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.

134529 questions
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Is there a fundamental theorem of calculus for improper integrals?

Let $f\,\,$ be a continuous function on $[a,\infty)$ such that $\int_a^\infty f(t)\,dt$ converges. Define the function $F\,$ on $[a,\infty)$ with $$F(x) := -\int_x^\infty f(t)\,dt \qquad\text{for all}\quad x\in[a,\infty).$$ Can we somehow deduce…
user18516
27
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4 answers

Can I ever go wrong if I keep thinking of derivatives as ratios?

I have been forewarned about it, I have read the answers here, but I haven't seen a counter example where it doesn't work. I know that it isnt really a fraction, but does it effectively get the same result in all cases, or are there counterexamples…
kuch nahi
  • 6,789
27
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4 answers

What is the difference between writing $f$ and $f(x)$?

I see a lot of professors in my calculus courses using $f$ and $f(x)$ in a way that looks interchangeable. Sometimes it drives me crazy because I always thought of them as being different. ($f$ means an independent variable, $f(x)$ means a variable…
Mohamed Moustafa
  • 381
27
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6 answers

Most general $A \subseteq \mathbb R$ to define derivative of $f: A \to \mathbb R$?

I was looking up the definition of the derivative in several books, and what was making me uneasy was the first sentence, generally along the lines of "let $f$ be defined on...". They don't seem to be able to agree on what $f$ should be defined on.…
user33661
27
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4 answers

how to prove the chain rule?

I have just learned about the chain rule but my book doesn't mention the proof. I tried to write a proof myself but can't write it. So can someone please tell me about the proof for the chain rule in elementary terms because I have just started…
user210387
27
votes
2 answers

Question about a rotating cube?

I read an article not too long ago that posed the following problem: What is the volume of the solid of revolution created by spinning a unit cube about an axis joining two opposing vertices? So the shape generated will be two cones and a…
Hautdesert
  • 1,606
23
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8 answers

Why is the number $e$ so important in mathematics?

I've heard a lot about this number $e$. Why is it so important? How does it fit into the 'bigger picture' of mathematics? How is it calculated and used?
Gordon Gustafson
  • 3,365
23
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8 answers

Maclaurin expansion of $\arcsin x$

I'm trying to find the first five terms of the Maclaurin expansion of $\arcsin x$, possibly using the fact that $$\arcsin x = \int_0^x \frac{dt}{(1-t^2)^{1/2}}.$$ I can only see that I can interchange differentiation and integration but not sure how…
mary
  • 2,374
23
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2 answers

How did Ramanujan get this result?

We know Ramanujan got this result $$\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots }}}=3$$ and he used the formula $$x+n+a=\sqrt{ax+{{(n+a)}^{2}}+x\sqrt{a(x+n)+{{(n+a)}^{2}}+(x+n)\sqrt{\cdots }}}$$ where $x=2,n=1,a=0$ ,we get the first result, but I don't know…
Daoyi Peng
  • 479
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5 answers

Proof of L'Hospitals Rule

No matter where I would look it would seem that L'Hospital's Rule has a strange proof-given that they teach it in high school, it seems troublesome that I can't find a solid proof at that level of knowledge. Does anyone have a proof that is fairly…
user82004
22
votes
4 answers

What is the difference between square of sum and sum of square?

What is difference between square of sum $(\sum_{i=1}^{n}x_i)^2$ and sum of square $\sum_{i=1}^{n}x_i^2$? I think square of sum is bigger than sum of square but i can not find a relation between them. I mean:…
achmed
  • 221
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1 answer

Prove that $\int_0^x f^3 \le \left(\int_0^x f\right)^2$

This problem comes from Calculus by Spivak, namely in Chapter 14- "The Fundamental Theorem of Calculus". Suppose that $f$ is a differentiable function with $f(0)=0$ and $0
Arpon
  • 1,246
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Partial derivatives inverse question

If $ \partial u/\partial v=a $, then $ \partial v/\partial u=1/a$?
pioneer
  • 388
22
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4 answers

Is there an epsilon-delta definition of the second derivative?

Is there an epsilon-delta definition for the second derivative? I know that there is such a definition for the first derivate $f'(x)$ which can be derived from the limit $f'(x) = \lim_{y\rightarrow x} \frac{f(y)-f(x)}{y-x}$ for a function…
Stephan Kulla
  • 7,091
22
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3 answers

Trig substitution; why can we ignore the absolute value?

If we have to integrate $$f(x)=\frac{x}{\sqrt{1-x^2}}$$ and we substitute $x=\sin \theta$ then we eventually have to take the square root of $\cos^2x$ which is equal to $|\cos x|$. But in my textbook and class lectures, we simply remove the…
Jason
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