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
4 answers

Proving a function is continuous for all real numbers

This a homework question: Prove $f(x) = 2x^3 - 4x^2 + 5$ is continuous for all real numbers. Which proof technique do I use for this? I don't really know where to start.
Logan Serman
  • 679
4
votes
2 answers

If $f(x)$ is uniformly continuous at $(0,1)$ then is it bounded at $(0,1)$?

Possible Duplicate: Uniform continuity I have a question which is: If ${f(x)}$ is uniformly continuous at ${(0,1)}$ then is it bounded at ${(0,1)}$? This sound like it's correct to me but I can't see why exactly (Or maybe it's wrong :P). Could…
Jason
  • 2,367
4
votes
2 answers

The derivative is a real limit?

In my classes, the derivative is usually defined as "the limit of the fractional incremental ratio". But I found out another way to define the derivative from an old book from Gardner & Thompson's "Calculus Made Easy". For example, if we have $f(x)…
Micro
  • 271
4
votes
7 answers

For $x>0$, Prove that $\dfrac{x}{1+x^2}<\tan^{-1}x < x$

Looking for an elegant way to do it. I know one way to do it, will post soon
Holy cow
  • 1,395
4
votes
4 answers

$f(x) = \frac1{\cos x}$ then $f^{(n)}(x)$ is ...?

$f(x) = \frac{1}{\cos x}$ $f'(x) = \frac{\sin(x)}{\cos^2(x)}$ $f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$ $f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$ $\vdots$ $f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$ Some of these are easy:…
futurebird
  • 6,248
4
votes
2 answers

Integral of the square of a function

I have an integral of the square of a function ($f(x)\cdot f(x)$ from $0$ to infinity) that's very hard to compute. I just need to know the sign of its value. Is it always non-negative? If not, what are the conditions for non-negativity?
Hass
  • 367
4
votes
1 answer

Tangent Line at given point. Help!

Find an equation of the tangent line at the given point. $7y^2 − xy^2 − x^3 =0$ the point is $(\frac72,\frac72)$ Ive found the derivative: $14y\frac{dy}{dx}-y^2-2yx\frac{dy}{dx}-3x^2=0$
Kels
  • 95
4
votes
1 answer

prove $f(\sqrt{2})\geq 0$ related $(g(x)f'(x))'+f(x)\geq 0$

Let $g(x)$ has continuous derivative,with $g(x)\geq 1,x\in \mathbf{R}$,if $f(x)\in \mathbf{C^{2}}(-\infty,+\infty)$,and $$ f(0)=f'(0)=\int_{-\pi}^{\pi}dy\int_{0}^{\pi}\frac{\cos(nx)-\cos(ny)}{\cos{x}-\cos{y}}dx\qquad n=2k+1,k\in \mathbf{N}^{+}…
pxchg1200
  • 2,050
4
votes
4 answers

Finding Integral of $y = e^{ax}\sin (bx)$

I was solving questions like $ \int e^{2x} \sin x dx. $ I decided to find the general term for $$\int e^{ax} \sin (bx) $$ which can be directly used in questions like (above). I hope it helps others :) (Just wanted to share my work. I will…
Kushashwa Ravi Shrimali
  • 1,412
4
votes
1 answer

calculus, equation of a tangent, an easy question

Find the equation of the tangent line : $$\ln{xy}= 2x $$ at point $( 1, e^2 )$ I end up with slope of $e^2$ so the equation will be $$ y= e^2(x-1) $$ $$ y = e^2x - e^2 $$ But the answer was just $$ y= e^2x $$ so appearantly they used $$ y=mx+b $$…
user157908
  • 255
  • 1
  • 4
  • 10
4
votes
2 answers

Calculus / substituing in f(x)+g(x) to find h'(x)

$$\begin{align} f(x) &=7\\f'(x)&=2\\ g(x) &=2 \\ g'(x)&=-5 \\ h(x) &= f(x) + g(x)\end{align}$$ Find: $h'(2)$ My attempt was: $2+7=9$ but it seems to be wrong.
John
  • 403
  • 1
  • 8
  • 19
4
votes
3 answers

Calculation of $ \lim_{x\rightarrow 1}\frac{(1-x)\cdot(1-x^2)\cdot(1-x^3)\cdots(1-x^{2n})}{\{(1-x)\cdot(1-x^2)\cdot (1-x^3)\cdots(1-x^n)\}^2} = $

Calculation of $\displaystyle \lim_{x\rightarrow 1}\frac{(1-x)\cdot(1-x^2)\cdot(1-x^3)\cdots (1-x^{2n})}{\{(1-x)\cdot(1-x^2)\cdot (1-x^3)\cdots(1-x^n)\}^2} = $ My Trial After simplification, we get $$\displaystyle \lim_{x\rightarrow…
juantheron
  • 53,015
4
votes
2 answers

Finding Equation of Tangent parallel to x & y axis

The question: A curve has equation $4x^2+8x+9y^2-36y+4=0$. (i) Find $\dfrac{dy}{dx}$. (ii) Write down the equation(s) of the tangent(s) to the curve that are parallel to $\qquad$(a) the $x$-axis $\qquad$(b) the $y$-axis. Answers $\qquad$ (i)…
Jiew Meng
  • 4,593
4
votes
1 answer

Showing properties of a surge function

I am working on the question below and I am getting stuck. Consider the surge function $y=axe^{-bx}$ with $a$ and $b$ positive constants. (a) Find the local maxima, local minima, and inflection points. (b) How does varying $a$ and $b$ affect the…
NehriMattise
  • 115
4
votes
1 answer

Given enough terms, does a taylor series become equivalent to the function it is approximating?

I've recently started learning about Taylor/Maclauren series and I'm finding it a bit hard to wrap my head around a few things. So, if $f(x)$ is not infinitely differentiable and we construct a polynomial $p(x)$ such that $p(a) = f(a)$, $f'(a) =…
miscmanman
  • 41
1 2 3
…
99 100