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Questions tagged [calculus]

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

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

134529 questions
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Limit of a sequence with indeterminate form

Let $\displaystyle u_n =\frac{n}{2}-\sum_{k=1}^n\frac{n^2}{(n+k)^2}$. The question is: Find the limit of the sequence $(u_n)$. The problem is if we write $\displaystyle…
user63181
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Is a function that has a vertical tangent line a function?

At, $a$, the function has a "infinite slope" or vertical tangent line. If the slope of the tangent line is considered to be the instantaneous rate of change, at that point, the function increases "straight up". Since the function increases…
asdf
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If $\int_0^x f^2(t)dt \le f(x)$ for all $x \in [0,1]$, then $\min_{[0,1]} f(x) \le 1$?

Suppose that $f$ is a continuous function on $[0,1]$ and $$\int_0^x [f(t)]^2dt \le f(x) \quad \text{for all} \quad x \in[0,1].$$ Prove or disprove $$\min_{0\le x\le 1} f(x) \le 1.$$ In case the desired inequality does not hold, what is the best…
Nuno
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Express $f_n(x)=\cos{(n\arccos{x})}$ as a polynomial.

I had an interesting problem on an exam a few days ago in elementary calculus. It reads: Show that for $n\geq 2,$ the function $f_n(x)=\cos{(n\arccos{x})}, \ x\in[-1,1]$ is a polynomial of degree $n$ and determine the coefficient for $x^n$. I was…
Parseval
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If the absolute value of an analytic function $f$ is a constant, must $f$ be a constant?

I've been thinking how to prove that an analytic function $f$ is a constant if the absolute value of $f$ is a constant, but I haven't figured it out yet. What I was thinking is to use Cauchy-Riemann equations, but it didn't work well... If this is…
Tengu
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Evaluating $\int_0^1 \sqrt{1 + x ^4 } \, d x $

$$ \int_{0}^{1}\sqrt{\,1 + x^{4}\,}\,\,\mathrm{d}x $$ I used substitution of tanx=z but it was not fruitful. Then i used $ (x-1/x)= z$ and $(x)^2-1/(x)^2=z $ but no helpful expression was derived. I also used property $\int_0^a f(a-x)=\int_0^a f(x)…
Sourav
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If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const

Problem: Given $f:[0,1] \rightarrow \mathbb{R}$ ($f$ continuous ) and $f(x^2) = f(x)$ $\forall x \in [0,1]$. Show that function $f$ is a const.
Ma.H
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True or false: If $f(x)$ is continuous on $[0, 2]$ and $f(0)=f(2)$, then there exists a number $c\in [0, 1]$ such that $f(c) = f(c + 1)$.

I was solving past exams of calculus course and I've encounter with a problem, which I couldn't figure out how to solve.The question is following; Prove that whether the given statement is true or false: Suppose that $f(x)$ is continuous on $[0,…
Our
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Calculus question with circle, and string tracing an area

The figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Find the area of the region, not including the circle itself that is traced out when the string…
eMathHelp
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Evaluation of $\sum^{\infty}_{n=1}\left(\frac{1}{3n+1}-\frac{1}{3n+2}\right)$

Evaluation of $$\sum^{\infty}_{n=1}\left(\frac{1}{3n+1}-\frac{1}{3n+2}\right)$$ $\bf{My\; Try::}$ Here I have solved it using Definite Integration, Like…
juantheron
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convergence of a tower power

Prove that the sequence of general term $(\frac 12)^{(\frac 13)^{(\frac 14)^{...\frac 1n}}}$ is convergent. the three dots are antidiagonal of course :) My try was to compare it with some easier one like the power tower of $\frac 12$. I conjectured…
Samir Karim
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10th derivative of a function

I want to find $f^{(10)}(0)$ where $f(x)=\ln(2+x^2)$. I know that it can be done "by hand", but I believe there is a smarter way. I think I should use Taylor series and the fact that $f^{(n)}(0)=a_n*n!$ , but I'm not sure how.
alex kur
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Examining the convergence of $\int_{0}^{1}\left(\left\lceil \frac{1}{x} \right\rceil-\left\lfloor \frac{1}{x} \right\rfloor\right) \, dx$

Okay so I'm trying to determine whether $\int_{0}^{1}\left(\left\lceil \frac{1}{x} \right\rceil-\left\lfloor \frac{1}{x} \right\rfloor\right) \, dx$ converges and if so, to what value? So the function $\left\lceil \frac{1}{x}…
lamyvista
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Second Mean Value Theorem for Integrals Meaning

The Second Mean Value Theorem for Integrals says that for $f (x)$ and $g(x)$ continuous on $[a, b]$ and $g(x)\ge 0$ $$\int_a^bf(x)g(x)\,dx=f(a)\int_a^cg(x)\,dx+f(b)\int_c^bg(x)\,dx$$ I have a difficult time understanding what this means, as opposed…
meiji163
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Ranges and the Fundamental Theorem of Calculus 1

I'm going over a chapter by chapter review for my calculus final and discovered this problem: $$y=\int_{\sqrt{x}}^{x^3}\sqrt{t}\sin{t}\;\mathrm dt$$ They split it up so that it became: $$-\int_1^{\sqrt{x}}\sqrt{t}\sin{t}\;\mathrm dt +…
ShrimpCrackers
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