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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Proof: If $f'=0$ then is $f$ is constant

I'm trying to prove that if $f'=0$ then is $f$ is constant WITHOUT using the Mean Value Theorem. My attempt [sketch of proof]: Assume that $f$ is not constant. Identify interval $I_1$ such that $f$ is not constant. Identify $I_2$ within $I_1$ such…
MathMathCookie
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Limit Rule $\lim f(x)^{g(x)}$

I want to know the following is true : If $$ c,\ d\in {\bf R},\ \lim_{x\rightarrow 0} f(x)=c>0,\ \lim_{x\rightarrow 0} g(x) =d>0$$ then $$ \lim_{x\rightarrow 0} f(x)^{g(x)} = c^d$$ In calculus book such formula cannot be found. Consider the…
HK Lee
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Are there any other functions that behave the same as $ce^x$ with respect to differentiation

$$\frac{d}{dx} ce^x = ce^x$$ Are there any other functions $f$ such that $$\frac{d}{dx} f(x) = f(x)$$ or is $ f(x) = ce^x $ the only one?
Cameron Martin
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Simplifying $\frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V}$

The problem given is this. If the variables $P, V$, and $T$ are related by the equation $PV = nRT$, where $n$ and $R$ are constants, simplify the expression $$\frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} \cdot \frac{\partial…
user9182
  • 81
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A mistake in Stewart's book

I'm making a revision of calculus and I'm using Stewart's book I think he is wrong in case $(a)$: In case $(a)$ the function is not defined in $x=2$, then we can't say that the function is continuous or not at this point, in fact this function is…
user42912
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nth derivative of a finite amount of composite functions

I was curious to see whether or not there is a formula for the $n$th derivative of $k$ composite functions. If $F(x)=(f_1\circ f_2\circ...\circ f_k(x))$ then is there a formula for $$\frac{d^n} {dx^n}(F(x)).$$ For the $n$th derivative of two…
user124862
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Evaluate: $\int \frac{1}{x^7-x}\ \mathrm{d}x$

Evaluate: $$\int \frac{1}{x^7-x}\ \mathrm{d}x$$ My approach to this question: $$\int \frac{1}{x^7-x}\ \mathrm{d}x = \int \frac{1}{x(x^6-1)}\ \mathrm{d}x$$ $$\int \frac{1}{x(x^6-1)}\ \mathrm{d}x = \int \frac{1}{x(x-1)(x+1)(x^2-x+1)(x^2+x+1)}\…
Sc4r
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$\sqrt{2\sqrt{2\sqrt{2\cdots}}}=2$

Show that $$\sqrt{2\sqrt{2\sqrt{2\cdots}}}=2$$ $$\sqrt{2}=\mathbf{2}^{1/2}$$ $$\sqrt{2\sqrt{2}}=\mathbf{2}^{1/2+1/2^2}$$ $$\sqrt{2\sqrt{2\sqrt{2}}}=\mathbf{2}^{1/2+1/2^2+1/2^3}$$ Show the limit of…
bsdshell
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finding examples for a non negative and continuous function for which the infinite integral is finite but the limit at infinity doesn't exist

Question: a. Find an example for a non-negative and continuous function s.t. $\int _0^\infty f(x)dx$ is finite but the following limit doesn't exist: $\lim_{x\to \infty} f(x)$. b. Is it possible that $\int _0^\infty f(x)dx$ is bounded but $f(x)$ is…
jreing
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Evaluation of $\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}$

Evaluate $$\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}.$$ $\underline{\bf{My\;\;Try}}::$ Let $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n} = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln…
juantheron
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How to find the center of mass of half a torus?

Consider a halved solid torus (half a donut). The radius of the torus are $R_1$ and $R_2$. I need to find its center of mass. The hint they give is that the center of mass of a homogeneous solid object $\Omega \subset \Bbb R^3$ is calculated…
Twnk
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Integral mean value theorem- relate a point in the derivative to an integral

Question: Let $f:[a,b] \to \Bbb R$ be a continuously differentiable function s.t $f(a)=f(b)=0$ Prove that exists a point $c \in (a,b)$ such that $$ |f'(c)| \ge \frac 4{(b-a)^2} \int ^b_a f(x) dx $$ What we did: We tried using the integral mean value…
jreing
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Fencing perimeters: why is the intuition wrong?

This is from a practice GRE problem. "A total of x feet of fencing is to form 3 sides of a level rectangular yard. What is the maximum area in terms of x?" I can do the calculations and take the derivative to see that the area is maximized when the…
Joe
  • 81
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Relation between torsion of a curve and the curl of a vector field

The torsion of a curve in $\mathbb{R}^3$ indicates how much it twists around. The curl of a vector field indicates how much the vector field twists around. Is there a relation between the curl of a vector field and the torsion of a curve through…
Davidac897
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Why isn't there a good product formula for antiderivatives?

Computing antiderivatives is more challenging than computing derivatives, in part due to the lack of a ``product formula''; namely, while $(fg)'$ can be expressed in terms of $f,f',g,g'$, there seems to be no way to express $\int fg$ in terms of $f,…
robinson
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