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
2 answers

proving continuity with monotonic functions

Let $f:\left(0,\infty\right)\longrightarrow\mathbb{R}$ be a monotonically increasing function. Let $g:\left(0,\infty\right)\longrightarrow\mathbb{R}$ , $ g\left(x\right)=\frac{f\left(x\right)}{x}$ is a monotonically decreasing function. How…
sony jimbo
  • 305
4
votes
1 answer

For what values $p,q$ does the improper integral $\int_0^1 x^p (1-x^2)^q dx$ converge?

Question: For what values $p,q$ does the improper integral $\int_0^1 x^p (1-x^2)^q dx$ converge? I am struggling as I'm not sure where to start. What is the best way to approach this question?
mathjacks
  • 3,624
4
votes
1 answer

Find $\sum _0^{\infty} \frac{1}{2^nn!}$

Find $$\sum_{n=0}^{\infty} \frac{1}{2^nn!}$$ I know it's less than $$2 = \sum_{n=0}^{\infty} \frac{1}{2^n}$$
Ilianna Chavez
  • 41
4
votes
1 answer

Stupidly simple calculus problem with unexpected answer...

Here is a simple calculus problem: The width of a rectangle is increasing at a rate of $\frac{1 \:cm} {min}$ and the height is decreasing at the rate $\frac{2 \:cm} {min}$. What is the rate of change of the area of a rectangle when it's width is…
user175755
  • 163
4
votes
3 answers

Prove that $\lim _{n\to \infty \:}\left(\sqrt{n^2+3}-\sqrt{n^2+1}\right)=0$

Prove that $\lim _{n\to \infty \:}\left(\sqrt{n^2+3}-\sqrt{n^2+1}\right)=0$. I’m new to the subject and the square roots are throwing me a bit off.
Mick
  • 41
4
votes
2 answers

The limit of a difference between a cube root and square root

I've been trying to evaluate the limit of $$\lim_{x \to \infty}[(x^3+x^2+1)^{1/3}-(x^2+x)^{1/2}]$$ I've tried using the property $$x^6-y^6 =(x-y)(x^5+x^4y+\cdots+xy^4+y^5)$$ where $x = (x^3+x^2+1)^{1/3}$ and $y=(x^2+x)^{1/2}$. But I can't seem to…
user211337
  • 601
4
votes
1 answer

Prove:$f(x)$ is monotonically increasing.

Prove:$f(x)=x \left(1-{2x \over \pi}\right) \tan x$ is monotonically increasing in$[0,{\pi \over 2})$. I find $f'(x)=\left(1-{4 \over \pi}x \right)\tan x+\left(1-{2 \over \pi}x \right){x \over \cos^2 x}$,how to prove $f'(x)>0$ in $[0,{\pi \over…
GEE20151011
  • 376
  • 1
  • 10
4
votes
1 answer

Relate Rates with Circle

Two runners at the same point begin running in opposite directions along a circular track of radius $100$m at a speed of $5$m/s. At what rate is the (shortest) distance between them growing after $10$sec?
Bryce
  • 171
4
votes
5 answers

Finding the limit of $\frac{1}{t\sqrt{1+t}} - \frac{1}{t}$ as $t$ tends to $0$

$$\lim_{t\rightarrow 0}\left(\frac{1}{t\sqrt{1+t}} - \frac{1}{t}\right)$$ I attemped to combine the two fraction and multiply by the conjugate and I ended up with: $$\frac{t^2-t^2\sqrt{1+t}}{t^3+{t\sqrt{1+t}({t\sqrt1+t})}}$$ I couldn't really work…
user138246
3
votes
3 answers

A formula for second derivative

I want to prove that if $f''$ is continuous at $x_0$, then $$f''(x_0)=\displaystyle\lim_{h\to 0}\dfrac{f(x_0+2h)+f(x_0)-2f(x_0+h)}{h^2}$$ Any hint to prove it? I can't use l'Hopital or Taylor. Thanks.
Surtan
  • 249
3
votes
2 answers

indefinite integral and inequality

Let $$ f(x) \leq g(x), \forall\, x.\hspace{0.5cm} (1)$$ Moreover, considering the indefinite integrals $$\int f(x)\,dx= F(x) + C_1 \hbox{ and } \int g(x)\,dx = G(x) + C_2.$$ My question: If we supppose (1), is true that $$ F(x) + C_1=\int f(x)\,dx…
Welljc
  • 145
3
votes
1 answer

Integral simplification

$$ \int_{-\infty}^{-1} e^{ikx} \left( \frac{-A}{-x-1+\sqrt{x^2-1}} \right)dx = \frac{A}{2}\int_1^\infty e^{-ikx} \left( 1 - \sqrt{\frac{x+1}{x-1}} \right)dx. $$ Hello, thank you very much for this website, I want to know how is it possible to…
copets
  • 31
3
votes
5 answers

Limit with square root

$$\lim_{x\to16}\frac{4-\sqrt{x}}{16x-x^2}$$ I am not sure what to do, I have tried factoring everything and using both conjugates, neither options gives me anything usable.
user138246
3
votes
3 answers

Find K for negative roots

Given $$m^3+m^2+2m+K=0 $$ how do I find the $K$ values that allow just negative roots? Edit: Even though some answers are already posted, I´ll clarify - as suggested in the comments - that this question arise from my earlier one: Find K values that…
Luis
  • 215
3
votes
1 answer

Spherical electrode diffusion problem; trying to get to planar system

I'm working through the derivation of current in a spherical electrode, and so far I've been able to get it into the following, starting from Fick's 2nd Law: Any help would be appreciated.
R Eaton
  • 31
1 2 3
…
99 100