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
4
votes
1 answer

Is there a solid algebraic motivation behind the derivative?

I work in my university's math help center and am often presented with questions rooted in poor conceptual or intuitive understanding understanding of various mathematical questions; esp. with the beginning calculus students I work with. From day…
Sawyer
  • 580
4
votes
3 answers

Limit related to $\zeta(x)$

I'm noticing some things: $$\lim_{n \to \infty} \left(\sum_{x=1}^{n} x^{-1/3}-\frac{3}{2}n^{2/3} \right)=\zeta(1/3)$$ Note $\int n^{-1/3} dn=\frac{3}{2}n^{2/3}+c$ $$\lim_{n \to \infty} \left(\sum_{x=1}^{n} x^{-1/2}-\frac{2}{1}n^{1/2}…
Ahmed S. Attaalla
  • 18,518
4
votes
1 answer

Related rates problem with bead sliding down a curve.

I keep getting the wrong answer on this problem. A bead slides down the curve $xy=10$. Find the bead's horizontal velocity at time $t=2$ if its height at time $t$ seconds is $y=400-16t^2$. I do…
Marshall Kendrick
  • 43
4
votes
5 answers

Is any finite sum of real numbers well-defined?

I am trying to find an argument or counterexample for the following proposition. Let $ (a_i)_{i=1}^{n} $ be a finite sequence of real numbers. Then, $ \sum \limits_{i=1}^{n}a_i $ is well defined. By "well-defined" I mean that the sum exists and…
Theorist Reflectionist
  • 488
4
votes
2 answers

If $g(x):=f(x, kx^m)$ is continuous at $0$, then $f(x,y)$ is continuous at $(0,0)$

If $g(x):=f(x, kx^m)$ is continuous at $0\;\;\;$ $\forall k\in R$, $\;\;\;\forall m\in N$, then $f(x,y)$ is continuous at $(0,0)$. I'm not quite sure what is meant here by at 0. This means $f(x, kx^m) = 0$? Seems there could be an issue in this…
Justin
  • 2,813
4
votes
2 answers

Is the determinant of the jacobian of a rotation matrix always equal to 1?

I saw on wikipedia that the determinant of a rotation matrix is always one (possibly by definition?), but it doesn't say anything about the determinant of the Jacobian of such a matrix. Since applying a rotation shouldn't change the integral of a…
user1736
  • 8,573
4
votes
2 answers

How does one show that $\lim_{n \rightarrow \infty}\int_{0}^{1}\frac{x^{n}}{1 + x^{n}}\, dx = 0$?

How does one show that $$\lim_{n \rightarrow \infty}\int_{0}^{1}\frac{x^{n}}{1 + x^{n}}\, dx = 0?$$ My idea is to evaluate the inner integral, but I can't seem to be able to do that.
ADF
  • 1,735
4
votes
1 answer

What's this symbol called?

What is this symbol called and how do you use it?
stuart
  • 513
  • 4
  • 10
4
votes
1 answer

Cancel partial derivative in fraction

The question is related to this one. Let $f(x,y) = xy$ and $g(x,y) = f(x,y)^2 = x^2y^2$. Now consider the fraction of partial derivatives \begin{align} \frac{\frac{\partial g}{\partial x}}{\frac{\partial f}{\partial x}} = \frac{2xy^2}{y} =…
clueless
  • 771
4
votes
2 answers

"$f$ is differentiable at $x_0$" implies...

Just making sure I understood: $$\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)$$ At a first glance I didn't understand why the above is true. It's because (in the case above) we can say that $x=x_0+\Delta x$; right?
Py42
  • 608
4
votes
3 answers

Applied Calculus Homework Question

The function $f(x) = 1-f(x-1)$, for positive interger, $x$. If $f(2) = 12$, compute $ f(2012)$
Ctrl
  • 99
4
votes
3 answers

Properties of a continuous function

I want to see if there exists an $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for every $\epsilon > 0,$ $$\int^{\epsilon}_{-2\epsilon} f(x) dx \geq 1.$$ This seemed like a difficult question, and so I went about playing with the properties of…
Sarah
  • 248
  • 1
  • 4
4
votes
7 answers

limit of $\lim_{x\to 0} \frac{(\sin x)^2 - (x \cos x) ^ 2}{x^4}$

I have question. I want to solve this limit. it's $\frac{0}{0}$ so we have to change it. there is two way with two different value. $\lim_{x\to 0} \frac{(\sin x)^2 - (x \cos x) ^ 2}{x^4}$ First way: before that we know that $\lim_{x\to 0}…
Amin
  • 595
4
votes
5 answers

Find a positive number $\delta$?

Find a positive number $\delta$ such that when $|x-1|<\delta$, then $|x^2-1| < 0.45$ Part 2: Find the LARGEST number $\lambda$ with the property that when $|x-1|< \lambda$, then $|x^2-1| < 0.45$ I don't understand this at all. Help?
Taylor
  • 53
4
votes
2 answers

smoothness of the functions in Hadamard lemma

For every smooth function $f\in C^\infty (\mathbb{R}^n)$ there are smooth functions $g_i$ such that $f(x)=f(0)+\Sigma x_ig_i(x).$ This is proved by defining $g_i(x) = \int_{0}^{1}\frac{\partial f}{\partial {x_i}}(tx)dt$. I think these $g_i$ are…
user53216
1 2 3
…
99 100