Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
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How to Solve $\int \frac{x + 1}{(x^2 + 6x + 14)^3} dx$?
I am trying to solve the following integral and would appreciate some guidance:
$$\int \frac{x + 1}{(x^2 + 6x + 14)^3} \, dx$$
I have attempted various methods, including substitution and integration ...
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$\lim_{n\to\infty} \{a_n\}=1$ , $\lim_{n\to\infty} \{\frac{a_n}{d_n}\}=1$, $\lim_{n\to\infty} \{\frac{a_n}{d_n^2}\}=\frac{3}{5}$
Let $\{x\}$ denote the fractional part of $x$ and $d_n=\text{lcm}(1,2,...,n)$. Does there exist a sequence with all positive terms $(a_n)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{a_n\}=1\ \text{,}\ \...
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Handling of algebra in differential calculus
3Blue1Brown"Essence of calculus" series called "Derivative formulas through geometry"- 3rd episode of chapter 3
I have considered the area gained to be $dx*(\frac{1}{x}-d(\frac{1}{...
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Arc length derivation [closed]
enter image description here
Why does the hypothetical X value Xi that would equal the avg slope of a section get replaced by X? I don't remember using the mean value theorem in the definition of the ...
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Where does $x\mapsto e^{-x}-(1-\frac xn)^n$ reach its maximum?
For all integer $n\ge2$, the map $f_n:[0,n]\to\mathbb{R},x\mapsto e^{-x}-(1-\frac xn)^n$ reaches its maximum at some $x_n$. I could prove that $1<x_n<2$.
I would like to show that the sequence $(...
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A problem about function properties.
Given $0 < p_1 \le p_2 \le p_3 < 1$ and $p_1 + p_2 + p_3 = 1$. Define $f : \mathbb{R} \to \mathbb{R}$ as $ f(x) := \ln(p_1^x + p_2^x + p_3^x). $
Prove that $f$ is an infinitely differentiable, a ...
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Does $\int_a^b g^2=0 $ imply $\int_a^b fg=0$?
I am trying to prove the Cauchy Schwarz inequality for integrable functions, which is as follows:
(Cauchy-Schwarz): If $f$ and $g$ are integrable on $[a,b]$, then $$\left(\int_{a}^{b}f^{2}\right)\...
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Limit of $\sqrt[n]{2-x_n}$
Let $x_n$ be the real positive root of equation: $$x^n=x^{n-1}+x^{n-2}+\ldots+x+1$$
Find $\lim{(2-x_n)^{\frac{1}{n}}}$
Here is what I tried:
Initial:
$$x^n > 1 \Rightarrow x > 1$$
It is easy to ...
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Example of a series where the ratio test $\limsup |a_{n+1}/a_n|$ can be applied, but $\lim |a_{n+1}/a_n|$ cannot
The ratio test asserts the absolute convergence of $\sum_{n\geq 1}a_n$ if $$\limsup \bigg |\frac{a_{n+1}}{a_n}\bigg |<1$$
In calculus, we learn the seemingly weaker form
$$\lim \bigg |\frac{a_{n+1}}...
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Fourier series problem... :( [closed]
How do you find the Fourier series of $f(x)=|\cos(x)|$ in $0<x<2\pi$?
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Proof that a function which is continuous on $[a,b]$ is uniformly continuous (Spivak)
I'm following Spivak's Calculus and in an Appendix, the following theorem is being proven:
If $f$ is continuous on $[a,b]$, it is uniformly continuous on $[a,b]$.
However, I don't understand a few ...
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A limit involving the $n$th derivative of the reciprocal of the Riemann Zeta function: $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ [closed]
Does the limit $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ exist? If so, how can we determine its precise value, or is there any way to approximate it with high precision? ($(1/\zeta)^{(n)}(2)$ is ...
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Computations with $\textsf{CD}$-densities
Given a continuous non-negative function $h:[a,b]\rightarrow(0,+\infty)$, define the convolution
$$h^\varepsilon:=\left(h^\frac{1}{N+1}*\rho_\varepsilon\right)^{N+1},$$
where $\rho_\varepsilon$ is a ...
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Calculate the surface integral of $F = \langle x,y,z\rangle$ over the surface given by $3x-4y+z=1$ [closed]
Calculate the surface integral of $F = \langle x,y,z\rangle$ over the surface given by $3x-4y+z=1$ for $0 \leq x \leq 1$ and $0 \leq y \leq1$, with an upward-pointing normal.
I'm not sure about how to ...
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Solution of equation with unknown under the integral
I have a problem which I have reduced to solving the following equation for the unknown $r_0$:
$$
1/2 = \int_0^D f(r)p(r,r_0)dr
$$
where $D \in \mathbb{R}$, and $f$ is continuous density function.
$p(...
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