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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Why ReLU function is not differentiable at 0?

I'm kind of rusty in calculus. Why is the ReLU function not differentiable at $f(0)$? $$ f(x) = \begin{cases} 0 & \text{if $x \leq 0$} \\ x & \text{if $x > 0$}. \end{cases} $$
rvimieiro
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Evaluate the double integral

$$\int_0^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} (x^2+y^2+\sin(\pi(x^2+y^2)))\,dy\,dx$$ *sorry if the mathjax is off, I'm new at it. Anyways, I can use the properties of double integrals to make it $$\int_0^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}…
user1766888
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The series $\sum_{n=2}^{\infty }\frac{1}{\log (n!)}$

I'm trying to find out whether this Series $\sum_{n=2}^{\infty } a_{n}$ converges or not when $$a_{n}=\frac{1}{\log (n!)}$$ I tried couple of methods, among them: d'Alembert $\frac{a_{n+1}}{a_{n}}$, Cauchy condensation test $\sum_{n=2}^{\infty }…
user6163
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Injective function that is not surjective

I'm trying to find an example of a continuously differentiable function from $\mathbb{R}$ to $\mathbb{R}$ that is injective but not surjective. I can easily find one from $\mathbb{Z}$ to $\mathbb{Z}$ but I'm having difficulty finding one for the…
Dan H
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Show that $x^{8}+5x^{2}=1$ has exactly $2$ real roots

How can we show that $x^{8}+5x^{2}=1$ has exactly $2$ real roots?
Tina
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convergence of a series involving $x^\sqrt{n}$

I was trying to prove the convergence of the series $\sum_{n=1}^{\infty}x^{\sqrt{n}}$, for $0
s_2
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Why $\sin\left( \frac 1 x \right) $ oscillates infinitely many times as $x \to 0$

Below, I have tried to prove why $\sin\left( \frac 1 x \right) $ starts oscillating infinitely many times as $x$ approaches zero. We know that the Sine function is an oscillating function. Let us assume a period $p$ such that, $$\sin\left(\frac 1…
R004
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Fundamental Theorem of Calculus and discontinuous functions

Will the Fundamental Theorem of Calculus be useful in evaluating: $$y=\int_{0.5}^{1} \left \lfloor t \right \rfloor dt?$$ The Riemann sums make this evaluation easy. I, however, would like to know how( if possible), can I use the theorem to…
R004
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partial derivative of an integral function

Given the function $f(x,y) = \int^x_y g(t) dt $, $g(t)$ continuous for all $t$. How to evaluate the partial derivatives of $f$ ? It is correct to do like this (by FTC) ?: $f_x$ = $g(x)x'$. $f_y = g(y)y'$. Thanks!!!
D.Rotnemer
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Good final exam bonus problem for calculus students?

I'm making a final exam for a first course in calculus at a university. I need some suggestions for a good bonus problem. By "good" I mean an interesting problem that some of the brighter students could solve in about 15 minutes. Hopefully it…
Forever Mozart
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A doubt on a proof of $\lim \frac{\sin x}{x}$ as $x\to 0$ provided in Simmons's Calculus with Analytic Geometry

I'm having difficulty understanding a proof of $\lim_{x\to0}\frac{\sin{x}}{x}=1$ provided in Simmons's Calculus With Analytic Geometry, pg. 72. The proof goes as follows: Let $P$ and $Q$ be two nearby points on a unit circle, and let $\overline{PQ}$…
Sasaki
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Computing a tricky limit

$$ \lim_{n \to \infty} \frac{1^m + 2^m + 3^m + \cdots + (2n-1)^m }{n^{m+1}} $$ I am kind of stuck since I cannot make it look into a form that would involve the integral of certain function. I know somehow it would be easy if we can compare this…
user139708
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How to guess the number of inflection points?

I am asked to find fast the number of possible inflection points of: $$y=(x-1)(x-2)^2(x-3)^4(x-4)^3$$ I know if the degree of any polynomial is even, its plot starts from the 2th quadrant to 1st quadrant of $\mathbb R^2$. This was what I could do…
Mikasa
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Assume $f$ is a continuous one-to-one function over an interval. Prove that $f$ is strictly monotone

Assume $f$ is a continuous one-to-one function over an interval. Prove that $f$ is strictly monotone. Attempt Since we know that $f$ is one-to-one, for every $f(x)$ there is exactly one element $x_0$ that maps to it. Thus, if $f(x) = f(y)$, then…
user19405892
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Construct a continuous function $f$ over $[0,1]$ satisfying $f(0) = f(1)$ but $f(x) \neq f(x+a)$

Suppose $0 < a < 1$ is not of the form $\dfrac{1}{n}$ for positive integer $n$. Construct a continuous function $f$ over $[0,1]$ satisfying $f(0) = f(1)$ but $f(x) \neq f(x+a)$ for all $x \in [0,1-a]$. This is a follow up question to this. I am…
user19405892
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