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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$\lim \limits_{h\to 0} \frac{e^h-1}{h}=1$ and the relation with $\sin'(x)=\cos(x)$

$$e^x=\sum _{n=0}^{\infty } \frac{x^n}{n!}$$ $$\begin{align*}\lim_{h\to 0} \frac{ e^{(x+h)}-e^x}{h}=e^x\color{blue}{\lim_{h\to 0} \frac{e^h-1}{h}}=e^x\tag{1}\end{align*}$$ How obvious about the following limit? I think it is obvious when using…
HyperGroups
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Average distance of 2 points on distinct lines

First, I need to find the function that finds the average chord length between $(t,0)$ and $(x,1)$ Then I can integrate this function to get the average distance of 2 points on opposite sides of a unit square The distance between $(t,0)$ and $(x,1)$…
Knocker379
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How to evaluate $\int_{0}^{1}\int_{0}^{1}\frac{(x+y+xy)\log{(x+y+xy)}}{x+y}dxdy$?

I am trying to evaluate this integral: $$\int_{0}^{1}\int_{0}^{1}\frac{(x+y+xy)\log{(x+y+xy)}}{x+y}dxdy.$$ It looks like this integral is symmetry with variables $x$ and $y$ but I can't find the way to exploit it. I tried to use…
OnTheWay
  • 2,669
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Find y'' using implicit differentiation

Pretty sure I messed up $$ x^3 - 3xy + y^3 = 1 $$ $$ 3x^2-3xy'-3y+3y^2y'=0 $$ $$ 3y^2y'-3xy'=3y-3x^2 $$ $$ (3y^2-3x)y'=3y-3x^2 $$ $$ y' = \dfrac{y-x^2}{y^2-x} $$ $$ y'' = \dfrac{(y^2-x)(y'-2x)-(y-x^2)(2yy'-1)}{(y^2-x)^2} $$ $$ y'' =…
James
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Limits of Integration in Trig Substitution

Lets say you must perform a simple integral with trigonometric substitution - we'll choose $\int{\sqrt{1-x^2}dx}$ from $0$ to $17$. Now we use $x = \sin(t)$ for our substitution. Here's my problem: the limits of integration for $x$ range from $0$…
John Machop
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Prove the exsistence of 3 zero points of a function

Assuming $a
Syrtis Major
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Evaluate $\lim_{x \to 0} \frac{\arccos(\frac{1-x²}{1+x²})}{\arcsin(x)}$

Evaluate: $$\lim_{x \to 0} \frac{\arccos(\frac{1-x²}{1+x²})}{\arcsin(x)}$$ I found this in a calculus book and I just can't seem to get it. I see that if we substitute $x$ for $\tan(y)$, we get $2y$ in the numerator, but then what do we do with…
Nick Larry
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How to write the equation of the function, involving a sine or cosine, whose graph is shown here

Write the equation of the function, involving a sine or cosine, whose graph is shown here. I think I need to use this form: f(t) = A cos(Bx) + C I found the amplitude (1) because that's the distance from the axis. I found the period (2) because…
marksawnson
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Why is $x\sin(1/x)$ discontinuous at $x=0$?

Why is $x \sin (\frac{1}{x})$ discontinuous at $x = 0$? I know that $\sin(\frac{1}{x})$ diverges at $x = 0$ but $\sin(\frac{1}{x})$ is between $-1$ and $1$ so isn't $x \sin(\frac{1}{x})=0$ at $x=0$?
Won
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calculate the derivative using fundamental theorem of calculus

This is a GRE prep question: What's the derivative of $f(x)=\int_x^0 \frac{\cos xt}{t}\mathrm{d}t$? The answer is $\frac{1}{x}[1-2\cos x^2]$. I guess this has something to do with the first fundamental theorem of calculus but I'm not sure how to use…
Kuai
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Calculating this double integral in polar coordinates

Calculate the double integral $\iint_D {(1+x^2 + y^2)ln(1+x^2+y^2)dxdy} $ where $D = \{(x,y) \in \mathbb R^2 | \frac{x}{\sqrt3} \leq y \leq x , x^2 + y^2 \leq 4\}$. I heard there is a way called Polar Coordinates but the more I looked and read…
TheNotMe
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How find this value $\frac{a^2+b^2-c^2}{2ab}+\frac{a^2+c^2-b^2}{2ac}+\frac{b^2+c^2-a^2}{2bc}$

let $a,b,c$ such that $$\left(\dfrac{a^2+b^2-c^2}{2ab}\right)^2+\left(\dfrac{b^2+c^2-a^2}{2bc}\right)^2+\left(\dfrac{a^2+c^2-b^2}{2ac}\right)^2=3,$$ find the value $$\dfrac{a^2+b^2-c^2}{2ab}+\dfrac{a^2+c^2-b^2}{2ac}+\dfrac{b^2+c^2-a^2}{2bc}$$ is…
math110
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$f(x+1)=f(x)+1$ for monotone continuous $f$ defined on $\Bbb R$, $g_n(x)=f^n(x)-x$, show $\lim_{n\to\infty}\frac{g_n(x)}{n}$ is independent of $x$.

$f(x+1)=f(x)+1$ for monotone continuous $f$ defined on $\Bbb R$, $g_n(x)=f^n(x)-x$, where $f^n(x)=\underbrace{f\circ \cdots\circ f}_n(x)$. Clearly, $g_n(x)$ is periodice with period $1$. How to show $\lim_{n\to\infty}\frac{g_n(x)}{n}$ exists and is…
xldd
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Finding a path where the function increases the most rapidly

Consider the differentiable function $z = f(x,y)$. Construct a path $\gamma: \mathbb{R} \to \mathbb{R}^3$ on the graph of $z=f(x,y)$ of $\textbf{ steepest ascent}$ that starts at some point $(x_0,y_0)$ in the graph of $z=f(x,y)$ thoughts: I know…
ILoveMath
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Integration of trig - reduction formula $\int \cos^4 x\; dx$

I am not sure where I am going wrong but this problem gets more complicated $$\int \cos^4 x\;dx $$ $u =\cos^3 x$ $du = -3\cos^2 x \sin x \;dx$ $dv = \cos x\;dx$ $v = \sin x$ $$\sin x \cos^3 x + 3\int \sin^2x \cos^2 x \;dx$$ Now I have two squared…
Dan the man
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