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
5
votes
1 answer

Evaluation of $ \int \sin (2015x)\cdot \sin^{2013}(x)dx$

Evaluation of $\displaystyle \int \sin (2015x)\cdot \sin^{2013}(x)dx$ $\bf{My\; Try::}$ Let $\displaystyle I = \int \sin (2015x)\cdot \sin^{2013}(x)dx = \int \sin (2014x+x)\cdot \sin^{2013}(x)dx$ So $\displaystyle I = \int \left(\sin 2014x\cdot \cos…
juantheron
  • 53,015
5
votes
3 answers

Calculate limit of ratio of these definite integrals

How do I evaluate the following limit? $\lim_{n\to\infty}\frac{\int_{0}^1\left(x^2-x-2\right)^n dx}{\int_{0}^1\left(4x^2-2x-2\right)^n dx}$
Maverick
  • 9,172
  • 2
  • 30
  • 61
5
votes
2 answers

Prove $D_x(\int_a^x f(t)dt) = f(x)$

Further to my understanding What makes a proof, I'm doing exercises. The purpose of this question is to compare the proof a certain textbook provides with the one that is logical to me, that I made up. If my proof is weak, why is it weak? Why is…
bobobobo
  • 9,502
5
votes
1 answer

Fundamental theorem of Calculus Part 1

If I'm taking the derivative with respect to $t$ of the integral: $$\int_{-\infty}^t (t-\tau)u(\tau)d\tau$$ Does the Fundamental theorem of Calculus result in the answer of $0$? I am trying to write the system in implicit form which is why I'm…
John
  • 51
5
votes
2 answers

Proving $f'(c_1)+f'(c_2)=2$ for $f$ such that $f(a) = a$ and $f(b) = b$

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a) = a$ and $f(b) = b$, show that there exist distinct $c_1$, $c_2$ $\in$ $(a,b)$ such that $f'(c_1)+f'(c_2)=2.$ My try: By applying Mean Value Theorem on interval $(a,b)$ one…
Daniel250
  • 51
5
votes
0 answers

Absolute Value in Linear Differential Equation

I have a question about dropping the absolute value sign when solving a linear differential equation. If $y'-y/x=1$ Integrating Factor $=e^{\int{-1/xdx}}=e^{-lnx}=1/x$ $y/x-y/x^2=1/x$ $[y/x]'=1/x$ $y/x=\int{1/x…
bkmoney
  • 185
5
votes
1 answer

Apostol (6.25.40): Find $\int {x^{-2}}\sqrt{2 - x - x^2} dx$

I've been struggling with this exercise from Apostol for some time (Section 6.25, Question 40). The integral is $$ \int\frac{\sqrt{2-x-x^2}}{x^2}\, dx $$ with a Hint of "multiply numerator and denominator by $\sqrt{2-x-x^2}$". NB: the answer…
emjay
  • 545
5
votes
3 answers

n-th derivative cardinal $\mathrm{sinc}(x)$ function.

I need to find formula for $n$-th derivative of $\mathrm{sinc}(x)$. The following question isn't related to homework but it's a question that seems very challenging to me, and I take some interest in it. I have been trying to find the $n$th…
Mark
  • 7,841
  • 6
  • 38
  • 72
5
votes
3 answers

$\epsilon - \delta$ definition of continuity?

trying to recall the $\epsilon, \delta$ definition of continuity, I came up with the following: A function is continuous at $x$ if $\forall \epsilon > 0 \; \exists \; \delta > 0: |f(x-\delta) - f(x+\delta)| < \epsilon$. This is very likely not…
vojta havlíček
  • 139
  • 6
5
votes
2 answers

Differentiating the function $x^3\cdot\min\{x,9\}$

Today I came across the function $x^3\cdot \min\{x,9\}$. My teacher differentiated it and wrote it directly again as $3x^2\cdot\min\{x,9\}$ I was wondering how come the $\min\{x,9\}$ part is not affected by differentiation. Any ideas please? Or am I…
Bhargav
  • 2,969
5
votes
2 answers

Finding a rare case where an incorrect rule works?

"A not uncommon error in calculus is to believe that the product rule for derivatives states that $(fg)' = f'g'$. If $f(x) = e^{3x}$, find a nonzero function g for which $(fg)' = f'g'$." I believe you can find the function(s) using algebra, I got…
Cains
  • 199
5
votes
1 answer

Formula for Alternating Geometric Series

I am aware of the following formula: $$\sum_{n=0}^{\infty}(-1)^nr^n=\frac{1}{1+r}$$ However, I am having difficulty understanding if there is a simple formula for the following equation: $$\sum_{n=0}^{\infty}(-1)^nr^{2n}={ }?$$ In addition, what…
Tyler Murry
  • 735
5
votes
3 answers

Show that $0=x^x$ has no solution in $\mathbb{R}$.

I want to show that $0=x^x$ has no solution for $x > 0$ in $x \in \mathbb{R}$. I know that there isn't a solution, but I don't know how to show it mathematically. EDIT: What I have finally written in my exercise as proof: $x^x = 0$ $\Leftrightarrow…
Basti Funck
  • 175
5
votes
3 answers

proving $\lim\limits_{x\to \frac{\pi }{2}}\frac1{\cos x}\ne \infty $

I have to prove the following equation for homework $$\lim_{x\to \frac{\pi }{2}}\frac1{\cos x}\ne \infty $$ The proof must be done by proving that exists a $M>0$ for which for every $l>0$ exists an $x$ so that $0<|x-(\pi/2)|
Jason
  • 2,367
5
votes
2 answers

IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$?

Given $$f(x) = \int_{0}^{\phi (x)} g(t) dt$$ How could we find $f'(x)$? Please explain your answer.
Quixotic
  • 22,431
1 2 3
…
99 100