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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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$\frac{\mathrm{d}}{\mathrm{d}x}$ Notation Explanation please?

I know how to derive, I know how to integrate. I know what to do when I see $\frac{\mathrm{d}}{\mathrm{d}x}$ and such but what does it really mean? I know it means something like derive in terms of $x$, but whats the difference between…
Captain Man
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Not sure about the derivative of the integral

Call me stupid, but I would like to know whether my understanding is okay: $$\frac{d}{dx}\left(\int_0^x f(s)ds\right)=\frac{d}{dx}F(x)=f(x)$$
luka5z
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How can I prove $\underbrace{\int \ldots \int}_{n} |x| dx = \frac{x^n |x|}{n+1}+C$?

So I was bored and decided to figure out the indefinite integral of the absolute value function, $|x|$. Using integration by parts ($u=|x|, dv=dx$, $dx = \text{sgn}(x)=\frac{|x|}{x}$), it can be shown that $\displaystyle\int |x| dx = \frac{x…
Eugene Bulkin
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Finding asymptotes to $y = \frac{2x^2 + 3x - 6}{2x + 1}$

I need to find the asymptotes of $y = \frac{2x^2 + 3x - 6}{2x + 1}$. The asymptote at $x = -1/2$ is clear. If one long divides they can easily see that there is an asymptote of $y = x + 1$ as $x$ goes to infinity. However, what is wrong with this…
Leduc
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When should the antiderivative of a rational function be defined as a piecewise function?

I'm doing basic problems on antiderivatives and there seems to be an inconsistency in my book. The instructions for these problems are: Find the most general antiderivative of the function. Number 11 is: 11. $f(x) = \dfrac{10}{x^9}$ So, I…
Matt Gregory
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Is $\sin(x)/ \tan(x) = \cos(x)$ at $0$?

We know that $\frac{\sin(x)}{\tan(x)} = \cos(x)$. But at $x = 0$, the LHS becomes $0/0$. So is the function undefined at that point?
brita
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Parity through Series expansion

By Maclaurin series; $\sin{x}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-....$, and we know that period of $\sin{x}$ is $2π$ because $\sin{(x+2π)} = \sin{x}$. But if we consider only the RHS of the above series then how we can tell that this expression…
wasim
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How I can calculate the derivative of a piecewise function like this?

How I can calculate the derivative of $$f(x) = \left\{ \begin{gathered} {x^2}\quad,\quad{\text{if}}\quad x \in \mathbb{Q} \\ {x^3}\quad,\quad{\text{if}}\quad x \notin \mathbb{Q} \\ \end{gathered} \right.$$ at some $x\in \mathbb{R}$?
mathsalomon
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If $(a_n^2)$ converges to $0$, then $(a_n)$ converges to $0$

I had a problem with this I could prove it by contradiction but, I wonder if you can do only algebraically can be done only with inequality? $$\left( {a_n } \right)^2 \to 0 \qquad\Longrightarrow \qquad \left( {a_n } \right) \to 0 $$ Thanks!
Daniel
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Maximize and Minimize a 12" piece of wire into a square and circle

A wire of length 12" can be bent into a circle, a square or cut into 2 pieces and make both a circle and a square. How much wire should be used for the circle if the total area enclosed by the figure(s) is to be: a) a Maximum b) a Minimum What…
OghmaOsiris
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What does a parametric equation mean?

I am following the last module of Differential Calculus on Khan Academy, that deals with Parameteric equations. Here are the parametric equations described in the lecture. $x(t) = 5t + 10$ $y(t) = 50 - 5t^2/2$ However, I really don't understand what…
desert_ranger
  • 335
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Calculus: The tangent line intersects a curve at two points. Find the other point.

The line tangent to $y = -x^3 + 2x + 1$ when $x = 1$ intersects the curve in another point. Find the coordinates of the other point. This was never taught in class, and I have a test on this tomorrow. This question came off of my test review…
Amanda
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How to find asymptotes of implicit function?

How to find the asymptotes of the implicit function $$8x^3+y^3-6xy-3=0?$$
user64494
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What is? $ \lim_{x \to \infty} \left(\frac{e}{\left( 1 + \frac {1}{x} \right)^x} \right)^x $

Find $$ \lim_{x \to \infty} \left( \dfrac{e}{\left( \left( 1 + \frac {1}{x} \right)^x \right)} \right)^x. $$ $ \lim_{x \to \infty} \left( \dfrac{e}{\left( \left( 1 + \frac {1}{x} \right)^x \right)} \right)^x = \lim_{x \to \infty} \left( \dfrac…
jujuju
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Calc III: Volume of the Intersection of two spheres

Question: I am not getting the correct answer. How do I get the solution (and why does my solution not work?) Find volume that lies inside both spheres: \begin{align} A: 4 &= (x+2)^2 + (y-1)^2 + (z+2)^2\\ B: 4 &= x^2 + y^2 + z^2\\ \end{align} My…
JDG
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