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
8
votes
6 answers

Product rule intuition

Can anyone think of an intuitive explanation of the product rule? I'm not asking for a derivation. To me it seems like quite an un-untuitive result, as apposed to the chain-rule (which is ironically harder to derive).
Daniel
  • 1,015
8
votes
1 answer

Generating an independent set

Suppose $f_1, f_2,...$ are a set of functions $\mathbb{R} \rightarrow \mathbb{R}$ so that each is the power of some non-constant function h. So $ f_ i=h^{n_i}$ for some natural number n. Is it possible for such a set to be linearly dependent? What…
Noteventhetutorknows
  • 696
8
votes
7 answers

Compute $ \int_{-\infty}^{\infty} \frac{x^2}{(1+x^2)^2} dx$

Compute $$ \int_{-\infty}^{\infty} \frac{x^2}{(1+x^2)^2} dx$$ Of course we have $$ \int_{-\infty}^{\infty} \frac{x^2}{(1+x^2)^2} dx = 2 \int_{0}^{\infty} \frac{x^2}{(1+x^2)^2} dx = 2 \int_{0}^{\infty} \left( \frac{x}{1+x^2} \right) ^2 dx = \lim_{…
Thomas
  • 2,556
8
votes
2 answers

Compute $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n^2 + k}{n^3 + k}$

Compute $$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n^2 + k}{n^3 + k}$$ I tried: $$ 1 \xleftarrow{n \to \infty} n \cdot \frac{n^2+1}{n^3 + n} \le \sum_{k=1}^{n} \frac{n^2 + k}{n^3 + k} \le n \cdot \frac{n^2 + n}{n^3 +1} \xrightarrow{n \to \infty}…
Thomas
  • 2,556
8
votes
3 answers

The negation of a limit condition, Spivak style.

I got stuck on this bit from Spivak's Calculus: When proving that $f$ does not approach $l$ at $a$, be sure to negate the definition correctly: If it is not true that for every $\varepsilon>0$ there is some $\delta>0$ such that, for all $x$, if…
bryanj
  • 3,938
8
votes
1 answer

Evaluating $\int \frac{1}{2+3 \sin\left(x\right)}$

I'm helping my daughter on her calculus homework and it has been many years for me. The problem is $$ \int \frac{1}{2+3 \sin\left(x\right)} dx $$ From WolframAlpha, the substitution should be $u = \tan\frac{x}{2}$. Once you have the substitution,…
user90855
  • 181
8
votes
2 answers

The intermediate value theorem implies a point at which function changes sign.

Let $f$ be continuous in $[a,b]$ and $f(a)<0,\ f(b)>0$. Then by the intermediate value theorem, $\exists c\in(a,b)$ such that $f(c)=0$. I was wondering if it is also true that there exists $d\in(a,b)$ such that $f(d)=0$ and $f$ changes sign of the…
user824904
8
votes
2 answers

Why do people say the Fundamental Theorem of Calculus is so amazing?

The fundamental theorem of calculus in the fifth edition of Stewarts Calculus text is stated as: Suppose $f$ is continuous on $[a,b]$. Then: 1). If $g(x) = \int_a^x f(t)dt$, then $g'(x)=f(x)$ 2). $\int_a^bf(x)dx=F(b)-F(a)$ where $F$ is any…
user637978
8
votes
5 answers

how to find x in $ax + e^x = k$

in my project i have faced with a formula that I can't solve it. a very simplified and basic version of that equation can be rewritten as $ax + e^x = k$. please help me to solve this elementary calculus equation.
Mahmoud Hasanloo
  • 81
8
votes
2 answers

Seeking formal explanation of definite integral over infinitesimal interval

Why does : $$\frac{\displaystyle\int_t^{t+h}f(s) \, ds}{h} = f(t)\text{ as }h \rightarrow 0\text{ ?}$$ Intuitively this makes sense, because the value of the integral is infinitesimally close to $f(t)$, you have $h$ of them, and you divide the…
tmakino
  • 697
8
votes
4 answers

Fun calculus problem I can't seem to solve

I've recently picked up a math book I haven't read since college (highly recommended reading by the way!). I was reviewing multi-dimentional derivatives and such, and I stumbled upon a problem I've been trying to solve for two days, and I can't get…
Phonon
  • 4,028
8
votes
2 answers

Integration of $\displaystyle \int\frac{1}{1+x^8}\,dx$

Compute the indefinite integral $$ \int\frac{1}{1+x^8}\,dx $$ My Attempt: First we will factor $1+x^8$ $$ \begin{align} 1+x^8 &= 1^2+(x^4)^2+2x^4-2x^4\\ &= (1+x^4)^2-(\sqrt{2}x^2)^2\\ &= (x^4+\sqrt{2}x^2+1)(x^4-\sqrt{2}x^2+1) \end{align} $$ Then…
juantheron
  • 53,015
8
votes
2 answers

Monotonicity of functions

Let $f(x) = xe^{x^2} + e^{-x^2}$ I'd like to prove that this function increases monotonically in the interval $(0,1)$. I was able to do it by taking the derivative and proving it was greater than zero. (though it took me quite some time to do…
Quark
  • 1,021
8
votes
4 answers

How do I evaluate $\lim_{n \rightarrow \infty} \frac{1\cdot2+2\cdot3 +3\cdot4 +4\cdot5+\ldots}{n^3}$

How to find this limit : $$\lim_{n \rightarrow \infty} \frac{1\cdot2+2\cdot3 +3\cdot4 +4\cdot5+\ldots+n(n+1)}{n^3}$$ As, if we look this limit problem viz. $\lim_{x \rightarrow \infty} \frac{1+2+3+\ldots+n}{n^2}$ then we take the sum of numerator…
Sachin
  • 9,896
  • 16
  • 91
  • 182
8
votes
8 answers

Is one of the reasons we know $ y = e^{2x}$ is always positive that it is a square?

One of my Calculus assignments asks how we can tell that $ y = e^{2x}$ is always positive. The options given are: It's an exponential function with a base $> 0$. It's a square. Its derivative is always positive. I said the only reason was 1. But I…
mowwwalker
  • 1,201
1 2 3
…
99 100