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
5
votes
3 answers

Is the optimal solution to this problem to row straight to the store?

For my homework, I was given this brainteaser: You’re sunbathing on the island shown on the map below. The island is six miles from shore at the closest point, and the nearest store is a convenience store seven miles down the beach. If you can…
mowwwalker
  • 1,201
5
votes
1 answer

Rotating parametric curve

Given parametric curve: $x=t\cos(t)$, $y=t^2$, how can i rotate the curve about the origin by an angle $\theta=\pi/3$?
Geetj
  • 53
5
votes
2 answers

for $x^2+y^2=a^2$ show that $y''=-(a^2/y^3)$

For $x^2+y^2=a^2$ show that $y''=-(a^2/y^3)$ I got that $y^2=a^2-x^2$ $y'=-x/y$ $y''=(-1-y'^2)/y$ But then I get stuck.
user341207
5
votes
1 answer

What is the rate of change of the height of water in a conical frustum bucket after one minute if it is being filled at a constant rate?

I have an embarrassing question to ask. Embarrassing in the sense that I should be able to confirm or refute the solutions I got but I just can't seem to rationalise the results I worked out. I am working on a tutorial for some of my students and I…
Pete Langshaw
  • 110
5
votes
0 answers

$\frac{dy}{dx}$, $\frac{d^{2}y}{dx^{2}}$, is there $\frac{d^{1.5}y}{dx^{1.5}}$?

Possible Duplicate: Is it meaningful to take the derivative of a function a non-integer number of times? So about derivative everyone knows we have the first derivative second derivative. Is there any notion of one and a half derivative? Thanks.
Hoang
  • 51
5
votes
1 answer

Why does this differentation output correct result?

I noticed something funny. If you differentiate $x^x$ treating the exponent as a constant, you get $xx^{x-1}=x^x$. If you treat the base as a constant, you get $x^x \ln{x}$. If you add these two bizzare and incorrect derivatives of $x^x$, you get…
user473686
  • 141
5
votes
2 answers

How find all values of $\frac{|x+y|}{|x|+|y|}+\frac{|y+z|}{|y|+|z|}+\frac{|z+x|}{|z|+|x|}$

$$A = \frac{|x+y|}{|x|+|y|}+\frac{|y+z|}{|y|+|z|}+\frac{|z+x|}{|z|+|x|}$$ Where $x, y $ and $z$ are real numbers and non-zero. How can we find all values of $A$? My try: From $x, y$ and $z$ at least two of have same sign $\longrightarrow A \geq…
Amiri
  • 61
5
votes
2 answers

Fundamental theorem of calculus (Spivak's proof)

I'm trying to understand one part of the fundamental theorem of calculus in the Spivak's calculus book. On page 282 he defined: $$m_h=\inf\{f(x);c\le x\le c+h\}$$ $$M_h=\sup\{f(x);c\le x\le c+h\}$$ Afterwards, he said $\lim_{h\to0}m_h=\lim_{h\to…
user42912
  • 23,582
5
votes
2 answers

Differentiating with respect to a function

I have a question regarding partial differentiation of a function of $x,y$ with respect to another function of $x,y$. Specifically, I was wondering whether my logic or technique would hold true for most…
jrand
  • 1,959
5
votes
2 answers

How did feynman compute $e^x$ with the accuracy he wanted?

Excerpt from the book Surely You're Joking, Mister. Feynman! Here Feynman calculates $e$ to a couple of powers. I understand that he luckily knew a couple of logs by heart. What I don't understand is the part where he adjusts the numbers to the…
delivosa
  • 401
5
votes
2 answers

On the monotonicity of the function $z \mapsto (z^{1-s}-1)/(z^s-1)$.

During my work I have met the function $$ z \mapsto \frac{z^{1-s}-1}{z^s-1}, $$ which I consider for $z>1$ (say). The number $s \in (0,1)$ is a given parameter. I would like to prove that it is monotonically decreasing when $1/2
Siminore
  • 35,136
5
votes
1 answer

Linear Differential Equation imaginary factoring inconsistancy

Recently I have been improving my skills in Linear differential equations and I came across a problem with a rather problematic solution. The problem is as follows: (Already converted from y prime form, I assure without error, I have checked five…
David
  • 73
5
votes
1 answer

A function that is less than its average on a neighborhood of every point is convex

Let $f: (a,b) \to \Bbb R$ be a continuous function such that $$\forall x \in (a,b)\; \exists \epsilon_0 \forall \epsilon < \epsilon_0: f(x) \leq \frac{f(x+\epsilon)+f(x-\epsilon)}{2}. $$ Is $f$ necessarily convex? By definition, $f$ is convex if…
Emolga
  • 3,527
5
votes
1 answer

Simple calculus question: is applying l'Hopital's rule to $\sin(x) / x$ really circular reasoning?

So I know this is a dumb question maybe, but it has been in my head recently. I once told someone, on the topic of introductory calculus, that using l'Hopital's rule to calculate $$\lim\limits_{x\rightarrow 0} \frac{\sin(x)}{x}$$ is circular…
user438666
  • 2,040
  • 1
  • 9
  • 23
5
votes
1 answer

What is wrong with my integral calculation?

I want to solve the integral $\int_1^\infty\frac{a}{a^2+x^2}dx$ where $a>0$ is a constant I first tried simplifying so \begin{align} \frac{a}{a^2+x^2} &=\frac{1}{a+\frac{x^2}{a}}…
arnoldschwarzenegger
  • 53
1 2 3
…
99 100