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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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Show that $P_{n}(x)\rightrightarrows 0\qquad (x\in[0,1])$

Let $P_{0}(x)=\sqrt{x}$, $P_{n+1}(x)=\frac{1}{2}P_{n}^{2}(x)+(1-\sqrt{x})P_{n}(x)$,Prove that $$ P_{n}(x)\rightrightarrows 0\qquad (x\in[0,1]) $$(the double arrows means Uniform convergence) my idea:it can deduce that $$ P_{n+1}(x)\leq…
pxchg1200
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How to prove that a conditionally convergent series can be rearranged to sum to any real number?

There is a theorem of Riemann to that effect. How to prove it? Note: This was asked by Kenny in the beta for "calculus".
user218
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Lipschitz at every point in $[a,b]$ but not Lipschitz on $[a,b]$?

The question is: Find a function on $[a,b]$ which is Lipschitz at every point of $[a,b]$, but not Lipschitz on $[a,b]$. I am having a hard time picturing this, and thus, having a hard time finding such a function... I have not learned a lot about…
MathMathCookie
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Does a differentiable everywhere function have a continuous derivative?

If a function $f$ defined on $[a,b]$ and differentiable everywhere, does it mean that its derivative $f'(x)$ is continuous everywhere on $[a,b]$? My understanding is 'yes'. Because if there were 'a gap' in $f'(x)$, we could integrate back to $f(x)$…
mosceo
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there exist two antipodal points on the equator that have the same temperature.

Argue that there exist at any time two antipodal points on the equator that have the same temperature. The temperature function can be assumed to be continuous. I am supposed to use the mean value theorem, but I don't see how. Thanks in advance.
kristof2014
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Epsilon-Delta Confusion

I don't understand the epsilon delta definition of a limit "According to the formal definition above, a limit statement is correct if and only if confining x to d units of c will inevitably confine f(x) to epsilon units of L." So, if we can…
nomlylni
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What exactly does $\frac{dx}{dy}$ mean?

I asked 3 professors at my university and none gave me a clear cut answer, but instead merely told me qualities of this notation. Here is what I understand so far from what they told me: 1)Treat the top variable as as variable when finding the…
chopper draw lion4
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Horse and snail problem.

A horse has a rubber band attached to it which can expand infinitely and is tied to a pole on the other end. At first the length of the rubber band is $l$. on the pole-side of the rubber band there is a snail. If both start walking at the same time:…
Asinomás
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Does the integral $\int_{1}^{\infty} \sin(x\log x) \,\mathrm{d}x$ converge?

I tried a couple of substitutions but have so far gotten nowhere. Can anyone guide me in the right direction? Thanks for your time.
Jack Smith
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Prove that $|f'(x)| \le \frac{A}2 \forall x \in [0,1] $

Let $f$ be twice differentiable in $[0,1]$ $f(0) = f(1) = 0$, $|f''(x)|\le A$. Prove that $|f'(x)| \le \frac{A}2, \forall x \in [0,1] $. Well this is what I came up with, $f'(c_1) = \dfrac{f(x) -f(0)}{x-0} = \dfrac{f(x)}{x}$ $f'(c_2) =…
guynaa
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Question about an integrable singularity

I've been trying to understand a concept of an integrable singularity. So far what I have discovered was that the case of the singularity occurs when there is a point where the integration becomes infinite. Am I right? Also, is there anything else…
GKED
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Is xy concave or convex or neither in strict positive orthant.

I understand that $f(x,y)=xy$ has a saddle point at $(0,0)$ and neither concave nor convex over the entire $\mathbb{R}^2$. But is that true as well when it's restricted to $\mathbb{R}^2_{++}$ (strictly positive orthant)? Thanks.
firemind
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Evaluating the Integral: $\int_0^\infty \frac{( \frac{1}{2} - \cos x )}{x} dx$

Evaluating the Integral: $\int_0^\infty\left[\frac{1}{2} - \cos\left(x\right)\right]\,{\rm dx \over x}$ I came upon this limit: $\lim_{x\rightarrow\infty} -Ci(x) + Ci(1/x) +\ln(x)$, is it $\gamma$ ? Here $ Ci(x) = \gamma + \ln x + \int_0^x…
Alan
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What is the meaning of $\frac{0}{0}$?

I asked my teacher what is the real meaning of $\cfrac{0}{0}$, and the answer I got was "nobody knows". I can't leave this subject "as is". I need a decent explanation, at least an explanation to why "nobody knows". I'm sure you'll come up with a…
yoniyes
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Evaluation of a series

Some hints to start the evaluation of this series? $$\sum_{k=0}^\infty \dfrac{2^{2k}(k !)^2}{(2k)!(2k+1)^2}\left(\dfrac{1}3-\dfrac{1}{4^{k+1}}\right)$$
Crazy_girl
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