that is the case. Thanks for every hint.
3
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2answers
49 views

Interval of θ for a Diagonal Line

I have a problem that I've been trying to solve for a couple of hours , but I'm just not understanding it. The problem is asking to take the polar equation $$r=\frac{4}{\cosθ + \sinθ}$$ and give ...
2
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0answers
23 views

Inequality of the Jacobian implies inequality of norm

Let $F: \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $J_{F}$ denoted the Jacobian of $F$. Similarly, let $G: \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $J_{G}$ be the Jacobian of $G$. If $h^{T}(J_{F}(x))^{T}(...
1
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1answer
43 views

Derivative of trace of a product containing an inverse matrix

What's the derivative of $$f(X)=\text{Tr}(YX^{-1})$$ with respect to $X$, where $X$ and $Y$ are square matrices of the same dimension? My first attempt is to apply the chain rule as: Let $h(X)=X^{-1}$...
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1answer
54 views

How to evaluate the following derivative of integral function [on hold]

$$\frac{d}{dt}\left(\int_{0}^{t}\frac{1 - e^{a(t-x)}\operatorname{erfc}\left(\sqrt{a(t-x)}\right)}{\sqrt{x}(x+b)}\,dx\right)$$
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1answer
60 views

How are derivatives and integrals exempt from the chain rule? [on hold]

A question I never considered when studying calculus is why the chain rule seems to fail when applied to derivatives and indefinite integrals. For example, according to the chain rule, the derivative ...
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0answers
38 views

Why does integration have to be the reverse operation of differentiation? [duplicate]

I am in High school and when I was learning calculus, we were taught that integration is nothing but the reverse of differentiation. But I really don't get it why does that needs to be the case. ...
2
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3answers
92 views

Explicit expression for $1/e$ as a limit

I am following an elementary math book: What is Mathematics and currently referring to infinite series representation of the exponent. In deriving the explicit formula for $e$ and $1/e$, the author ...
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0answers
6 views

Argmin using differentiation with respect to inverse matrix

I'm trying to understand the following step in a calculation: My problems: (1) If we want the argmin with respect to $R$, why are we not differentiating with respect to $R$? I assume differentiating ...
3
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1answer
45 views

Product rule: why not $δ_{xy}=xδ_y+yδ_x+δ_xδ_y$?

I'm reading V.I. Arnold's Huygens and Barrow, Newton and Hooke and he gives this diagram as part of his discussion of the derivation of the product rule, and I can't see what's wrong with it, except ...
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1answer
20 views

Expansion of $\sum_{n=0}^{\infty} (-1)^n \left(\frac{x^{n+\frac{1}{2}} - x^{-n-\frac{1}{2}}}{x^{\frac{1}{2}} - x^{-\frac{1}{2}}} \right) $

For $x\neq 0$, and probably with $|x|<1$, I want to show the following equation $$ \sum_{n=0}^{\infty} (-1)^n \left(\frac{x^{n+\frac{1}{2}} - x^{-n-\frac{1}{2}}}{x^{\frac{1}{2}} - x^{-\frac{1}{2}}...
6
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1answer
103 views

Evaluate $\int_{0}^{\frac{\pi}{4}}\tan xdx $ using idea of Riemann Sum

Evaluate $$\int_{0}^{\frac{\pi}{4}}\tan x\,dx$$ using Riemann Sum. My Attempt: $$\int_{0}^{\frac{\pi}{4}}\tan x\, dx=\frac{\pi}{4}\int_{0}^1\tan\left(\frac{\pi}{4}x\right)dx=\frac{\pi}{4}\lim_{n\to\...
3
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5answers
75 views

Solving $\int \frac{e^{2x}}{1+e^x} \, dx $

Problem: $\int \frac{e^{2x}}{1+e^x} \, dx $ My book says to divide in order to solve by getting $\int e^x-\frac{e^{x}}{1+e^x} \, dx $ but how am I supposed to divide? I tried long division but ...
2
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1answer
57 views

Generalizing a function that operates on functions

I've not studied much beyond Calculus so my notation and terminology may be rough/incorrect. I've got a 'function' $\mathfrak F$ that uses other functions and goes like this: $$ \mathfrak F(f,n) = ...
3
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1answer
46 views

Why is a Maclaurin series the function itself for a polynomial $f(x)$?

It's been about two years since I last took Calc 2, and I'm trying to brush up on some of the old material using Khanacademy. I watched a couple intro videos on the subject and understand Taylor ...
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1answer
34 views

what's the difference between the limit as x approaches zero and an infinitesimally small number [on hold]

When x approaches zero doesn't it become infinitesimally small or in other words become an infinitesimal? I was told that they are distinct concepts,for example wikipedia states that there two types ...
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0answers
72 views

How to solve this improper integral $ \int_1^{\infty}\frac{\ln x}{x^x}$? [on hold]

How to solve the following integral? $$ \int_1^{\infty}\frac{\ln x}{x^x}$$ I have tried the following: --> I have tried to simplify this integral by setting the derivative of function beside and ...
-3
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1answer
63 views

Can an equation be reverse engineered? [on hold]

Can an equation be reverse engineered. Like as the parametric equation of a line can be drawn through general equation of the line. can we reverse engineer the parametric equation to get the general ...
7
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3answers
82 views

Finding maxima and minima of $f(x,y)=x^4+y^4-2x^2$

Finding maxima and minima of $f(x,y)=x^4+y^4-2x^2$ I tried studying this exam problem but I need help in understanding it. I found $f_{x}=4x^3-4x \\ f_y=4y^3$ From this, I get the stationary points ...
1
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1answer
33 views

Is this Change of Variable Theorem fine as it is, or should we specify the domain that $f$ is continuous on?

One of my instructors provided me with the following Change of Variable Theorem: Let $a < b$. Let $f$ be a continuous function. Let $g$ be a function with a continuous derivative on $...
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0answers
23 views

Separation of variables for linear PDEs: determining the separation constant

Take as a simple example: $$u_{xx}+u_{yy}=0$$ With boundary conditions: $u(x,0)=0, u(H,0)=u_m$ $u(0,y)=0, u(0,W)=u_m$ With Ansatz $u(x,y)=X(x)Y(y)$ we get: $$\frac{X''}{X}=-\frac{Y''}{Y}=\lambda$...
2
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4answers
68 views

How to solve two inequalities that which simultaneous answers

I have two inequalities here, and I must find the answer for both of them simultaneously (joint answer): $\left\{ \begin{aligned} \dfrac{2}{x-3} \gt \dfrac{5}{x+6} \\ \dfrac{1}{3} \lt \dfrac{1}{x-2}...
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0answers
25 views

Where to start with an intergral involving two Heaviside step functions?

This question is a follow-up to a previous question. I am dealing with the following type of integral: $$ f(\vec x) = \int_{\mathbb R} \text{d}^3y\; (\vec x\cdot \vec y)^n H(A-|\vec y-\tfrac{1}{2}\...
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0answers
44 views

Ordinary differential equation solution ideas. [on hold]

Let $f$ be a differentiable function and $\mu>0$. Is there an explicit solution $y$ of the following first order ODE? $$ |y'(x)| = \exp\left(-\mu\sqrt{f^2(x) + (y(x)-x)^2}\right), \ \ x\in \ ]-\...
0
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2answers
21 views

Example of a function that is non-differentiable on an open interval

I am studying a course on single variable calculus. During a lecture, the professor mentioned in passing that there can be functions that are non-differentiable on an entire open interval. Can anyone ...
0
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1answer
34 views

Integrals Definite substitution from this? [on hold]

Integrals Definite substitution from $$\int^{\pi/4}_{0}\frac{8 \cos(2t)}{\sqrt {9-5 \sin t (2t)}}~dt ~?$$? u = 9 - 5 sin t(2t) du = -10(t cos (t) + sin (t)) , this correct for du = -10(t cos (t) ...
1
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1answer
55 views

Expressing $f(f(x))$ when $f(x)$ is a piecewise function

This is a question from MIT OpenCourseWare about calculus. I cannot find any explanation online since I have no idea what the keywords are. The question states that: $f(t)=\begin{cases} 2t & \...
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1answer
35 views

The proof for the Lebesgue differentiation theorem

Im looking at the proof of the Lebesgue differentiation theorem on wikipedia: https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem#Proof I don't see why this line is true. This looks like ...
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0answers
48 views

simplification for an challenging expression [on hold]

Can someone please help me how to simplify this $$ \left(1 - \frac{1}{\cos{\theta}}\right) a + \tan{\theta} \exp{(-i\phi)}b = 0 $$ expression to reach to the following? $$ -(\sin{\frac{\theta}{2}}\...
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0answers
25 views

Define complex antiderivative up to constant in terms of real definite integral

I know there's a complex analogue of the real antiderivative that is allowed to produce multivalued functions. For instance, the complex logarithm has some relationship to $z \mapsto \frac{1}{z}$ . ...
1
vote
1answer
21 views

How to apply steady state solution into this question

This is what my study guide defined as the steady state solution If $h(a)=0$ for some constant $a$, then the constant function $y=a$ is a solution of the DE. We sometimes called this a steady state ...
1
vote
3answers
64 views

$n$th derivative of $e^{ax}\sin(bx+c)$

How can we substitute $r \cos \alpha$ and $r \sin \alpha$ for $a$ and $b$? How, on successive differentiation, is there another $r$ multiplied?
1
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1answer
31 views

How to derive volume element for spherical coordinate fast (maybe informally)?

Is there a handy informal argument to derive change of variable volume element for spherical coordinate $r^2 \sin\phi$. Possibly some sort of geometric interpretation? The purpose is to not memorize ...
0
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2answers
37 views

the limit $\lim_{x \to 0^+} \frac{\sin(\arctan(\sin x))}{\sqrt{x} \sin 3x +x^2+ \arctan 5x}$

a hint to start to find the limit of this $\lim_{x \to 0^+} \frac{\sin(\arctan(\sin x))}{\sqrt{x} \sin 3x +x^2+ \arctan 5x}$ without using L'Hospital rule
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1answer
14 views

reparametrization function

I am trying to understand some computation, but it seems something is not going well. You will find this slide in the following link: https://www.asc.ohio-state.edu/kurtek.1/Lecture3_Srivastava.pdf ...
2
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1answer
32 views

From binomial theorem to differential calculus

I first posted this over at HSM, without much uptake. I'm trying to understand the development of the calculus. Does this sound plausible as one of the stages? Newton knows the binomial theorem, ...