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.

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Calculating arc length of a curve, stuck on dy/dx part (algebra mostly)

The equation is: $$x=\frac{1}{8}y^4 + \frac{1}{4}y^{-2},\qquad 1\leq y\leq 2.$$ I have the formula. I'm not sure how to write it out but this is what it says: Length is equal to the integral (with $b$ and $a$ for limits) of the square root of…
Ryan
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Proving one function is greater than another

How can I prove $f(x)$ $>$ $g(x)$ for all $x > 0$ given $f(x) = (x+1)^{2}$ and $g(x) = 4qx$ where $q$ is a constant in $(0, 1)$? My approach was to show that $(x+1)^2 > 4qx$ for the interval endpoints, e.g. $q=0$ and $q=1$. E.g. $(x+1)^2 \geq 4x$…
Vanush Vee
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What is the limit of the multidimensional integral?

What is the limit of the integral $$\int_{[0,1]^n}\frac{x_1^5+x_2^5 + \cdots +x_n^5}{x_1^4+x_2^4 + \cdots +x_n^4} \, dx_1 \, dx_2 \cdots dx_n$$ as $n \to \infty ?$
user64494
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Find the equation of the tangent line to the polar curve at given $(x,y)$.

Find the equation(s) of the tangent line(s) to the curve given $(x,y)$ point. $$r=1-2\sin(\theta )$$ at $(0,0)$. I am not sure how to go about find the the tangent line. Do I need to convert from polar to rectangular? Thanks!
EhBabay
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If $f$ and $g$ are continuous and for every $q\in \mathbb{Q}$ we have $f(q)=g(q)$, then $f(x)=g(x)$ for every $x\in \mathbb{R}$

Possible Duplicate: Can there be two distinct, continuous functions that are equal at all rationals? Hello guys, Let $f$ and $g$ be continuous functions, $f,g:\mathbb{R} \to \mathbb{R}$, such that for every $q\in \mathbb{Q}$ we have…
user6163
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A geometric/intuitive proof of why shortest distance between two non-intersecting curves lie along the normal

I saw this post but it was too technical for me. I am looking for proof similar to what is shown in this Quora post by Nicholas Halderman. The part I found confusing/ hard of the proof was where he proves the lemma:'a small displacement in this…
tryst with freedom
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When the polynomial $mx^3-nx^2+5x-1$ is divided by $x+2$ the remainder is $-39$?

When the polynomial $mx^3-nx^2+5x-1$ is divided by $x+2$ the remainder is $-39$. When the polynomial is divided by $x-1$ the remainder is $3$. Find the values of $m$ and $n$. My attempt?: I feel I did something incorrect $$-8m -n -10 - 1 = -39\\ m…
user73122
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A bound on a sum of five sines

If $0
sonia
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I wrote a proof of L'Hospital's rule, am I right?

Suppose $f$ and $g$ are differentiable and $g'(x)\neq 0$ on an open interval $I$ that contains $a$. Suppose that $$\lim_{x\to a}f(x)=L=\lim_{x\to a}g(x)\quad L=0\,or\,\infty\,or-\infty$$, then $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to…
Yang
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Is the sum of two functions, one of which is non differentiable, necessarily non-differentiable?

Suppose $f(x) = g(x) + h(x)$ where $g(x)$ is known to be non-differentiable. When $h(x) \ne -g(x)$ is some other function (differentiable or not), can $f(x)$ ever be differentiable? We can assume $f, g, h$ are real valued functions. edit: I should…
ijems
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How many roots has the following equation?

I would please like your guidance to find, depending on $a$, how many roots the following equation has: $x^4+4x+a=0$, where $a$ is a real parameter. I tried to use Bolzano's Theorem for a specific interval but I am afraid that my approach would not…
Vasilis
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Nonzero derivative implies function is strictly increasing or decreasing on some interval

Let $f$ be a differentiable function on open interval $(a,b)$. Suppose $f'(x)$ is not identically zero. Show that there exists an subinterval $(c,d)$ such that $f(x)$ is strictly increasing or strictly decreasing on $(c,d)$. How to prove this? I…
XLDD
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Prove or disprove : there exist at most two root of $f(x)=f'(x)$.

Let $f(x)$ be differentiable over $[0,+\infty)$, and $f'(x)$ be increasing and convex over $[0,+\infty).$ If $f(0)=f'(0)=0$, then there exist at most two roots of $f(x)=f'(x)$ over $[0,+\infty).$ Apparently, $x=0$ is already a root, hence we…
mengdie1982
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when is the particle speeding up and when is it slowing down

Based on this graph i have to figuere out when the particle is speeding up and when it is slowing down. My understanding is that when velocity and accelaration have the same sign then we are speeding up. however if velocity and accelaration have…
Miguel
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$\int \frac{dx}{x\log(x)}$

I think I'm having a bad day. I was just trying to use integration by parts on the following integral: $$\int \frac{dx}{x\log(x)}$$ Which yields $$\int \frac{dx}{x\log(x)} = 1 + \int \frac{dx}{x\log(x)}$$ Now, if I were to subtract $$\int…
providence
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