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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Simplification of equation gives different domain and range

I am trying to find the domain and range of: $$f(x) = \frac {x^2 - 4x + 3 }{x - 1}$$ In the book I am using, it says that the domain (x) is the set of all real numbers except 1 and that the range (y) is the set of all real numbers except -2.…
Paco G
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First derivative of $f(x)= \frac{2}{x+1} +3$

I'm struggling to find the first derivative of $f(x)= \frac{2}{x+1} +3$ using the limit definition of derivative. I keep coming up with $f'(x) = \frac{-5} {(x+1)^2}$ but I should be getting $f'(x) = \frac{-2}{(x+1)^2}$ \begin{align} \lim_{x\to 0} &=…
rujoesmith
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Differential operator.

I'm not familiar with the terminology "differential operator", so when I encounter it in this exercise I'm quite stumped. Am I right to assume that, given a polynomial $p(x)$ of real coefficients, what $D$ does is to carry out this…
user533068
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Prove that if $\dfrac{ax^2+2bx+c}{\alpha x^2+2\beta x+\gamma} (\alpha\neq 0)$ has $3$ inflection points, then all of them lie on one line?

Prove that if the rational function $f(x)=\dfrac{ax^2+2bx+c}{\alpha x^2+2\beta x+\gamma} (\alpha\neq 0)$ has three inflection points, then all of them lie on one line? (All the parameters are real numbers. It's an exercise problem. And there is a…
闫嘉琦
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How to choose the most significant denominator

I have to see if $$\int_{0}^{\infty}\frac{1}{\sqrt{x^3+x}}dx$$ is convergent or not. I know I have two improper points so $$\int_{0}^{\infty}\frac{1}{\sqrt{x^3+x}}dx = \int_{0}^{1}\frac{1}{\sqrt{x^3+x}}dx +…
Favolas
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Solving an equation $x + x^{0.925} = 15$

I have been trying to solve this equation for a couple of days now but am getting stuck. $x + x^{0.925} = 15$, find $x$. I went in the direction of taking log on both sides of the equation but that does not help and I cannot simplify further. Also,…
SeattleOrBayArea
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How do I finish finding the volume of this solid torus?

Use cylindrical shells to find the volume $V$ of the solid torus (the donut-shaped solid shown in the figure) with radii $r$ and $R$. This is as far I have come. How can I solve further?
None
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Two different answers from integrating $\int\frac{dx}{x\sqrt{x^2-1}}$ in two ways. What did I do wrong?

I want to calculate the answer of the integral $$\int\frac{dx}{x\sqrt{x^2-1}}$$ I use the substitution $x=\cosh(t)$ ($t \ge 0$) which yields $dx=\sinh(t)\,dt$. By using the fact that $\cosh^2(t)-\sinh^2(t)=1$ we can write…
user528935
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Radial (not Polar!) planimeter theory...

I am investigating the theory of a radial (not a polar) planimeter - when tracing a closed contour with the "pole" of the device outside of the contour (normally, the "pole" would be inside the contour). This has led me to this horrible expression…
bobg1756
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Prove positivity of $f$ when $f'' > f$

I'm given that $f:\mathbb{R} \to \mathbb{R}$ is twice differentiable and $f(0) = f'(0) =1$. Assuming that $f''(x) > f(x)$ everywhere show that $f(x) > 0$ for all $x$. I know that $f$ and $f’$ are continuous ($f’’$ exists). Since $f(0) = f’(0)= 1$…
WoodWorker
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"Multiplying" by a dt then cancelling?

Today I encountered a proof of a physics kinematic equation which contained the following steps: $$ \frac{dx}{dt} = v $$ $$ dt \frac{dx}{dt} = v\,dt $$ $$ dx = v\,dt $$ Now I have read the previous questions which state putting a dt on both sides…
jaynp
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Show that $f=0$ given $\int_{a}^{b} x^n\,f(x)\,dx=0$

Let $f:[a,b]\rightarrow \mathbb {R} $ a continious function such that $$ \int_{a}^{b} x^n\,f(x)\,dx=0$$ for all $n \in \mathbb N$. Show that $f$ is identically $0$. I notice that $f$ is bounded and it touches its bounds. But I don't know how to…
rafa
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Minimum value of $\dfrac{a+b+c}{b-a}$

$f(x)= ax^2 +bx +c ~ ~~(a0$ and $c>0$ I am unable to utilize these things to find…
Archer
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Find the derivative at (1,2)

$$f(x) = x^2 \sqrt{5 - x^2}$$ Find the derivative at $(1, 2)$. \begin{align} \frac{d}{dx} \left[ x^2 \sqrt{5 - x^2} \right] & = \frac{d}{dx} \left[ x^2 (5 - x^2)^{1/2} \right] \\ & = x^2 \frac12 (5 - x^2)^{-1/2}(-2x) + (5 -…
ilovetolearn
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When does the floor function has a limit and when it does not?

I have been asked to tell in which points in $\mathbb R$ the limit does and does not exist. $$\lim_{x\to x_0}\lfloor x\rfloor$$ Now, I have been thinking about first showing that the limit exists in all $x\in(\mathbb R / \mathbb Z)$ In order to do…
Yair Shemer
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