Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
138,162 questions
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Will one big ball of ice melt at the same rate as many small balls of ice (assuming they both have the same volume)?
This question is based of a real problem I am facing. Over the winter holidays, I need to travel to a different country. I need to transport medicine, and the medicine has to remain below a certain ...
- 4,193
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clarification about second derivative test on the real functions
I want to help my son with critical points and unconstrained optimization.
From my understanding, for functions of one variable $f:\mathbb{R} \to \mathbb{R}$, we start by locating the
critical points, ...
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Cubic equation for closest point to a function
We have $y = x^2 + 1$ and are trying to find the point closest to $(4, 0)$. For context, this is for a high school calculus class.
Clearly, $l = \sqrt{(x - 4)^2 + ((x^2 + 1) - 0)^2}$, from which we ...
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Computing the Positive Real Zero of the Symbolic Antiderivative of the Riemann-Zeta Function
Taking the indefinite integral of the infinite series definition of the Riemann-Zeta function gives this generalized antiderivative:
$$
\int\sum\limits_{n=1}^\infty \frac{1}{n^x} dx = x - \sum\limits_{...
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Average distance between all the points on 3d surface
I am trying to calculate the average distance a particle passing through a cylinder experiences. There is both a top and a bottom and the dimensions of the cylinder are known. Particles can exit any ...
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1
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How are these integral behaviour approximations different
Q1.
$$\int \frac{3x^{2} + 4x - 1}{(x^{2} + 1)^{2}\sqrt{x+1}}\, dx$$
$\textbf{A. }\frac{\sqrt{x+1}}{x^{2}+1} + C$
$\textbf{B. }-\frac{2\sqrt{x+1}}{x^{2}+1} + C$
$\textbf{C. }-\frac{x}{(x^{2}+1)\sqrt{x+...
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What is $\frac{xf'(x)}{f(x)}$ useful for?
Let $f$ be a differentiable function on some domain. Define the operator $E$ as:
$$E[f](x) = \frac{xf'(x)}{f(x)}, f(x) \ne 0$$
I came up with $E$ as a “What's the degree of this polynomial monomial?” ...
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Continuity and Differentiability of $|x|^x$
I am working on the following problem from the L’Hôpital’s Rule section of Stewart Calculus, and I would appreciate feedback on whether my approach is sound, as well as clarification on a few things.
...
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Is the incremental ratio minus the derivative a continuous function?
Let $f:\mathbb{R}^n\to \mathbb{R}$ be a $C^\infty$ function. If needed you can assume it has compact support.
I would like to prove that the function
$g:\mathbb{R}^{n+1}\to\mathbb{R}$
$$\quad (h,x)\...
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the perimeters of the regular $n$-gons inscribed in and circumscribed about a circle of diameter $1$ and $\pi$. Teiji Takagi' Introduction to Analysis
I am reading Introduction to Analysis (in Japanese) by Teiji Takagi.
Exercise (2) Let $a>0$, $b>0$; define
$$
a_1=\frac12(a+b), \qquad b_1=\sqrt{a_1b},
$$
and in general,
$$
a_n=\frac12(a_{n-1}+...
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On $\int_1^\infty{e}^{-(x+{\small1}/x)}dx=K_{_{1}}(2)+1/(2e^2)$
Show that:
$$ {\int_{1}^{\infty}\frac{dx}{{\large{e}}^{\,x+{\small1}/x}}}={{K_{{_1}}{(2)}}+\frac{1}{2e^2}}\tag{1} $$
$\,K_{\alpha}(x)$: Modified Bessel of the 2nd Kind.
Proposed by a colleague. I was ...
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Help with solving a weird looking maths problem [closed]
I've found this weird looking problem somewhere on the internet:
$\displaystyle 2\cos\left ( \frac{\pi}{2} + \frac{\pi}{6} \right)(1+e^{2i\pi})^2+ \ln \left( \sqrt{\left(\frac{1}{\operatorname{sech}(1)...
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Convexity of the function $ f(x) = (1 - r^{1/x})^{x}$ on $[1,2]$ [closed]
How can I verify that the function
$$
f(x) = \bigl(1 - r^{1/x}\bigr)^{x}, \qquad 0<r<1,
$$
is convex on the interval $x \in [1,2]$?
For example, one may take $r = 2/3$.
Numerically the function ...
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Is it wrong to write “infinity” instead of “undefined” in a trigonometry table?
I am a high-school math teacher and I have a question about the correct notation for trigonometric tables.
In standard mathematics, the tangent function is not defined at ninety degrees (or at pi over ...
- 1,186
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Proving that a sine function with countable discontinuities is Riemann integrable (without measure zero criterion)
How would I prove that the function $f\colon [0, 1]\to\mathbb{R}$ defined by
\begin{equation}
f(x) = \begin{cases}
1 & \text{if }x = \tfrac{1}{n} \text{ for some } n\in\mathbb{N}, \\
\sin (x) &...
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