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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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Derivative of function is bigger??!

Today I had a lesson about derivatives and our professor showed us a derivative that is bigger than his function. This doesn't seem legit, can somebody help me with that? Any explanation will be appreciated. By the way can somebody tell what is so…
Gil
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Prove the sum of squares of two functions equals 1

If you have $f'(x)=g(x)$, $g'(x) = -f(x)$, $f(0)=0$ and $g(0)=1$, how do you prove that $f^2(x)+g^2(x) = 1$?
paul
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Application of mean value theorem

$f$ is differentiable in $[a,b]$, and satisfies $f(a)
stph
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please check my proof and comment it

Let $\left(\frac1{(2k+1)!}\right)$ be an infinite sequence. I want to show the following limit $\lim \limits_{k \to \infty}{\frac{1}{(2k+1)!}}=0.$ Below is my proof. Please check it, I'm not confident about my ability. For all nonnegative integer of…
shuxue
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Spivak's Calculus exercise, possible error in text. Chapter 10, problem 24(a).

Working through all the problems in Spivak's Calculus (3E) and hit a snag here. Given $c, d \in R$, and distinct $x_1, ..., x_n \in R$, show that for any given $1 \leq i \leq n$ there exists a polynomial function $f$ of degree $2n-1$ such that…
Saigyouji
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Volume of a wine barrel

This is a famous calculus problem and is stated like this Given a barrel with height $h$, and a small radius of $a$ and large radius of $b$. Calculate the volume of the barrel given that the sides are parabolic. Now I seem to have solved…
N3buchadnezzar
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Find All Points on a Paraboloid where Tangent Plane is Parallel to a Given Plane

Find all points on the paraboloid $z=x^2+y^2$ where tangent plane is parallel to the plane $x+y+z=1$ and find equations of the corresponding tangent planes. Sketch the graph of these functions. I have its answer. I don't really understand such…
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How many points in the xy-plane do the graphs of $y=x^{12}$ and $y=2^x$ intersect?

The question in the title is equivalent to find the number of the zeros of the function $$f(x)=x^{12}-2^x$$ Geometrically, it is not hard to determine that there is one intersect in the second quadrant. And when $x>0$, $x^{12}=2^x$ is equivalent to…
user9464
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Trig substitution $\int x^3 \sqrt{1-x^2} dx$

$$\int x^3 \sqrt{1-x^2} dx$$ $x = \sin \theta $ $dx = \cos \theta d \theta$ $$\int \sin^3 \theta d \theta$$ $$\int (1 - \cos^2 \theta) \sin \theta d \theta$$ $u = \cos \theta$ $du = -\sin\theta d \theta$ $$-\int u^2 du$$ $$\frac{-u^3}{3}…
Dantheman
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A constrained extremum problem

Find the maximum possible value of $$A = a^{333} + b^{333}+c^{333}$$ subject to the constraints $$a+b+c=0$$ and $$a^2+b^2+c^2=1,$$ where $a,b,c\in \mathbb{R}$ Thank you for helping me.
Marisa
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A question on differentiation :

Let $$f(x)=\sin^{-1}(2x\sqrt{1-x^2})$$ I found out $f'(x)$ in three methods and got three different answers ! 1) Putting $x=\cos\theta$, we get $f(x)=2\cos^{-1}x$, on differentiating this we get $$f'(x)=\frac{-2}{\sqrt{1-x^2}}$$ 2) Putting…
Akshay Krishnan
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Area vs Volume Paradox

Possible Duplicate: How can a structure have infinite length and infinite surface area, but have finite volume? Hi, I have this question that I quite cant explain why. So the area under the curve $$y=\frac{1}{x}$$ from 1 to infinity…
Kartik
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Can you propose any hack for remembering the common derivative and integral formulas?

Calculus cheat sheet Remembering the following formulas has been a nuisance for me for years now. Common Derivatives Common Integrals They are too many in numbers Intuition doesn't work I mix up derivatives and integrals frequently Can anyone…
user366312
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$\frac{1}{x-a} + \frac{1}{x-b} + \frac{1}{x-c} = 0 $ has precisely two real roots

Prove that given $ a < b < c $ this equation: $$\frac{1}{x-a} + \frac{1}{x-b} + \frac{1}{x-c} = 0 $$ has precisely 2 real roots. I understand there are 3 point of discontinuities, but I have no idea how to prove this. Can you give me a hint?…
Royaltea
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How to integrate $\int_{-\infty}^{+\infty} e^{-x^2}\cos x \, dx$

Can you help me with this? $$\int_{-\infty}^{+\infty} e^{-x^2}\cos x \, dx$$
Dariusz
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