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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.

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Is the derivative of this function bounded?

This is related to my previous post. Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is a $C^1$ function which satisfied the following differential inequality: $$\frac{df}{dt}\leq C(f+f^{\frac{3}{2}}).$$ If $f>0$ and $f(t)\rightarrow 0$ as…
Paul
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Continuous function calculus

Are there any examples satisfy the continuous function $f:\Bbb R\rightarrow \Bbb R$ such that the image of the closed interval $[0,\infty)$ under $f$ is the open interval $(-1,1)$..like $f([0,\infty))=(-1,1)$? A picture is also ok.
Sssm
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If f is a function of $x$ and $t$ where $x$ itself is a function of time does this mean $\frac{\partial f}{\partial x}=0$?

$\frac{\partial f(x(t),t)}{\partial x}=0$? I suspect it probably doesn't but I can't justify it to myself.
JackFielding
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Convexity of $x^a e^{-x}$

This is a calculus homework. Given is $$f(x)=x^a e^{-x}$$ where $x \in ]0;\infty[$ for $a \lt 0$, $x \in [0;\infty[$ for $a \geq 0$. The task is to first find local or global minima and maxima depending on a, then decide on the convexity of f, again…
mafu
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Why don't the limits of integration matter when differentiating both sides of and equation?

Someone asked this question: and I am very interested in the answer, but don't understand it and don't have enough reputation to comment on it directly. The question asks if you can solve something like: $$ g(x) = \int_a^b f(x) \,dx$$ by…
Anda
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integrate $\int_0^12\arctan(1+a\ln(x)) \ dx $

show $\int_0^12\arctan(1+a\ln(x)) \ dx = \frac{\pi}{2} - a -a^2 -a^3 + O(a^4) $ for $a \ll 1$ I've never seen this type of integral before, but it looks very familar to something to do with taylor, so I attempted this: $I(x) =…
Warz
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How to evaluate the following summation

I am trying to find the definite integral of $a^x$ between $b$ and $c$ as the limit of a Riemann sum (where $a > 0$): $I = \displaystyle\int_b^c \! a^{x} \, \mathrm{d}x.$ However, I'm currently stuck in the following part, in order to find S: $S =…
arcbloom
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Finding inverse of a function.

Could someone please help me finding the inverse of the following function: $$f(x)= \frac{x-1}{\ln(x)}$$ where $x>0$? Thank you.
nicole
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What is the tangent line to the point on the curve $2x^2 - y^2 = 1$ at $x=5$?

Consider the curve described by $2x^2 - y^2 = 1$. What is the equation of the tangent line to the point on the curve in the fourth quadrant with x-coordinate $x = 5$? My solution: Derivative by $x$: $4x-2yy'=0$, and the point in the 4th quadrant is…
zhaoyin.usm
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The limit $\lim_{n \to \infty}\left(2 + 1/n\right)^{n}$

How can one show that $\displaystyle{\lim_{n \to \infty}\left(2 + {1 \over n}\right)^n}$ ?. I am trying to shoehorn the definition of $e$ somewhere but fruitlessly so.
user119114
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Finding limits of rational functions as $x\to\infty$

Possible Duplicate: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials $$\displaystyle \lim_{x\to\infty}\frac{(2x^2+1)^2}{(x-1)^2(x^2+x)}.$$ I do not know where to start on this, I tried multiplying it out and that didn't help…
user138246
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Calculus one question (critical points).

Let $f(\theta) =\cos^2(\theta)-2\sin(\theta)$, find the local maximums, local minimums, or neither. My solution is: $y'=-2\sin(\theta)\cos(\theta)-2\cos(\theta)=0\Rightarrow \theta=2k\pi+\pi /2,…
zhaoyin.usm
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Evaluating $\int_{0}^{\infty }(2e^{-3x}+4e^{-7x})^2dx$

I'm currently teaching myself calculus and I'm probably trying to run before I can walk, but I've been working on this problem.. I managed to find the correct result for: $$\int_{0}^{\infty }(2e^{-3x}+4e^{-7x})^2dx$$ by expanding it…
mash
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A simple question about Bohl functions, functions that are linear combinations of $t^ke^{\lambda t}$ where $\lambda \in \Bbb C, k\in\Bbb N_{> 0}$

A Bohl function is a linear combination of terms of the form $t^ke^{\lambda t}$ where $k$ is a non negative integer and $\lambda \in \Bbb C$. We denote the set of exponents of a Bohl function $p$ as $\sigma(p)$. For instance if $p(t) = te^t +…
Slugger
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Proving $\cot x =\alpha x$ has a solution $\forall \alpha>0$ in $(0,\frac\pi2)$

Prove $\cot x =\alpha x$ has a solution $\forall \alpha>0$ in the section $(0,\frac\pi2)$. Well let's define: $g(x)=\cot x -\alpha x$ I know that $\cot$ goes to infinity as x go to zero, and go to zero as x go to $\pi/2$. Also, as $-\alpha x$ gets…
GinKin
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