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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Orthogonal Trajectories

I am asked to show that the given families of curves are orthogonal trajectories of each other. $$x^2+y^2=ax$$ $$x^2+y^2=by$$ I know that two functions are called orthogonal if at every point their tangents lines are perpendicular to each other. If…
Kurt
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Show that for any positive integer $n$ there is $x \in [0,1-\frac{1}{n}]$ for which $f(x) = f(x+\frac{1}{n})$

Suppose $f$ is a continuous function over $[0,1]$ such that $f(0) = f(1).$ Show that for any positive integer $n$ there is $x \in [0,1-\frac{1}{n}]$ for which $f(x) = f(x+\frac{1}{n})$. We seem to be saying that $f(x+\frac{1}{n})$ is periodic with…
user19405892
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How to find $\lim_{t\to0}\bigg(\int_0^1[bx+a(1-x)]^tdx\bigg)^{1/t}$

If $0\lt a\lt b$, I'm trying to find: $$\lim_{t\to0}\bigg(\int_0^1[bx+a(1-x)]^tdx\bigg)^{1/t}$$ Solving the integral by substitution we have: $$\lim_{t\to 0}\bigg(\frac{1}{b-a}\cdot\frac{b^{t+1}-a^{t+1}}{t+1}\bigg)^{1/t}$$ I don't how to proceed.…
user42912
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Problem with a Definite Integral

I'm studying for finals and having issues with a question about the Fundamental Theorem of Calculus. The question is: $$\int_1^6 \frac {\mathrm{d}t}{4t+23}.$$ I took the integral of $$\frac{\mathrm{d}t}{4t+23}$$ and got $$\ln\left|4t+23\right|$$ I…
daniel furhang
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Prove that $2 < e < 4$ using upper and lower Riemann sums and the definition of $\ln{x}$

Prove that $2 < e < 4$ using upper and lower Riemann sums and the definition of $\ln{x}$ I think I understand the concept of what I need to do, but I am having some trouble implementing a solution. I guess this would be equivalent to showing…
stariz77
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Derivative of an implicit function

I am asked to take the derivative of the following equation for $y$: $$y = x + xe^y$$ However, I get lost. I thought that it would be $$\begin{align} & y' = 1 + e^y + xy'e^y\\ & y'(1 - xe^y) = 1 + e^y\\ & y' =…
Guy
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How to evaluate $\lim\limits_{t\to 0} \frac{e^{-1/t}}{t}$?

How can I evaluate \[ \lim_{t\to 0} \frac{e^{-1/t}}{t}\quad ? \] I tried to use L'Hôpital's rule but it didn't help me. Any hints are welcome. Thanks.
Jr.
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How do I show that this function is always $> 0$

Show that $$f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} > 0 ~~~ \forall_x \in \mathbb{R}$$ I can show that the first 3 terms are $> 0$ for all $x$: $(x+1)^2 + 1 > 0$ But, I'm having trouble with the last two terms. I tried to…
stariz77
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solve equation $y^{\alpha} + y^{1 + \alpha} = x $

how to solve this equation: $y^{\alpha} + y^{1 + \alpha} = x $ where $\alpha \in (-1, 0)$ Is there trick to solve it? EDIT. I want to find $y(x)$.
ashim
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Always a differentiable path through a convergent sequence of points in $\mathbb{R}^n$?

I've come up with this question in trying to solve a vaguely related exercise: If $x_n$ is any sequence of points in $\mathbb{R}^n$ with $x_n \longrightarrow 0$, is there a path $\gamma(t)$, $\gamma(0)=0$ that goes through all $x_n$ and which is…
Weltschmerz
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Evaluate the integral: $\int\frac{xe^{2x}}{(1+2x)^2}\ dx$

$\int\frac{xe^{2x}}{(1+2x)^2}\ dx$ This an integration by parts problem. I am asking for assistance with my method. I was taught to use the LIATE (Logarithmic function, Inverse Trig Function, Algebraic Function, Trigonometric Function, and…
BenCarson2016
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Help with verifying integral inequality.

I am looking at problems from a released Fall 14 mock exam. The question in particular is number 2: Let $f$ be a continuous function in $[0,1]$ satisfying the condition: $$ \int_x^1 f(t) dt \geq \frac{1-x^2}{2}$$ for $x \in [0,1]$ Prove…
Dair
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Limit of a product $(1 + \frac{k}{n})$

I am trying to find the limit of $$\displaystyle \lim_{n\to\infty}\left(\prod_{k=1}^n\left(1+\frac{k}{n}\right)\right)^{1/n}$$ I have that it is equivalent to…
abelian3
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Power series expansion of $e^{-1/x^2}$ at a point different from 0

The function $f(x)=e^{-1/x^2}$ ($f(0)=0$) does not have a power series expansion at $z_0=0$. Now my question: Is there a power series for $f$ centered at $z_0\neq0$ with convergence radius greater than $|z_0|$?
user271114
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Evaluate integral inside integral

I am struggling in evaluating the follow type of integral: $$\int_{-\infty}^{x}f(z)\left[\int_{z}^{\infty}g(y)\,\mathrm dy\right] \,\mathrm dz.$$ I did some research and found out that the above expression is equivalent to the following, which might…
Ocean
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