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
6
votes
3 answers

Intuitive Understanding About the Implicit Function Theorem

In multivariable calculus, given a function like $F(x,y,z) =0$, the implicit function $z=f(x,y)$ exists if and only if $\frac{\partial F}{\partial z} \not = 0$. And the implicit function is given by $\frac{dz}{dy}=-\frac{\frac{\partial F}{\partial…
Strin
  • 1,539
6
votes
4 answers

How can I get unblocked on learning Calculus?

I love math and science. In fact paid for $1/2$ my college tuition by tutoring algebra and trigonometry. But when it came to calculus, I became blocked. I understand the concepts of speed and rate of change, but when I start seeing all the symbols…
Bertha
  • 61
6
votes
7 answers

The box has minimum surface area

Show that a rectangular prism (box) of given volume has minimum surface area if the box is a cube. Could you give me some hints what we are supposed to do?? $$$$ EDIT: Having found that for $z=\frac{V}{xy}$ the function $A_{\star}(x, y)=A(x, y,…
Mary Star
  • 13,956
6
votes
4 answers

L'Hospital's rule with a denominator that goes away

I am having trouble with a homework question that seems really simple, and I get the wrong answer. $$\lim_{x\to 0} \frac{\sqrt{1+2x}-\sqrt{1-4x}}{x}$$ I then get the derivative of the top and bottom and I get $$(1+2x)^{\frac{-1}{2}} -…
user138246
6
votes
3 answers

Is there an easier way of finding a Taylor series than just straight computing the formula?

Let's say the task is to find the Taylor series at the origin of the function $$f(x) = \frac{3x}{1-x-2x^2}$$ The formula is $T^n_0 =\sum^n_{k=0} \frac{f^{(k)}(0)}{k!}x^k$. If I follow this formula, I need to at least compute the 4th or 5th…
ElleryL
  • 1,583
6
votes
5 answers

Prove: $\int_0^{\frac{\pi}{2}}t(\frac{\sin nt}{\sin t})^4dt<\frac{\pi^2n^2}{4}$

I have a question about integral. Prove: $$\int_0^{\frac{\pi}{2}}t\left(\dfrac{\sin(nt)}{\sin(t)}\right)^4dt<\dfrac{\pi^2n^2}{4}$$ I have tried several methods including $\sin(t)\geq\frac{2t}{\pi}$, but I can't work it out.
89085731
  • 7,614
6
votes
2 answers

Prove $F(x)=\frac{1}{x}\int_0^xf(t)$ is convex

Assume $f(x)$ is convex in $[0,\infty)$, Prove $F(x)=\frac{1}{x}\int_0^xf(t)dt$ is convex
89085731
  • 7,614
6
votes
2 answers

Prove that $\varlimsup _{n\rightarrow \infty}(u_n)^{\frac{1}{n}}=1$ where $u_{n+1}=\frac{2u_n^3+2u_n^2+u_n}{2u_n^2+3u_n+1}$ and $u_0=1$.

Prove that $$\varlimsup_{n\rightarrow \infty} (u_n)^{\frac{1}{n}}=1,$$ where $u_0=1$ and $$u_{n+1}=\frac{2u_n^3+2u_n^2+u_n}{2u_n^2+3u_n+1}\;.$$
impartialmale
  • 826
6
votes
3 answers

Is there a "closed" form of sequence $u_{n+1}=\frac{u_n^2}{u_n+1}$

Let $u_0=1$ and $u_{n+1}=\frac{u_n^2}{u_n+1}, \forall n\in \mathbb{N}$. a) Find the formula of $u_n$? b) Calculate the limit $\displaystyle\varlimsup_{n\rightarrow \infty} (u_n)^{\frac{1}{n}}$.
impartialmale
  • 826
6
votes
2 answers

Series sum is approximately $b\log n$ implies the terms are approximately $b/n$?

Let's say I have a sequence $a_n \ge 0$ such that I know: $$b \log n - C \le \sum_{i=1}^n a_i \le b \log n + C$$ for some constants $b$ and $C$ larger than 0. How can I prove that: $$a_n = \frac{b}{n} + o(1)\ ?$$ This intuitively seems correct…
harmonic
  • 127
6
votes
2 answers

Find $\lim\limits_{t\to 0^+} \int_0^{\infty} \frac{t f(x)}{t^2+x^2} dx$

The question is: Let $f(x)$ be bounded and continuous on $[0,\infty)$. Let $\displaystyle F(t)=\int_0^{\infty} \frac{t f(x)}{t^2+x^2} dx$ for $t>0$. Find $\displaystyle \lim_{t\to 0^+} F(t)$. If I set $|f(x)|\leq M$, then I can obtain $|F(t)|\leq…
Sun
  • 95
  • 4
6
votes
2 answers

Improper integral: $\int_1^{+\infty}\frac{\mathrm dx}{x(x+1)(x+2)\cdots(x+n)}$

I've tried many ways, but it seems that it didn't work: $$ \int_1^{+\infty}\frac{\mathrm dx}{x(x+1)(x+2)\cdots(x+n)} = \int_0^1\frac{x^{n-1}}{(x+1)(2x+1)\cdots(nx+1)}\mathrm dx = \cdots $$ Any help would be appreciated!
Shine Mic
  • 835
  • 4
  • 16
6
votes
1 answer

Signing $y''$ from $\log(\frac{x+y}{x})=x+y$

Suppose that $x,y>0$ are positive reals such that $y$ is defined implicitly in terms of $x$ via: $$ \log\left(\frac{x+y}{x}\right)=x+y.\tag{$\star$} $$ I would like study the sign of $y''$. Attempt: Write ($\star$)…
yurnero
  • 10,505
6
votes
3 answers

Prove: $\lim _{x \to \infty}\sum_{1}^{\infty}\frac{x^2}{1+n^2x^2}=\sum_{1}^{\infty}\frac{1}{n^2}$

I want to ask you if can it be so simple to prove that $\lim _{x \to \infty}\sum_{1}^{\infty}\frac{x^2}{1+n^2x^2}=\sum_{1}^{\infty}\frac{1}{n^2}$ by divide the numerator and denominator with $x^2$ and that's it? If it this simple indeed you can…
Jozef
  • 7,100
6
votes
2 answers

basic calculus proof - using theorems to prove stuff

A function $f(x)$ is defined and continuous on the interval $[0,2]$ and $f(0)=f(2)$. Prove that the numbers $x,y$ on $[0,2]$ exist such that $y-x=1$ and $f(x) = f(y)$. I can already guess this is going to involve the intermediate value theorem.…
nofe
  • 1,015
1 2 3
…
99 100