Mathematics Stack Exchange
Mathematics Stack Exchange

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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What is $\frac{d}{dx}\left(\frac{dx}{dt}\right)$?

This question was inspired by the Lagrange equation, $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$. What happens if the partial derivatives are replaced by total derivatives, leading to a situation where a…
Ben Lerner
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What is the indefinite integral of the function $f(x) = \frac{\sin(x)}{x}$

I want to know if the function $f(x) = \frac{\sin(x)}{x}$ is integrable and if it is, then what's its integral? My high school book says its a non-integrable function while WolframAlpha says its integral is Si$(x) +$constant. Please shed some light…
Bazinga
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Proving that $A=\{(-2)^n : n \in \mathbb{N} \}$ is unbounded

I am trying to prove that $A=\{(-2)^n : n \in \mathbb{N} \}$ is unbounded. What I did was first to show that for every $n \in \mathbb{N}$ if $n$ is even then $(-2)^n = 2^n$ and if $n$ is odd then $(-2)^n = -2^n$ (I did it by induction on…
yotamoo
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Minimizing fuel usage for small boat between given points

I'm 12 years old and my I was getting bored this afternoon so my dad gave me this math problem (he said it was supposed to be hard, and that I should do some research to learn how to solve it). "A small boat moving at $V$ km/h uses fuel at a rate…
user290960
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Evaluation of $ \lim_{n\rightarrow \infty}\left[\prod^{n}_{r=1}\left(1+\frac{n}{r}\right)^{\frac{r}{n}}\right]^{\frac{1}{n}}$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\left[(1+n)^{\frac{1}{n}}\cdot \left(1+\frac{n}{2}\right)^{\frac{2}{n}}\cdot \left(1+\frac{n}{3}\right)^{\frac{3}{n}}.........2\right]^{\frac{1}{n}}$ $\bf{My\; Try::}$ Let $$ y =…
juantheron
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Discontinuous function with continuous derivative

Differentiability implies continuity, but what in this case ? $$ f(x) = \frac{1}{\sqrt{2}}\arctan\left(\sqrt{2} \tan{x}\right) $$ It's discontinuous at $x = \pi/2 + n\pi$, $n$ is integer. But, $$ g(x) = \frac{1}{1+\sin^2{x}} $$ It's continuous…
Mohamed Mostafa
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An example of an infinitely differentiable function with compact support

Could anyone give me a function infinitely differentiable on the real line and having a compact support? And the function must be nonnegative and normalized, i.e. the integration of the function on the real line must be one. I tried to think of one…
Keith
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What do convergence and divergence mean? And why do they matter?

I understand that when a series diverges, y doesn't approach 0 when x approaches infinity, and converging series do. But what does this say? I just want to understand some applications. NOTE: I'm still on integration by parts, so try to explain in…
PhilTheLawyer
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formula for the $n$th derivative of $e^{-1/x^2}$

$f(x) = \begin{cases} e^{-1/x^2} & \text{ if } x \ne 0 \\ 0 & \text{ if } x = 0 \end{cases}$ so $\displaystyle f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} \frac {e^{-1/x^2}}x = \lim_{x \to 0} \frac {1/x}{e^{1/x^2}} = \lim_{x…
jxhyc
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How Differential got into calculus

I am confused what differentials are and how they become related to derivatives and integrals, as neither my textbook explains them nor my teacher. What we do to solve differential equations is to convert derivative to a "differential form" like…
Haider Rehman Butt
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Can help me to find $\sum_{n=1}^{\infty }\frac{1}{(4n-1)^3}$?

Can help me to find $\sum_{n=1}^{\infty }\frac{1}{(4n-1)^3}$?
E.H.E
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Calculus in a discrete universe

Suppose we ascertained that space and time are discrete and the units are Planck's. Would that affect calculus? I know that integration does not require a continuum, but about differentiation I read contrasting views, w.r.t. physical functions. In…
user168605
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A problem on Mean Value Theorem

If $f''(x)$ exists on $[a,b]$ and $f'(a)=f'(b)$, then : $$f(\frac{a+b}{2})=\frac 1 2[f(a)+f(b)]+\frac{(b-a)^2}{8}f''(c)$$ for some $c\in(a,b)$. I tried but was unable to think of a function and was unable to use the given condition except for…
evil999man
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Premises of the Mean Value Theorem

Why does the statement of the mean value theorem requires that: (1)The function $f$ be continuous on the closed interval $[a,b]$ (2)Differentiable on the open interval $(a,b)$. Couldn't we just require (2) and the the first premise will be met…
Mohamed Zaki
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Equations of lines tangent to an ellipse

Determine the equations of the lines that are tangent to the ellipse $\displaystyle{\frac{1}{16}x^2 + \frac{1}{4}y^2 = 1}$ and pass through $(4,6)$. I know one tangent should be $x = 4$ because it goes through $(4,6)$ and is tangent to the ellipse…
Invader
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