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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$\int(x+1)dx$ yielding different results with $u$-substitution and termwise integration

Considering two methods of integrating the very easy: $\int(x+1)dx$ First just going term by term: $\int(x+1)dx = x^2/2 + x + C$ Or by making a u-subtitution. Let $u = x+1$, then $du = dx$ and the integral becomes $\int u du = u^2/2$ = $\frac…
IntegrateThis
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How do you set up this Tricky u-sub?

Tricky u-sub. Can you point me in the right direction? $$\int{{x^3}\sqrt{5-2x^2}}dx$$ $u = 5-2x^2$ $ du = -4x dx$ Obviously, this does not match fully. I tried breaking up the $x^3 = x*x^2$ and I continued with: $u=5-2x^2$ $2x^2 =…
JackOfAll
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How to integrate, $\int_{0}^{\pi/2}\sin (\tan\theta) \mathrm{d\theta}$?

Integrate, $$\int_{0}^{\frac{\pi}{2}}\sin (\tan\theta) \mathrm{d\theta}$$
HOLYBIBLETHE
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Proof of e as a limit

I'm reading this text: A few questions: What's the importance of them going from $h$ to $x$ in the first line? What is the difference? How did they go from $$\lim_{x \to 0} \frac{\ln(1+x) - \ln(1)}{x}$$ to $$\lim_{x \to 0} \left[\frac{1}{x} \cdot…
Jwan622
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Let f be a continuous and differentiable function such that f(a)=f(b)=0

Let $f:[a,b] \rightarrow \mathbb{R} $ be a continuous function in $[a,b]$ and differentiable in $(a,b)$ such that $f(a)=f(b)=0$ . Show that $7f(c)+cf'(c)=0$ for some $c \in (a,b)$. I tried to use rolle's theorem and the mean value theorem, but…
Eliads
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Why is the intermediate value theorem so important?

I would like to know why the intermediate value theorem is so important. So my questions are: Which important theorems do we prove using the intermediate value theorem? Are there direct applications of the intermediate value theorem outside…
user42912
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Prove that a continuous function $f:\mathbb R\to \mathbb R$ is injective if and only if it has no extrema

I have a homework problem where I need to prove that a continuous function $f:\mathbb R\to \mathbb R$ is injective if and only if it has no extrema (local or global). So far what I have is: We'll assume that $f$ is injective and assume that it has…
Daniel
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${{2\pi i=0}}$?

I was trying to solve for $i^i$. I got distracted and did the following: $e^{i \theta} = \cos(\theta) + i\sin(\theta)$ $e^{i 2\pi} = cos(2\pi) + i\sin(2\pi) = 1$ ||| Take ln() of two sides of equation: $\ln(e^{i2\pi}) = \ln(1)$ $i2\pi = 0$ Clearly…
Shawhin Layeghi
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$\int 5\sqrt{x^3+2x}\, \textrm{d}x$

I can not find any method to solve this, why do the typical methods fail? I have tried subtituion, parts, strange subitution, etc. How does one solve this integral?
yiyi
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Find degree of polynomial satisfying given condition

Question: If $f(x)$ is a polynomial of degree $n$ such that $$1+f(x)=\frac{f(x-1)+f(x+1)}{2} \forall x\in R$$ then find $n$. My attempt: I first started off by trying to prove $f(x)$ to be periodic, as I always do whenever I spot…
Gaurang Tandon
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Help make Wonder Woman's box big.

Wonder Woman wants a box for her lasso. It is to be built from a rectangular piece of steel measuring 25 cm by 40 cm by cutting out a square from each corner and then bending up the sides. Find the size of the corner square which will produce a…
yiyi
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Different methods, different answers.

If $\int_{\pi/2}^\theta\sin x\,dx=\sin2\theta$, then the value of $\theta$ satisfying $0<\theta<\pi$, is (a) $3\pi/2$ (b) $\pi/6$ (c) $5\pi/6$ (d) $\pi/2$ Method 1: I apply Leibniz rule and differentiate both the sides with respect to…
MrObjectOriented
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Lagrange: Show that there are two points where $f'$ equals zero

Let $f: \mathbb R \to \Bbb R$ a differentiable function. Let $T>0 \in R$ such that $f(x+T)=f(x) \forall x \in \Bbb R$ Show that the interval $[0,T)$ has two points where the function $f'$ get equals to $0$(means that $f'(x) = 0$ What I've done: If I…
user21312
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How to prove $\nabla\cdot \vec{B}=0 \Rightarrow \exists \vec{A}:\vec{B}=\nabla \times \vec{A}$

Suppose $\vec{B}$ is a differentiable vector field defined everywhere such that $\nabla\cdot \vec{B}=0$. Define $\vec{A}$ by the integral $$A_1=\int_0^1 \lambda(zB_2(\lambda x,\lambda y,\lambda z)- yB_3(\lambda x,\lambda y,\lambda z))…
Freeman
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A calculus limit problem

The problem is: $\lim_{x\to0}$ $\frac{sin(\frac{1}{x})}{sin(\frac{1}{sin(x)})}$ my intuition tells that the answer equal to $1$ by the limit equality : $\lim_{x\to0}$ $\frac{sinx}{x}=1$. But we can easily see that the limit of numerator and…
superman
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