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
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Finding the value of Antiderivative $F(x)$ at $x=2$, where $f(x)=\frac{x^3+\sin(x)}{x^2+2}$

First of all, many thanks for this great website. Let $F(x)$ be the antiderivative of $$f(x)=\frac{x^3+\sin(x)}{x^2+2}.$$ If $F(5)=\pi$, then what is $F(2)$? This question was in Calculus exam, I know that $F(x)=\int f(x)\ \mathrm dx +C$, but I am…
F.Turgut
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Finding the value of $b$ so $\int_1^b (x-2)^3 dx =0$

Please how do find $b\gt 1$ so that $$ \int_1^b (x-2)^3~dx =0?$$ This question is on a chapter dealing with antiderivatives and I'm not sure how to go about it. At this point it is assumed that I don't know how to integrate yet. I'm also not…
Gorg
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How to evaluate $\int^\infty_0 \frac{\sin(\sin(x))}{x} e^{\cos(x)} dx$

I saw this problem on a Facebook post. I tried to look for a solution but it was really a mess. The only thing I could see is that even though $\frac{\sin(\sin(x))}{x} e^{\cos(x)}$ is not defined for $0$, $\lim_{x\to0}\frac{\sin(\sin(x))}{x}…
AJMC2002
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Infinite Factorization Power Series of $\sin(x)$

My teacher has given us a rather long problem and the last part is stumping me. How would one go about factoring the power series of sin(x)? Where: p(x) = x - $\frac{x^{3}}{3!}$ + $\frac{x^{5}}{5!}$ - $\frac{x^{7}}{7!}$ ... Better expressed as: p(x)…
LucasS
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How to prove that $2\sqrt{3}$ is greater than $\pi$

Without calculator, how to prove that $2 \sqrt{3} > \pi$? The level is baccalauréat grade. I confirm it's not a school exercise at all, as I left school like 35 years ago.
TiLapiot
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When and why do we take into account the curvature of the curve at infinitesimal scale in calculus?

Background We know that the integral $\int f(x)~dx$ represents the area under the curve, where the area is represented by the sum of individual areas of elemental strips of the width $dx$ and height $f(x)$. As $dx$ is infinitesimal, we can ignore…
Mihail
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Range of $a$ in $x^2-a=\sqrt{x+a}$

The equation $x^2-a=\sqrt{x+a}$ has real or imaginary roots depending on the values of $a.$ Then range of $a$ for which the equation. $(a)\;\; $ No real roots $(b)\;\; $ One real root $(c)\;\;$ Exactly two real roots $(d)\;\;$ At least two real…
jacky
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Suppose h is twice differentiable on $[0,2b]$ for some $b > 0$ with $h(0) = 0$ and $h''(x) < 0$.

The question asks us to show that $h(x+y) \le h(x) + h(y)$ for all $x,y \in [0,b]$, I move $h(x)$ to the left side and divide $y$ on the both side. Like $$\frac{h(x+y)-h(x)}{y} \le \frac{h(y)}{y}$$, and I try to take limits on the both side, but it…
Iloveolaf
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Find a polynomial of degree $4$ with the coefficient $x^2$ equal to $6$, and zeros $-3$, $ 2$, $ -1$, $-2$

I started off with: $$f(x)= a(x-(-3)) (x-(2)) (x-(-1)) (x-(-2))$$ $$f(x)= a(x+3) (x-2) (x+1) (x+2)$$
user68053
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How to prove $\sum_{n=-\infty}^ \infty {\rm sinc}\bigl( \pi(t-n)\bigr) = 1$?

Thank you by avance for your help. So, I found on this website, that $\sum_{n=-\infty}^{\infty} {\rm sinc}( \pi n)= 1$. But I could not find any way to prove it. I know it’s about fourrier, but I don’t know how to do so... Does anyone know how to do…
mathenfolie
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Solve $\frac{1}{x}\cdot \cos x + \ln x \sin x = 0$

I was working on the following function: $$f(x) = \frac{\ln x}{\cos x}$$ I tried to find values of x where derivative will equal to zero. After taking derivative of $f'(x)$, I got: $$\tag 1f'(x) = \frac{\frac{1}{x} \cdot \cos x + \ln x \cdot \sin…
Stokolos Ilya
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Value of $(3\beta^2-4\beta)^{\frac{1}{3}}+(3\beta^2+4\beta+2)^{\frac{1}{3}}$ if $\beta$ is the root of $x^3-x-1=0$

If $\beta$ is the root of the equation $x^3-x-1=0$, find the value of $$(3\beta^2-4\beta)^{\frac{1}{3}}+(3\beta^2+4\beta+2)^{\frac{1}{3}}.$$ This is what I tried: $x=\beta$ is a root of $x^3-x-1=0,$ so getting $\displaystyle…
jacky
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Please help Calculate this limit

Help me find: $$ \lim_{n\rightarrow\infty}\sqrt[n]{2^n\cdot3^0+2^{n-1}\cdot3^1+2^{n-2}\cdot3^2+\cdots+2^0\cdot3^n} $$
Matema Tika
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Cutting a can from a metal sheet to maximize volume

Question: Consider the following square sheet of metal (grey) with the side s = 10 cm. From it, we want to cut the parts of a cylindrical can with a lid and a bottom (bblack). Determine the radius and height of the can when the volume is…
MathInferno
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Third derivative at point is greater than 3

f : $I \to \Bbb R$ is differentiable 3 times in open interval $I$ which contains the closed interval [-1,1]. $f(0)=f(-1)=f'(0)=0$ and $f(1)=1$ show that exists a point $c \in (-1,1) s.t. f^{(3)} (c) \ge 3$ What I did: I used Rolle's theorem to prove…
jreing
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