Mathematics Stack Exchange
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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.

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Area enclosed by polar curve

I can't get the text answer using standard method of integration of a polar equation. Yet when I use a symmetry method I do get their answer. Can you assist in clarification? Find the area of the region enclosed by $r=4cos(3 \theta)$. I use $ \frac…
user163862
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Proof of orthogonality of unit vectors in orthogonal curvilinear coordinate system

Context Suppose $\mathbf{R}(q_1,q_2,q_3)=\mathbf{r}(x,y,z)$ represents a position vector in physical space in a curvilinear coordinate system defined by $q_i=q_i(x,y,z)$ for $i=1,2,3$. The reverse mapping $x=x(q_1,q_2,q_3)$ (similarly for $y$ and…
Zxcvasdf
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Determining $\lim_{(x, y) \to (2y, y)} \exp(\frac{|x-2y|}{(x-2y)^2})$

Find the limit of $$\exp\left(\frac{|x-2y|}{(x-2y)^2}\right)$$ when $(x,y) \to (2y,y)$. I have considered two cases: $(x-2y)<0 $ and $(x-2y)>0$. But in first case the limit turns out to be $0$ and in the second case limit is undefined. I am not sure…
Kashmira
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Finding $\lim_{x\to\pi/2}\left(\frac{1-\sin x}{(\pi-2x)^4}\right)(\cos x)(8x^3 - \pi^3)$ using algebra of limits

Let $$ F(x) = \left(\frac{1-\sin x}{(\pi-2x)^4}\right)(\cos x)(8x^3 - \pi^3) $$ Then find the limit of $F(x)$ as $x$ tends to $\pi/2$. How can we find the limit using algebra of limits? The limit of $\dfrac{1-\sin x}{(\pi-2x)^4}$ as $x$ tends to…
Mathaddict
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Taylor series equation

Let $f(x)=\displaystyle \sum_{n=0}^{\infty}\frac{x^{3n}}{(3n)!}$ and $g(x)=\displaystyle \sum_{n=0}^{\infty}\frac{x^{3n+1}}{(3n+1)!}$ and $h(x)=\displaystyle \sum_{n=0}^{\infty}\frac{x^{3n+2}}{(3n+2)!}$ Show that…
i.a.m
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Weaker Conditions for Integral Test (for series)

My calculus textbook (as well as wikipedia and other online sources) list three conditions to verify before one can establish: $\sum\limits_{n=a}^\infty f(n)$ converges $\iff \int\limits_{a}^\infty f(x)dx$ converges: 1) $f(x)$…
Terence C
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Prove $\int^{\infty}_0e^{-ax}\frac{1-\cos x}{x^2}\mathrm{d}x=\arctan\frac{1}{a}-\frac{a}{2}\ln(a^2+1)$, where $a>0$

How do you prove $\int^{\infty}_0e^{-ax}\frac{1-\cos(x)}{x^2}\mathrm{d}x=\arctan\frac{1}{a}-\frac{a}{2}\ln(a^2+1)$, where $a>0$? I have no idea how to start.
koifish
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Why does $a_n = (1+\frac{2}{n})^{n}$ converge to $e^2$?

Determine whether $a_n = (1+\frac{2}{n})^{n}$ converges or diverges. If it converges, find the limit. So I tried to say that $a_n = (1+\frac{2}{n})^{n} \Rightarrow \ln(a_n) = n\ln(1+\frac{2}{n})$. Unfortunately I don't know what the next…
ghshtalt
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When do you know when you have to check for positive AND negative limits?

When I see a simple limit question like "find the limit as $x \rightarrow \infty$ when $f(x) =$ $3x+7 \over x+2$ I know that all I have to do is factor out x which makes $3 \over 1$ $\cdot$ $7 \over x$ / $2 \over x$, which both turn into 0 as I…
ming
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What is the PEMDAS for $e^{x^2}$

$e^{x^2}$ Is this read as $(e^x)^2$ or $e^{(x^2)}$ Why does one exponent take precedence over the other? Is it because of sequential ordering of the same operation?
JackOfAll
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I need assistance in integrating $ \frac{x \sin x}{1+(\cos x)^2}$

Find the integral $$ \int_0^{\pi} \frac{x \sin x}{1+(\cos x)^2}$$
Sabrina Ahmed
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Best undergraduate calculus book?

I'd like to know which are the best undergraduate calculus books for mathematicians. I'm looking for a complete and rigorous book that allows a Mathematics student to fully understand the undergraduate calculus courses. Is there a better book than…
Ludwig Varg
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Maximum value of area of rectangle

Tried attempting by using altitude and similarity of triangles, but the problem is that variables are not getting elliminated.
maveric
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What's the integration of $\int \sin^4 (x)dx$?

What's the integration of $\int \sin^4 x \,dx$? I don't see the approach to this question. I have a issue with this question as well: $$\int \sin x \cos x (\sin x+\cos x) \,dx.$$ I simplify this to $\sin^2 x \cos x + \sin x \cos^2 x$ and set $u=…
hah
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how to find $f(2)$

Hello im trying to solve a homework problem and i'm stumped. here is the problem. the question states suppose $f$ and $g$ are continuous functions such that $g(2)=6$ and $\lim_{x\to 2}$ $[3f(x)+f(x)g(x)]=36$ find $f(2)$ i have tried solving it by…
Miguel
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