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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Why is $\sqrt{x}$ a function?

How $\sqrt{x}$ can be a function when $\sqrt{4}$ is equal to $-2$ and $2$?
omidh
  • 876
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Why is indefinite integral called so?

Two questions that are greatly lingering on my mind: 1. Integral is all about area(as written in Wolfram). But what about indefinite integral? What is the integral about it?? Is it measuring area?? Nope. It is the collection of functions the…
user142971
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6 answers

Proving that $e^x>1+x^2$

I am trying to prove that $e^x>1+x^2$ for any $x>0$ for my homework assignment. However I have run into trouble doing this. I was trying to probe that $\ln {{e}^{x}}>\ln (1+{{x}^{2}})$ is true for $x>0$ and then that would mean that $e^x>1+x^2$ is…
Jason
  • 2,367
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Why is the antiderivative of $\frac{1}{1+x^2}=\tan^{-1}(x)$?

My textbook says the antiderivative of $\frac{1}{1+x^2}$ is $\tan^{-1}(x)$. To confirm this to myself I took the derivative of $\tan^{-1}(x)$ expecting to get $\frac{1}{1+x^2}$ , but instead I ended up with $-\frac{1}{\sin^2(x)}$. So why is…
Asker
  • 101
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Why does substitution work for a maclaurin series but not a taylor series?

Say I was trying to calculate the taylor expansion of $\sin(x^2)$ around $x = 0$. I could assume that $u = x^2$ and solve for taylor expansion around $x=0$ of $\sin(u)$. I would just need to substitute $x^2$ back in for $u$ when I am completed. I…
Corbin McNeill
  • 83
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Lipschitz implies bounded derivative?

Is there any function which is Lipschitz but has an unbounded derivative ? I guess there isn't but I have not found one.
Dave Nguyen
  • 438
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Question about substitution

[Note:] The error described here has been corrected in the meantime in response to this question. When checking wikipedia on substition they say that $\int f(g(t))g'(t) dt = \int f(x)dx$ with x = g(t). Which is true. However, when checking example 1…
Admar
  • 61
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How to prove that this function will converge to $1$?

I have an assignment for tomorrow that ask me to prove that the sequence/function $f(x) =\begin{cases}\dfrac x2\quad\quad\quad\text{if } x\text{ is even}\\3x+1\quad\;\text{if }x\text{ is odd}\end{cases}$ (where $x$ is a natural number) will converge…
Jerry Taylor
  • 71
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Area under a curve with polar coordinates

We have the function $r=\sin(3\theta)$, and I want to calcule the area under its curve. I think this is easy if we use a double integral. It should be: $3\displaystyle\int_{0}^{\pi/6}\int_{0}^{\sin(3\theta)}rdrd\theta$ Am I right? But now, I want to…
Kaze16
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Why is $e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$

While looking through an example in Carothers' Real Analysis, I came across the following: $$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$ and of course I noticed that it looks similar to $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$, so…
Sujaan Kunalan
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Solving a $1^\infty$ indeterminate form.

I'm preparing for my calculus exam and I can't solve this limit: $$\lim_{x\rightarrow\infty}\left(\frac{1+\tan(1/x)}{1-\tan(1/x)}\right)^x$$ The limit tends to $1^\infty$, which is indeterminate. I've tried several things and I couldn't solve…
Alejandro
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Root Mean Square of Sum of Sinusoids with Different Frequencies

In circuit analysis it is stated that the RMS (Root Mean Square) value of a waveform which consists of a sum of sinusoids of different frequencies, is equal to the square root of the sum of the squares of the RMS values of each sinusoid. Therefore,…
user137035
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2 answers

Converting a Riemann sum to an integral

Given this sum: $$ \frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n-1}$$ I am trying to convert (approximate) it to an integral. This is what I have so far: $$ \frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n-1}…
yotamoo
  • 2,753
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Show that one-sided limits always exist for a monotone function (on an interval)

Show that one-sided limits always exist for a monotone function on an interval $[a,b]$. Me attempt: 1) If a function is monotone on an interval $[a,b]$, then $f(a)\le f(x) \le f(b)$ for $x\in[a,b]$. Therefore if there exists left-hand (right-hand)…
luka5z
  • 6,359
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2 answers

How can you absolutely prove that a function has an infinite power series representation?

I am beginning to grasp onto the idea of a power series and the way that we can "hunt" for the constants in a maclaurin series expansion. However, what I don't yet understand is how we know that there is an infinite representation in the first…
louie mcconnell
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