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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Finding maximum area of a pentagon

Out of all the pentagons in the shape of a rectangle overlapped by an isosceles triangle, with perimeter P fixed, determine the dimensions of the one with maximum area.
Pedro Gabriel Carvalho
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How to define minimum value for given number?

I want to say minimum value for X is 1000. now X can be anything. so M would be the multiplier of X so that M*X >= 1000. What would be the mathematic way of doing this without using if statement? M = 1; X = 1000; if(X < 1000) M = 1000/X; X = M*X; I…
M.kazem Akhgary
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A question regarding Czarnowski's cat function

The Czarnowski cat function $c(x, n) = \sin^n(x) + \cos(x)$ gives the so-called Czarnowski's cats. There are infinitely many of these cats, and legend has it that Czarnowski created a clever mathematical naming scheme to give them all unique names.…
Edi Madi
  • 311
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The condition of fixed point theorem.

Theorem (a). If $g$ is continuous on $[a,b]$, $g(x) \in [a,b]$. Then $g$ has a fixed point in $[a,b]$. (b). In addition, if there exists $k<1$ such that $|g'(x)| \leq k $. Then $g$ has unique fixed point in [a,b]. Why we can not modify the…
sinoky
  • 299
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Calculate integral $\int_{-1}^{1} (1-x^2)^n dx $

How to show that : $I_n = \int_{-1}^{1} (1-x^2)^n dx $ is equal to $\dfrac{2^{2n+1}(n!)^2}{(2n+1)!}$ ? With integration by parts ? I don't know how to prove this equality. Someone could help me ? Thank you in advance :)
Mélanie De la Cheminée
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Is $(x^2)^x$ differentiable at $0$?

Have to grade a midterm where one of the true/false questions boils down to whether or not $f(x)=(x^2)^x$ is differentiable at 0. I'm not sure of the answer. For one thing, the continuity of $f(x)$ is author-dependent since it hinges on what $0^0$…
anon
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Where do I go wrong when solving this integral involving a quadratic?

Simple question. I have an arithmetic mistake somewhere in here but can't find it. $\int \frac{3x-1}{x^2+10x+28}dx$ $\frac{1}{3}\int \frac{3x-1}{\frac{1}{3}(x+5)^2+1}dx$ Set: $u=x+5$ $x=u-5$ $dx = du$ $\frac{1}{3}\int…
Korgan Rivera
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Explanation of a substitution in $\int \sqrt{8 - x^2} \mathrm{dx} $.

I saw that substitution should be $u = \arcsin \frac{x}{2^{3/2}}$, but i can't see how can i come to that conclusion. I'd appreciate a little push in the right direction.
MathsLearner
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Integral of a two variable function $e^{-(5x^2-6xy+5y^2)}$

How to compute $$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(5x^2-6xy+5y^2)} \, dx \, dy \;?$$ I have seen the integral first time. I can't find any hint how to even proceed? Kindly HELP. Thank You!
Subhash Chand Bhoria
  • 966
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The Limit Comparison Test V1

I'm currently learning Calculus 2, more specifically I'm learning about sequences and series. I'm not enjoying this section as much as I thought I would, this is because I'm having to learn all these different tests to determine the convergence and…
user405274
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Question about calculating $\lim\limits_{n \to\infty} \dfrac{6n^5+(\sin(8n))^2}{n^4+6} $

I have the following limit: $$\lim\limits_{n \to\infty} \frac{6n^5+(\sin(8n))^2}{n^4+6} $$ My fist question is, can I solve this by dividing by the highest n power? The reason I was uncertain about doing this was because of the $(\sin(8n))^2 $…
melm
  • 315
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Find the derivative of the inverse function to $f(x)=\frac{1}2\sin(2x) + x$

Find the derivative of the inverse function of: $$ f(x)=\frac{1}2\sin(2x) + x $$ I already know that this function is one-to-one. What I've done: $$ y=\frac{1}2\sin(2x) + x $$ $$ 2y - 2x = \sin(2x) $$ $$ \frac{\arcsin(2y - 2x)}2 = x $$ Is this a…
Curtain
  • 415
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Product of two differentials (infinitesimals)

In a physics derivation today my lecturer ended up with two differentials multiplied together, like this term in a longer summation: $p^2 q^2 \mathrm{d} p ~ \mathrm{d} q$ And then he said this term becomes zero because the two differentials are…
Thomas
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What's the maximum area of the rectangle inscribed under the curve $y =\cos x$ in $[\frac{-\pi}{2}, \frac{\pi}{2}]$?

What I tried was pretty basic. $$A=2xy=2x\cos x$$ $$A'=2\cos x-2x\sin x$$ $$A' = 0 \implies 2\cos x-2x\sin x=0$$ $$x=\frac{\cos x}{\sin x}=\cot x$$ But I got stuck here. I didn't know what to do next.
Eyad H.
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Finding volume of a region in three space

I want to calculate the volume of $V$ where V is the region $$ V = \left\{ (x,y,z) : \left( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \right)^2 = \frac{ x }{h} \right\} $$ Try: The approach would be to compute $$ \iiint_V dV$$ but I am…
user139708
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