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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how to find surface area of a sphere

could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area…
bantus
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$\frac{\sqrt{1+\sqrt{x}}+\sqrt{1+\sqrt{1-x}}}{\sqrt{1-\sqrt{x}}+\sqrt{1-\sqrt{1-x}}}$

Find the value of $\frac{\sqrt{1+\sqrt{x}}+\sqrt{1+\sqrt{1-x}}}{\sqrt{1-\sqrt{x}}+\sqrt{1-\sqrt{1-x}}}$, if $x\in \left(0,\frac{1}{2}\right)$. I know it is equal to $\sqrt{2}+1$, but I don't know how to prove it?
CSG
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Is this true for all polynomials

I took $3$ random polynomials with non zero roots one having even degree and two having odd degrees $f(x)=\color{red}{4}x^2-(4\sqrt3+12)x+12\sqrt3$ having roots $\color{blue}{3,\sqrt3}$ and leading coefficient $\color{red}{4}$ and calculated values…
user593646
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"Calculus 4th Edition" by Michael Spivak -- Chapter 11 Problem 59

In Michael Spivak's "Calculus" 4th edition, Chapter 11 Problem 59 reads Suppose that the function $f > 0$ has the property that $$(f')^2=f-{1 \over f^2}.$$ Find a formula for $f''$ in terms of $f$. (Why is this problem in this chapter?) This seems…
David M
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Calculating $H'(x)$ given $H(x) = \int_{x^3 + 1}^{x^2 + 2x} e^{-t^2} dt$

Given $\displaystyle H(x) = \int_{x^3 + 1}^{x^2 + 2x} e^{-t^2} dt$, we want to find $H'(x)$. First, we rewrite $H(x)$ as follows: $$\begin{align} &= \int_0^{x^2 + 2x} e^{-t^2} dt + \int_{x^3 + 1}^0 e^{-t^2} dt \qquad &\text{Properties of integrals}…
Chrisuu
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Prove that $ f(\xi)=f'(\xi)\int_{0}^{\xi}{f(x)dx} $

Let $f:[0,1]\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative such that $$ \int_{0}^{1}{f(x)dx}=\int_{0}^{1}{xf(x)dx} $$ How can we prove that there exists $\xi\in(0,1)$ such that $$ f(\xi)=f'(\xi)\int_{0}^{\xi}{f(x)dx}…
pxchg1200
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Typical Calculus BC Separation of Variables Question

I was told that I have a cylindrical water tank $10$ ft tall that can store $5000 $ ft$^3$ of water, and that the water drains from the bottom of the tank at a rate proportional to the instantaneous water level. After $6$ hours, half of the tank has…
Niwde Aup
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Why do differentiating and integrating 'work'?

Why do you get a function's (changing) slope when you take its derivative and why do you get the area under the function when you take its integral? What is the easiest reasoning behind this?
Ylyk Coitus
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Function example? Continuous everywhere, differentiable nowhere

Possible Duplicate: Are Continuous Functions Always Differentiable? If such a function exists, can anyone give an example of a function $f(x) : \mathbb{R} \longrightarrow \mathbb{R}$ that is continuous for all $x \in \mathbb{R}$ but…
e_tothe_ipi
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Understanding operator under a subtitution

in my notes, I have the following phrase: With $x = e^t$ and $x \frac{d}{dx} = \frac{d}{dt}$ How, how come we get $x \frac{d}{dx} = \frac{d}{dt} $? I know if I diferentiate with respect to $t$ I obtain $$ \frac{d}{dt} x = x $$ I do not understand…
James
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Is it true that $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}=\pm\infty\ \implies\ \lim_{x \to a}f'(x) = \pm\infty$?

Is it true that $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}=\pm\infty \implies \lim_{x \to a}f'(x) = \pm\infty$? Here, $f$ is a function defined on some open interval $I$, and $a\in I$. Assume $f$ is continuous at $a$ and differentiable around $a$. I…
Euler's Friend
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Is there a way to find this limit algebraically? $\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$

I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped. $$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$ You'll notice that using…
Jae Swanepoel
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Integral yielding part of a harmonic series

Why is this true? $$\int_0^\infty x \frac{M}{c} e^{(\frac{-x}{c})} (1-e^{\frac{-x}{c}})^{M-1} \,dx = c \sum_{k=1}^M \frac{1}{k}.$$ I already tried substituting $u = \frac{-x}{c}$. Thus, $du = \frac{-dx}{c}$ and $-c(du) = dx$. Then, the integral…
jrand
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Solving equation with $\arctan(x)$ and $\ln(x)$

Last week my group had math exam and this questions popped up. I approximated $x$ using Newton-Rhapson method ($x\approx 0.341$). However, after the exam professor told us that we should have found an exact solution. I tried however; I couldn't make…
Soulmass
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Need an $ f $ so that $ 0 < -\left(f'(x) + \frac{f''(x)}{f'(x)}\right) < \epsilon/x $ and $ e^{f(x)}/x \stackrel{x \to \infty}{\longrightarrow} 0 $.

First-time poster. Please forgive me if I do something unorthodox. To be more specific, I need a function $f:[0,+\infty)\to[0,+\infty)$ so that $f$ is nondegenerate, nondecreasing, continuous, $$\lim_{x\to+\infty}\frac{e^{f(x)}}{x}=0$$ and…
danzibr
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