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 recognize an improper integral?

I think $\displaystyle \int ^{4} _0 \frac{\sin x}{x} dx$ is not an improper integral. Is this true? If not true, then how can one recongnise improper integrals in general?
Christian Prince
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Differentiability in R^n

If $f: U \rightarrow \mathbb R^n$ ($U \subset \mathbb R^m$ is an open set) is differentiable and $f(x) \neq 0$ $\forall x \in U$ $\Rightarrow$ $\varphi: U \rightarrow \mathbb R$, $\varphi(x) = \frac {1}{||f(x)||}$ is differentiable. I know how to…
KatyF
  • 61
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Rearranging Pokemon Experience Formula to make Level the Subject

As the title suggests, I am trying to rearrange some of the formulas for calculating experience based on level to be the other way around (calculating level based on experience). I am having trouble with rearranging this formula for n (n being…
Hayden
  • 171
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Why can I multiply both sides by $dx$?

When we start learning about differential equations sometimes we "multiply" both sides of the equation by a differential and then integrate. Example: $\frac{dy}{dx}=x$ then $dy=x*dx$ and so on. I have always thought this is a "shortcut" since…
Moises Arizpe Rojo
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Proving continuity of $f$

Here is a problem I encountered some time back: Suppose $f$ is bounded for $a\leq x\leq b$ and for every pair of values $x_1$ and $x_2$ with $a\leq x_1\leq x_2 \leq b$, $f(\frac{1}{2}(x_1+x_2))\leq \frac{1}{2}(f(x_1)+f(x_2))$. Prove that $f$ is…
Eisen
  • 2,434
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Finding $\frac {a}{b} + \frac {b}{c} + \frac {c}{a}$ where $a, b, c$ are the roots of a cubic equation, without solving the cubic equation itself

Suppose that we have a equation of third degree as follows: $$ x^3-3x+1=0 $$ Let $a, b, c$ be the roots of the above equation, such that $a < b < c$ holds. How can we find the answer of the following expression, without solving the original…
Javad Kouhi
  • 671
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Example of a smooth function with zero derivative that is not constant

One of false beliefs in this question on Math Overflow is "If f is a smooth function with df=0, then f is constant". What is a counterexample to this statement? Can it be made correct by adding some restriction, e.g. that f is a function from reals…
ripper234
  • 329
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How to prove $|F(x)|\leq\frac{M(b-a)^2}{8}$

$f(x)$ is derivable in $[a,b]$, $|f^{'}(x)|\leq M$. $\int_a^b f(x)dx=0$. Let $F(x)=\int_a^x f(t)dt$. Try to prove $|F(x)|\leq\frac{M(b-a)^2}{8}$ I want to use Taylor expansion at $f(\xi)=0$, but I can't continue.
89085731
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Prove: $f(t)\leq t+1$

Let $f(x)$ be a strictly positive continuous function in $[0,1]$ such that $f^2(t) \leq 1 + 2\int_0^t f(s)\, \mathrm{d}s$. Prove that $f(t)\leq t+1$. Obviously, if $f(x)$ can be differentiated, it is true. But I can't deal with the case that it is…
89085731
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6
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2 answers

The discussion about $\int_a^b P^2(x)f(x)dx=0$

$f$ is Riemann integrable in $[a,b]$,and $\int_a^b f(x)dx>0$. If the polynomial $P(x)$ satisfies $\int_a^b P^2(x)f(x)dx=0$. Prove $P(x)=0$.
89085731
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anyone can help me with solving this $x^{x^{3}}=3$?

Find the value of $x$ : $x^{x^{3}}=3$ I tied with "log" but I couldn't. any help?
SAM
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Integral with specific method

Evaluate $$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx }$$ Consider : $$\displaystyle F(m,n)=\int _{ 0 }^{ \pi /2 }{ \sin ^{ 2m-1 }{ x } \cos ^{ 2n-1 }{ x } dx } $$ $$\sin^{2}x = t$$ : $$ \displaystyle F(m,n) =\frac{1}{2}…
User1234
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What's the relation between different antiderivatives?

If a function $f(x)$ has different forms of antiderivatives: $\frac { d }{ dx } { F }_{ 1 }(x)=f(x)$ $\frac { d }{ dx } { F }_{ 2 }(x)=f(x)$ What's the relationship between $F_1$ and $F_2$, is that ${F}_{1}(x)-{F}_{2}(x)=constant$ correct? For…
apollonian
  • 107
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Integration of $\sec(x+2) \tan(x+2)/\sqrt{x+2} $

What is $\int\frac{\sec(x+2)\tan(x+2)}{\sqrt{x+2}}dx$? I tried the $u=\sqrt{x+2}$ but just get $\int 2\sec(u^2)\tan(u^2)du$, which I am stuck on. Also tried $w=x+2$ which gives a similar problem, $\int\frac{\sec w\tan w}{\sqrt{w}}dw$. So anyone who…
Thomas
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5 answers

How to integrate with respect to $x^2$?

$$\int x\,dx^2$$ I'm having trouble comprehending this question. I can perform substitution, but when I do I come out wrong. Here's how I've happened to handle this: $\int x\,dx^2$ $u=x^2$ $du=2x\,dx$ $dx=\frac {du}{2x}$ $\frac12 \int…
PhilTheLawyer
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