Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, integrals, and their applications, mainly of one-variable functions. For questions about convergence of sequences and series, this tag can be use with more specialized tags.

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One definite integral-multiple answers?

So I came across the integral $\displaystyle \int \frac{1}{1+x^2}\, dx$ Now the conventional way is to solve this via trigonometric substitution, and you get the answer as $tan^{-1} x + C$. However, ...
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A question on the "intermediate value property" for disconnected sets!

We know that a continuous function on a connected space satisfies the Intermediate value property. Let $\mathbb{R}^+\times Z_{\alpha}=X_{\alpha}$, where $$Z_{\alpha}=\{\alpha+n: n\in \mathbb{N}\}$$ ...
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Prove that the equation has solutions in two different intervals [-1,1] and [1,2]

$a,b,c \in \mathbb{R}$ and $a,b,c$ positive numbers $\frac{(a+b)\cdot x+a-b}{x^2-1}+\frac{c}{x-2}=1$ Show that the equation has a solution in the interval $[-1,1]$ and a solution in the interval $[1,2]...
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How to express a function into powers of $(x-1)$ and $(y-2)$ using Taylor's formula?

Use Taylor's formula to express the following in powers of $(x-1)$ and $(y-2)$: $f(x,y)=x^3 + y^3 + xy^2$ Solution: $f(1,2)=1 +8 + 4=13$ $f_x (1,2) = 3 + 4=7$ $f_y (1,2) = 12 + 4=16$ $f_{xx} (1,2) = 6$...
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$\int_0^\infty x^{-1/2}e^{-(x-a)^2}dx$

(Edit: Some comments below correspond to $\int_0^\infty x^{-1/2}e^{(x-a)^2}dx$ which indeed blows up) Here is an integral I encountered in the wild $\int_0^\infty x^{-1/2}e^{-(x-a)^2}dx$ If we do $...
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Sum of binomial coefficients on $n$ instead of $k$

There are lots of results on sum of binomial coefficients over $k$. How do we show that $$\sum_{n=k+1}^s \binom{n}{k}=\binom{s}{k+1}$$? Thank you in advance!
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Primal-Dual Pair Feasible solution implies optimal solution

Hello I am trying to understand one solution for the following problem: 17.16 Consider the problem $$ \begin{aligned} \operatorname{minimize} \ & \boldsymbol{c}^{\top} \boldsymbol{x}, \quad \...
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Number of differentiable functions satisfying $y' = 2 \sqrt{y}$

The following question was asked in KVPY $2021$ held on $22$nd May $2022$: The number of differentiable functions $y:(-\infty, +\infty) \to [0, \infty)$ satisfying $y' = 2\sqrt{y}$, $y(0) = 0$ is (A) ...
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What is the minimum value of the function $f(x)= \frac{x^2+3x-6}{x^2+3x+6}$?

I was trying to use the differentiation method to find the minimum value of the person but it did not give any result, I mean when I differentiated this function and equated to zero for finding the ...
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How to proove $\lim_{x \to a}[f(x)g(x)] = 0$ if $\lim_{x \to a}g(x) = 0 $

I am working with "Calculus with Analytic Geometry" from the author Leithold and I came to the execrcise where I have to proove that $\lim_{x \to a}[f(x)g(x)] = 0$ for $\lim_{x \to a}f(x) = ...
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You have seen that $ \int_0^{\infty} f(t) e^ {-t}dt $ = $ \int_0^{\infty} f'(t) e^ {-t}dt $ is sometimes true. Find some non-polynomial examples

You have seen that $$ \int_0^{\infty} f(t) e^ {-t}dt = \int_0^{\infty} f'(t) e^ {-t}dt $$ is true for some functions $f(t)$. Find some non-polynomial examples. Is this problem meant for me to take ...
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Convergence Test Validation

Is the convergence test approach to verify the series's convergence properly used? Please help me validate this answer.
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For $a<x<b$, given definition of convexity as $\frac{f(x)-f(a)}{x-a}<\frac{f(b)-f(a)}{b-a}$, is $\frac{f(b)-f(x)}{b-x}>\frac{f(b)-f(a)}{b-a}$ true?

Given the following definition of a convex function $f$ A function $f$ is convex on an interval if for $a,x$, and $b$ in the interval with $a<x<b$ we have $$\frac{f(x)-f(a)}{x-a}<\frac{f(b)-...
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Flux through surface of revolution

I'm trying to solve the following problem Let $C$ be the curve in the $xy$ plane given in polar coordinates by $r = 2-\sin(\theta),\ 0 \leq \theta \leq \pi$ and let $S$ be the surface given by ...
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Concavity of a Derivative from a Graph

Is it possible to know the concavity of a derivative $f'$ given the graph of $f$? For example, I was given this graph here: How can I know the concavity of the function $f'$ over $(0,1)$ by simply ...
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Calculus: Differentiating Related Functions

problem that uses the y-term with DX/DT and the x-term with DY/DT How do you decide to which variable term (x, y) you're going to derive with DX/DT or [problem that uses the y-term with DX/DT and the ...
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2 votes
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Series Equaling Euler–Mascheroni Constant

Is there a known series equaling the Euler–Mascheroni Constant? And if there is this, wouldn't that imply that the Harmonic series plus this new series equal a series that is exactly $\ln(x)$? I have ...
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1 answer
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Two Power Series Identify on an Open interval not Containing Zero

I searched a lot for a convincing answer for this question but failed to find one (That is formally complete). I wonder if the following claim is true, and if so, for a formal proof. Claim: Let there ...
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Spivak's Calculus, Ch. 11, **69b: $f$ increasing at every $a \in [0,1]$. Prove $f$ increasing on $[0,1]$.

A function $f$ is increasing at $a$ if there is some number $\delta>0$ such that $$f(x)>f(a) \text{ if } a<x<a+\delta$$ and $$f(x)<f(a) \text{ if } a-\delta<x<a$$ (a) Suppose ...
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Uniform convergence of $\sum_{n=1}^{\infty}(\sum_{k=0}^{m}kn^k)x^n$ $m\in \mathbb{N}$ Some constant

How can I prove or disprove uniform convergence $\sum_{n=1}^{\infty}(\sum_{k=0}^{m}kn^k)x^n$ $m\in \mathbb{N}$ Some constant
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wedge volume problem

Find the volume of the wedge cut from the first octant by the cylinder $z = 12 - 3y^2$ and the plane $x+y=2$. What I did- sketched parabola and repeated in all of x axis, drew the plane and found the ...
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1 answer
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Finding a solution of partial derivative of a standard derivative

Considering the formula mentioned below, I arrived to the expansion as stated after performing a partial derivative with respect to the x co-ordinate: $$\nabla \left({\frac{dB}{dt} B}\right) = \frac{...
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Doubt regarding finding the gradient of of a scalar field

I am new to vector calculus. I watched few you tube videos and came to the conclusion that directional derivative is something like slope with direction and its ...
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2 answers
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Calculate the integral by Riemann $ \int _{-1}^{4}( x+1) dx$

Calculate the integral by Riemann $\displaystyle \int _{-1}^{4}( x+1) dx$. We will choose equal parts: Distance will be $ \Delta x=\frac{4-( -1)}{n}=\frac{5}{n},$ with the left point $a_{k} =-1+ \...
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2 answers
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How do we conclude that $\int_{-\infty}^\infty x/(1+x^2)dx$ is not convergent?

In computing the integral $\int_{-\infty}^\infty \frac{x}{1+x^2}dx$ I get the following answers: $0$ due to the symmetry of the integrand. $\left.\ln(\sqrt{1+x^2})\right|_{-\infty}^\infty$ from brute ...
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What is the formula for the orbital velocity of the Earth from x, y, z coordinates of ephemerides at set time intervals?

For example for step size 10080 minutes x y z v 1721057.5 B.C. 0001-Jan-01 -5.83E-01 7.93E-01 3.65E-03 1721064.5 B.C. 0001-Jan-08 -6.78E-01 7.16E-...
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1 answer
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Spivak's Calculus, Ch. 11, **69a: $f$ continuous and increasing at every $a \in [0,1]$. Prove $f$ increasing on $[0,1]$.

**69. A function $f$ is increasing at $a$ if there is some number $\delta>0$ such that $$f(x)>f(a) \text{ if } a<x<a+\delta$$ and $$f(x)<f(a) \text{ if } a-\delta<x<a$$ (a) ...
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Derivation of Area theorem (conformal mapping).

While solving the Area theorem , i'm facing trouble in understanding the equation in these two black boxes, i know how they write $\displaystyle A=\frac{1}{2}\int_{c} R^2 d\phi$, but how they got its ...
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Optimization problem involving simultaneous solution of $k$ partial derivatives

My Question is how to maximize F in the following setup: $$F(\beta) = \prod_{i \in A}p_{i} \prod_{i \in B} 1-p_{i}$$ $$p_{i} = \frac{1}{1+e^-(x_{i}\beta)}$$ Where $A,B \subset N$ and $A \cap B = 0$. ...
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2 answers
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How to solve (not numerically) the inequality $x^a-x^b -c>0$?

Let $x>0$ and $a, b, c\in\mathbb{R}^*_+$ with $b>a$. I am wondering if there is a way (not numerical) to solve the inequality $$x^a-x^b -c>0.$$ Could someone please help or give some hints? ...
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How to prove that there exists $C_1>0$ such that $ \max\{\|f_{n+1}-f_n\|_{\infty}, \|g_{n+1}-g_n\|_{\infty}\}\le 2^nC^nC_1^n\frac{T^n}{n!} $?

I have two iterated update upper bound for two sequences of continuous functions $\{f_n\}$ and $\{g_n\}$, that is there exists $C>0$ such that $$ \|f_{n+1}-f_n\|_{\infty}\le C\int_0^T|f_n(s)-f_{n-1}...
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Understanding numerical integration

I need help, with the fourth line where they introduced $t$. What does a "dummy" variable mean and why are they taking the integral with respect to $y$? Secondly, why have they added the &...
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1 answer
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Rewriting the sum of a geometric series as a function

I have a question about geometric series: The task wants you to find a function $S(x)$ for the sum of a geometric sequence: $$2\sum_{n=1}^\infty (x-1)^{2n-1} $$ My first thought is to use: $$\sum_{n=...
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4 votes
1 answer
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Evaluating $\iiint\frac{xyz\,dx\,dy\,dz}{\sqrt{a^2x^2+b^2y^2+c^2z^2}}$ over $\{(x,\,y,\,z)\in[0,\,\infty)^3\mid x^2+y^2+z^2\le R^2\}$

Evaluate $$\iiint\frac{xyz\,dx\,dy\,dz}{\sqrt{a^2x^2+b^2y^2+c^2z^2}}$$ for $a>b>c>0$ where $\Omega:=\{(x,\,y,\,z)\in[0,\,\infty)^3\mid x^2+y^2+z^2\le R^2\}$ What do we do with the ...
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1 vote
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Proof check: Version of IVT from Spivak's Calculus

Theorem 7-1: If $f$ is continuous on $[a,b]$ and $f(a) < 0 < f(b)$, then there is some number $x \in [a.b]$ such that $f(x) = 0$. I have rewritten Spivak's proof using my own understanding of ...
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2 votes
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Spivak's Calculus, Ch. 11, **68: $f(x)=\alpha x+x^2\sin{1/x}$ for $x \neq 0$, $f(0)=0$. Prove $f$ is not increasing in an interval around $0$.

Two asterisks on a problem in Spivak's Calculus signal a potentially very tricky problem. I solved the following two asterisk problem from chapter 11, "Significance of the Derivative". I am ...
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3 votes
4 answers
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Minimum possible sum of squares of two numbers with sum $k$?

If the sum of two numbers is k. Find the minimum value of the sum of their squares. This is my calculations so far. ...
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3 answers
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Intuition behind Lagrange remainder term in Taylor's Theorem

Let $f:\mathbb R\to\mathbb R$ be an $n+1$-times differentiable function. Taylor's Theorem states that for each $x\in\mathbb R$, $$ f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\dots+\frac{f^{(n)}(0)}{n!}x^n+\...
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The concept of Power series in Strang's Calculus Lectures : how to understand " matching at $x=0 \space f(0), f'(0), f''(0), f'''(0) ...$"?

Source : https://www.youtube.com/watch?v=N4ceWhmXxcs In his Calculus Course , Pr. Strang kindly takes the pain to explain to the layman the concept of Power series. The basic idea, he says, is to ...
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1 vote
1 answer
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Question about calculating flux with different coordinate systems

For a question, I am asked to find the flux of $F=\langle 3x,0,2\rangle$ across the surface of $x^2+y^2+z^2=4, x>0, \ y<0,\ z<0$. I tried solving this with cylindrical and polar coordinate ...
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1 answer
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What is $f'(\frac{1}{f(x)})$ in Leibniz's notation where $y=f(x)$ and $f(x)$ is differentiated with respect to $x$

I'm trying to find differentiable function whose reciprocal equals its inverse $f^{-1}(x)=[f(x)]^{-1}$. I read that there is no such function, but I still wanted to try. If the equality is true, then $...
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Compute $\displaystyle\int_0^{\pi}\dfrac{{\rm d}{\theta}} {\left({\epsilon}\cos\left({\theta}\right)+1\right)^2}$ [closed]

$$\int_0^{\pi}\dfrac{{\rm d}{\theta}} {\left({\epsilon}\cos\left({\theta}\right)+1\right)^2}$$
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-2 votes
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Integral of $\{x\}$ from $e$ to $\pi$ [closed]

The question is: $$\int_e^\pi \{x\}\,dx$$ I have no idea where to get started, so any help is appreciated!
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Are there nonlinear differentiable functions that are positively homogeneous of order 1?

A function $f:\mathbb{R}\mapsto \mathbb{R}$ is positively homogeneous of order 1 if $f(tx) = tf(x) \quad \forall t>0$. For instance, $f_{\alpha}(x) = \alpha x$ is a positively homogeneous funnction ...
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Find volume of solid bounded by $z=x+y,(x^2+y^2)^2=2xy,z=0,(x>0,y>0)$

Moving into polar coordinates find volume of solid bounded by given surfaces. $z=x+y,(x^2+y^2)^2=2xy,z=0,(x>0,y>0)$ Moving into polar coordinates we get. $z=r(cos\phi+sin\phi),r^2=sin(2\phi),z=0$...
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proof that Riemann function is Darboux Integrability in the segment [0,1] [closed]

proof that Riemann function is Darboux Integrability in the segment [0,1] definition of the function: $R(x) = \begin{cases} \frac{1}{q}, & x=\frac{r}{q}\in \mathbb{Q} \\ \ 0, & x\...
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2 votes
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If $P$ is a partition of $[a,b]$, Is there always a choice of points $q$, for $P$ is such that $\sigma(f,P,q) = \underline{s}(f,P)$?

The title says it all. If $P$ is a partition of $[a,b]$, does there always exist a choice of points $q$, for the associate partition $P$ is such that $$ \sigma(f,P,q) = \underline{s}(f,P)\;? $$ Here ...
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Riemann Sum Problem Explanation f(x)=mx on left endpoints using xk

I am learning Riemann when I encountered this question and its solution. Question A curve f(x)=mx in closed interval [a,b] where m>0 and a>=0. Calculate riemann sum of f(x) using xk as left ...
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manifold criterion with implicit function theorem

I'm studying Spivak's Calculus on manifolds, chapter 5.1. I cannot understand the following theorem. Let $A \subset \mathbb{R^n}$ be an open and let $f : A \rightarrow \mathbb{R^p}$ be a ...
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1 vote
2 answers
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How to prove the function $x^3 + 3x^2 + 4x + 1$ is increasing for all values of $x$?

If I have the function $x^3+3x^2+4x+1$, how can I prove it is increasing for all values of $x$? I tried setting it equal to zero after differentiating it $3x^2+6x+4$, but I got an imaginary number. ...
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