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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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Calculus question relating to the mean value theorem

Suppose $f$ is a twice-differentiable function with $f(0) = 0$, $f\left(\frac12\right) = \frac12$ and $f'(0) = 0$. Prove that $|f''(x)| \ge 4$ for some $x \in \left[0,\frac12\right]$. I've been stuck on this question for a while now without any idea…
Mimi
  • 201
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Finding an equation of a plane

Find an equation of the plane that passes through the line of intersection of the planes $x-y=1$ and $y+2z=3$ and is perpendicular to the plane $x+y-2z=1$. I'm having a hard time with this. I'm very confused by this section in the book, as the…
Snowman
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Insane, out of control boat problem

I had this problem on my Calc 1 exam today and found it to be a bit difficult. I'll walk you through the problem and my attempt at solving it, hopefully you guys will be able to help me! At noon, boat A is 20 miles west of boat B. Boat A is…
Defacto
  • 231
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$x^x=(1-x)^{(1-x)}$

I have this equation $x^x=(1-x)^{(1-x)}$ and I want to find $x$. The solution $x=1/2$ is clear and from the function plots ploted with, for example, wolfram alpha, is obvious this is the only situation. But how can I prove that this is the only one?…
Jane Doe
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What is the difference between the first derivative and the second derivative test ?

So a bit confused when I see a question say: "Use the second derivative test to find all relative extrema" When using the second derivative test are we not looking for concavity and points of inflection. So far, in order to find relative extrema,…
Siddart Fredrick
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silly question about convergent sequences

Suppose $(a_n)$ is convergent, say $a_n \to L$. Then $a_{n+1} \to L$ as well. Here is my confusion: IS $a_{n+1}$ meant to be the $(n+1)st$ term of the sequence? or is it the subsequence $(a_2,a_3,a_4,....)$?? Reason of the question: In my book, the…
user139708
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Definite integral of even and odd functions proof

Let $f$ be continous on [-a,a] a) prove : $\int^{a}_{-a} f(x) dx = 0$ Because $f$ is odd $f(-x) = -f(x)$ $$\int_{x=-a}^0 f(x) \, dx = \int_{x=-a}^0 -f(-x) \, dx$$ letting $t = -x, dt = -dx$ $\int_{x=-a}^0 -f(-x) \, dx = \int_{t=a}^0 f(t) \,…
Kasmir Khaan
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Find the smallest real number $a\gt 0$ for which the equation $a^x=x$ has no real solutions

As the title says, We seek the smallest real number $a\gt 0$ for which the equation $a^x=x$ has no real solutions. This is inspired by this question. I must admit that I did not have much luck with this..Any suggestions?
MathematicianByMistake
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Using the definition of derivative to evaluate $(\sin^2(x))'$

I know that $\lim_{x\to 0} \dfrac{\sin x}{x} = 1$. But I'm stuck in using the definition of derivative to evaluate $(\sin^2 x)'$. Will appreciate any helpful input. Thank you.
uohzxela
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Assigning a finite value to a divergent integral

Can someone help me to regularize the following divergent integral? $$ \int_0^{1/2}\, \frac{d x}{x^{3/2} (1-x)^{3/2}} $$ Guys, thank you very much for your answers. Thus if I have understood your procedure, the regularized result of this divergent…
Andrea
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Does $\int_1^\infty \sin^2 (x^2) \, dx$ converge?

Does the following integral converge? $$\int_1^\infty \sin^2 (x^2) \, dx$$ I tried $$\int_1^\infty \sin^2(x^2) \, dx=\int_1^\infty \frac{1-\cos(2x^2)}{2} \, dx = \frac{\sqrt{2}}{4} \int_1^\infty\frac{1-\cos(u)}{2\sqrt{u}} \, du$$ The idea is that, I…
user34183
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Rayleigh quotient for real matrices, critical points

I have found an exercice in a calculus book, which I have problems to solve. $$\text{Let}\, R:\mathbb{R}^n-{0}\to \mathbb{R},\quad R(x)=\frac{x^tAx}{x^tx} = \frac{\langle x,Ax\rangle}{\langle x,x\rangle},$$ where $\langle \cdot,\cdot \rangle$ is…
Lucien
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High School Trigonometric Integration

I am wondering which step has gone wrong. Is it wrong to use $u=\sin x$ ? $$ \int \cos^3x\ \sin x\;\mathrm{d}x $$ $$=\int\cos^2x \sin x \cos x\;\mathrm{d}x $$ $$=\int(1-\sin^2x)\sin x\;\mathrm{d}(\sin x)$$ $$=\int\sin x\;\mathrm{d}(\sin…
Tiszt
  • 93
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functional equation of type $f(x)-f(y) = f\left(\frac{x-y}{1-xy}\right)$

If $f(x)-f(y) = f\left(\frac{x-y}{1-xy}\right)$ and $f$ has domain equal to $(-1,1)$, then which of the following is the function satisfying the given functional equation? Options: (a) $2-\ln\left(\frac{1+x}{1-x}\right)$ (b)…
DXT
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How to evaluate $\lim\limits_{x\to 0} \frac{\arctan x - \arcsin x}{\tan x - \sin x}$

I have a stuck on the problem of L'Hospital's Rule, $\lim\limits_{x\to 0} \frac{\arctan x - \arcsin x}{\tan x - \sin x}$ which is in I.F. $\frac{0}{0}$ If we use the rule, we will have $\lim\limits_{x\to 0}…
Authawich Narissayaporn
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