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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Meaning of $f^n(x)$

I'm in what seems the first year of college (I'm in France so it doesn't have the same name) In calculus and other math classes I came across multiple meaning/definition for a single notation. I know the context of the exercise and teacher…
DrinRchd
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Show (via differentiation) $1-2+3-4+\cdots+(-1)^{n-1}n$ is $-\frac{n}{2}$ for $n$ even, $\frac{(n+1)}{2}$ for $n$ odd.

i) By considering $(1+x+x^2+\cdots+x^n)(1-x)$ show that, if $x\neq 1$, $$1+x+x^2+\cdots+x^n=\frac{(1-x^{n+1})}{1-x}$$ ii) By differentiating both sides and setting $x=-1$ show that $$1-2+3-4+\cdots+(-1)^{n-1}n$$ takes the value $-\frac{n}{2}$ if n…
salman
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Calculate Subtotal from Total and Percentage

If I know my tax rate, and I know the total, I'm looking for the subtotal. So if (subtotal + (subtotal * taxrate)) = total, and I have total and taxrate, how do I get subtotal?
jeremib
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Let $f$ be continuous on $[0,1]$. Find the limit $\lim_{n \rightarrow \infty}(n+1) \sum_{k=0}^n \int_0^1 x^k(1-x)^{n-k} f(x) \mathrm{d} x$.

$\int_0^1 x^k (1-x)^{n-k} dx$ is a Beta function. The probability density function of the Beta distribution is given by $f(x; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}$. Therefore, $x^k (1-x)^{n-k} = B(\alpha,…
Yuejin Xie
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Convergence of $(1+x^n)^{1/n}$

I have the sequence of functions $f_n = (1+x^n)^{1/n}$ for $0 \leq x < \infty$. I can easily see that (as $n$ approaches $\infty$) it is pointwise converging to $1$ for $x\leq 1$ and to $x$ for $x>1$. I'm trying to figure out whether or not it is…
NBP
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Limit of exponential integral $ \lim{t \rightarrow \infty}{(te^t\int_{t}^{\infty}\frac{e^{-s}}{s}ds)} $

Evaluate the following limit: \begin{equation} \lim\limits_{t \rightarrow \infty}{(te^t\int\limits_{t}^{\infty}\frac{e^{-s}}{s}ds) := I} \end{equation} That's the Exponential integral, which is not an elementary function. Considering $…
user85663
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Prove that $f(x)=\sqrt{x(1-x)}$ is continuous on $(0,1)$

Attempt: By definition, it is to prove $ \forall a \in (0,1),\exists \delta>0, s.t.\lvert x-a \rvert < \delta$ implies $\lvert \sqrt{x(1-x)} - \sqrt{a(1-a)} \rvert < \epsilon$, which is equivalent to $ \frac{\lvert x-a \rvert \lvert 1-(x+a)…
aphrodite
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How to evaluate $\int \cos{x}/\cos{x}\,dx$ without cancelling?

In my (mostly physics based) education, when confronted with an integral like the following: $$\int \frac{\cos{x}}{\cos{x}}\,dx$$ I've typically been taught to cancel functions that appear in both the numerator and denominator, so that it can be…
Cdizzle
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If $y=x^{2}$ then what is $\frac{d}{d y}\left(\frac{d y}{d x}\right)$?

Here's what I did: $\begin{array}{l} y=x^{2} \\ \Rightarrow \frac{d y}{d x}=2 x \end{array}$ and we know that $x=\pm \sqrt{y}$, let's take the $+$ for the sake of the example. so $\frac{d y}{d x}=2 \sqrt{y}$ so $\begin{array}{c} \frac{d}{d…
elie makdissi
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Continuity and arc length

I have recently looked back on the formula for arc length $$\int_a^b \sqrt{1+f'(x)^2}\ dx .$$ In the case of a circle we have the interval $[-r,r]$ and can rearrange $ r^2 = x^2 + y^2 $ to $y=\sqrt{r^2-x^2}$. Since we consider the principal square…
Adam S
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Related rates problem

I'm learning single variable calculus; I finished a section on related rates several weeks ago. I'm sure the novelty of related rates and simple optimization problems will wear off eventually, but right now I'm having a lot of fun solving these…
Ryan
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Is $f(x)=\sqrt{9x^2 +173x+900}- \sqrt{9x^2 +77x+900}$ increasing or decreasing for $x \in \mathbb R^+$?

What is an effective way to find that if the function, $f(x)=\sqrt{9x^2 +173x+900}- \sqrt{9x^2 +77x+900}$ is increasing or decreasing for $x \in R^+$ I was asked to find the range of $f(x)$ for $x \in \mathbb R^+$. I could find the limit of f(x)…
q123LsaB
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Integral condition implies derivative greater than $4$

Let $f:[0,1]\to \mathbb{R}$ be a differentiable function such that $\displaystyle \int \limits _0^1f(t)\,dt=1$, $f(0)=0$, $f(1)=0$. Prove that there exists an $x_0\in (0,1)$ such $|f'(x_0)|\geq 4$. I'm trying to use mean value theorem on this but…
q123LsaB
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What does $\mathrm d^2 x$ exactly mean?

I am learning radiometry and one of the equation is radiance which is given as the radiant flux per unit projected area per unit solid angle. In equation: $$L = {d^2\Phi \over {cos(\theta)dAd\omega}} (eq. 1)$$ Now further in the book I read they use…
Marc Ourens
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Limit question using L'Hospital rule

Here is the limit I am trying to do $$ \lim\limits_{x \to \infty} \frac{x^2 + \mathrm{e}^{4x}}{2x- \mathrm{e}^x} $$ Now, here first, I am trying to identify the indeterminate form so that I can use L'Hospital's rule. Numerator tends to $\infty$ as…
user9026
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