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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Volume of intersection between a cone and cylinder

Find the volume above the x-y plane inside the cone $z=2-(x^{2}+y^{2})^{1/2}$ and inside the cylinder $(x-1)^{2} + y^{2}=1$ Now using calculus this is actually a rather difficult integration using a Df matrix the bounds are rather un-intuitive to…
Faust
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Proof of Riemann integrability

Let the unbounded function $f(x)$ on the interval $[0,1]$ be defined as $$f(x)=\left\{\begin{array}{l l}\frac{1}{x}&x\text{ in }(0,1]\\ f(x)=0&x=0\end{array}\right.$$ Show that $f(x)$ is not Riemann integrable.(Hint:infinity is not a real number.)
Iclal Demir
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Problems with functions.

I have been working on the following problem relating to functions: "Consider a line $y=kx+b$ ($k<0$, $b>0$) which is tangent to the parabola $y=x^2−4x+4$. Find the maximal value of the region bounded by the lines $y=kx+b$, $y=0$ and $x=0$". The…
Aaron
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Other method other than integration by parts

I would like to integrate this $$\int x \ln x\sin x\mathrm dx$$ can I do it with another quick method than using integration by part?
user515213
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Evaluate the limit $\lim\limits_{n\to\infty}{\frac{n!}{n^n}\bigg(\sum_{k=0}^n{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}}\bigg)}$

Evaluate the limit $$ \lim_{n\rightarrow\infty}{\frac{n!}{n^{n}}\left(\sum_{k=0}^{n}{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}} \right)} $$ I use…
pxchg1200
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Integral $\int_{-1}^{1} \frac{\mathrm dx}{x}$

Consider the following integration $$\int_{-1}^1\frac{\mathrm dx}{x}$$ B thinks this expression should be written as $$\int_{-1}^0\frac{\mathrm dx}{x}+\int_0^1\frac{\mathrm dx}{x}$$ which is not convergent, but A thinks it is zero because it is an…
STUDENT
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Epsilon Delta proof of a floor function

I have to provide a proof for a function but I'm struggling to grasp the main concept. $$\lim_{x \to 3} \left\lfloor \frac{x}{2}\right\rfloor = 1 $$ Here is what I've come up with: $$\frac{x}{2} - 1 \lt \left\lfloor \frac{x}{2}\right\rfloor \lt …
Cu7l4ss
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Can the sum of a differentiable and a non differentiable function be differentiable?

Can the sum of a differentiable and a non differentiable function be differentiable? In one of the solutions to a question, my book's author used the fact that the sum of a differentiable and non differentiable function cannot be differentiable. I…
Flaneur Fission
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Prove that $P^{(n)}(x) = a_{n}n!$ if $P(x) = a_{n}\cdot x^{n} + a_{n-1}\cdot x^{n-1} + \cdots + a_{0}$

Let $P(x) = a_{n}\cdot x^{n} + a_{n-1}\cdot x^{n-1} + \cdots + a_{0}$ be a polynomial with real coefficients. How can we prove that its $n$th derivative is:$$P^{(n)}(x) = a_{n}n!$$
Gerry
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What is the domain of the function $f(x)=\sqrt[3]{x^3-x}$?

Let $f$ be: $f(x) = \sqrt[3]{x^3 -x}$, an exercise book asked for the domain of definition. Isn't it over $\mathbb R$. The book solution stated $Df = [-1,0] \cup [1, +\infty[$ I don t get it. Can you explain?
Papa
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Showing that $x^3-x^2-1$ has only one REAL root.

So, I need to prove that $x^3-x^2-1$ = 0 has only one root. Here's what I have so far: We have $f(-2) = -13 < 0, f(2) = 3 > 0$. Because f is a polynomial, it is continuous on all $\mathbb{R}$ and there is c s.t. $-2 < c < 2$ and $f(c) = 0$ by I.V.T.…
Hellowhatsup
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Conditions for level sets to be closed curves

Im wondering if there is any result to guarantee that the level sets of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ are closed curves? Specifically if $f\in C^1(\mathbb{R^2},\mathbb{R})$ and the level set $\{(x,y)\mid f(x,y) = C\}$ is…
OgvRubin
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Derivative of a product and derivative of quotient of functions theorem: I don't understand its proof

I'm studying for math exam and one of the questions that often appears is related to derivative of a product of two functions. The theorem says that $(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$. The proof goes like…
AndrejaKo
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The sum of the series $ \cos(x)-\cos(2x)+\cos(3x)-...$

In the book "The spirit of mathematical analysis" of Martin Ohm, the author gives an example of differentiating an infinite series and obtaining an absurd result (page 2) From the…
James Warthington
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Prove $y=\ln(2x-1)/\ln(x)$ is a decreasing function

Given $y=\ln(2x-1)/\ln(x)$, prove $y$ is decreasing for $x>1$. While this is obvious by couple computations, the usual differentiation method to show this is true is not getting me anywhere since finding the $y'=0$ point is rather nuisance with ln…
user9115
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