Questions tagged [calculus]

For basic questions about limits, derivatives, integrals and applications, mainly of one-variable functions.

Filter by
Sorted by
Tagged with
-1
votes
1answer
53 views

how to find minimum form piecewise function

Equation: $$(1+6.64x)^2+4(1+8.89x)^2-8(1+6.64x)-16(1+8.89x)-(\frac{1}{((1+6.64x)+(1+8.89x)-5)}+\frac{1}{((1+6.64x)-3)}-\frac{1}{(1+6.64x)}-\frac{1}{(1+8.89x)})$$ plot and solution form wolframalpha ...
2
votes
2answers
46 views

Approximations using derivatives

I came across the following definitions in my textbook: The differential of $x$, denoted by $dx$, is defined by $dx = \Delta x$ The differential of $y$, denoted by $dy$, is defined by $dy=f'(x) dx$ ...
-2
votes
1answer
32 views

Please explain this integration relation [closed]

For $a>0,{\ }b>0$, $$ \lim_{T\to\infty}\int_{a}^{b}\frac{-e^{-Ty}}{y}\,\mathrm{d}y = \lim_{T\to\infty}\int_{Ta}^{Tb}\frac{-e^{-y}}{y}\,\mathrm{d}y $$
0
votes
0answers
29 views

Parametrizations of a straight line

Let $p$ and $v$ in $\mathbb{R}^n$ two vectors and let $\Gamma$ be the subset of $\mathbb{R}^n$ $$ \Gamma = \{p+tv : t\in \mathbb{R}\}. $$ Is it possible to characterize/describe all the curves $\gamma:...
0
votes
0answers
36 views

regular curves and ordinary cusps

Let $\Gamma$ be the subset of $\mathbb{R}^{2}$ given by $$ \Gamma = \{(t^2,t^3) : t\in \mathbb{R}\} $$ Does there exist a regular curve of class $C^{3}$ (3−times continuously differentiable), say $\...
1
vote
1answer
81 views

Trying to find roots of $\arctan(2(x − 1)) − \ln |x| = 0$ analytically

I have found the roots graphically and also numerically. Apparently there is also a calculus root to finding them analytically. I was thinking to use the derivative but it doesn't seem to work. ...
-2
votes
2answers
52 views

Splitting $~x~$ and $~y~$ and solving for $~y~$

Solve for $~y~$: $$3xy+5y=2x+7$$ I have to do this for as an assignment going into Calculus, the problem is the teacher wants us to research how to do the problems on our own, and I don't know what ...
1
vote
1answer
23 views

Swapping charge time for a capacitor doesn't behave as expected

This is a very basic question about equations but reading online courses about equations didn't clear up what I'm missing. TLDR; How to get from $T = RC$ to $R = \frac{T}{C}$ Working on a Raspberry ...
1
vote
0answers
32 views

Exchange of differentiation and limit on a closed interval [duplicate]

For a differentiable function $f(x,y)$, is the following true,$$\lim_{x\to 0}\frac{\partial f(x,y)}{\partial y} = \frac{\partial \lim_{x\to 0} f(x,y)}{\partial y}$$ on a closed interval $y\in[0,\bar{y}...
0
votes
0answers
27 views

A series involving spectrum of $- \Delta_{\mathbb{S}^2}$

Infinite series involving eigenvalues of the Beltrami-Laplace operator on Riemannian manifolds as well as $L^p$-estimates of eigenfunctions arise in the study of the nonlinear Schrödinger equation (...
2
votes
1answer
32 views

How to find the derivative of $x{^2/3} - (x - 1)^{1/3}$ by the limit calculation?

I had to solve an exercise and was going to ask for help here, but as I typed, I spotted an error myself. I felt sorry to throw away all that MathJax stuff I typed for more than an hour, so I decided ...
4
votes
1answer
76 views

Finding global extrema of $a\left(\frac{1}{2}-b\frac{\sin(cx)}{x} \right) - b(1-\cos(cx))$, for $x\geq 0$

I have the following equation: \begin{equation} f(x) = a\left(\frac{1}{2}-b\frac{\sin(cx)}{x} \right) - b(1-\cos(cx)), \quad x\geq0, \end{equation} where a, b and c are strictly positive constants....
1
vote
0answers
70 views

L'Hospital and rate of change of definite integrals

I'm trying to evaluate a rational function that involves the following limit in the denominator and the same limit and integral albeit with different limits of integration in the numerator. $\...
0
votes
1answer
79 views

$\int_{-\infty}^\infty \frac{\sin(x-\frac 1x)}{x+\frac 1x}dx$ via complex analysis

I'm a little new to complex analysis, so bear with me: First, I defined a function $f(x)= \frac{\sin(x-\frac 1x)}{x+\frac 1x}$, so that I could define a new function $f(z)= \frac{e^{i(z-\frac 1z)}}{z+...
5
votes
1answer
83 views

A quick way to determine interval of convergence for power series

I would start with an example.. Consider the power series $$\sum_{n=1}^\infty\frac{(n+4)(x-2)^n}{7^n(n^2+11)}$$ Determine the interval of convergence of this power series. If the interval is ...
1
vote
2answers
60 views

How can I prove this :$\lim_{x\to \infty } \left(x(x+1) \log \left(\dfrac{x+1}{x} \right)-x\right)=\frac12$ for high school level?

I have tried to evaluate this limit: $$\lim_{x\to \infty } \left(x(x+1) \log \left(\dfrac{x+1}{x} \right)-x\right)=\frac12$$ using $\lim_ {x\to \infty }\left(1+\dfrac{1}{x}\right)^{x}=e$, and using ...
2
votes
2answers
48 views

Prove something like If $\lim_{x\to 1}\frac{P(x)}{(x-1)^2}$ exists, then $(x-1)^2\mid P(x)$ holds

I want to ask how to prove a little property about limits. In calculus course such property is frequently used: If $\lim_{x\to 1}\frac{P(x)}{(x-1)^2}$ exists, where $P(x)$ is a polynomial, then $(x-1)^...
3
votes
4answers
175 views

Prove $x^2-2x+\sin(\frac{\pi}{2}x) \le 0$

I have been searching non-trigonometric approximations for some trigonometric functions and have found myself in need of showing that, $$x^2-2x+\sin\left(\frac{\pi}{2}x\right) \le 0,$$ in the range $...
0
votes
1answer
43 views

Why do all integrals equal to zero when substituting $ u = x(x-a-b)$. [duplicate]

My friend showed me this substitution for an integral $$\int _a^b f(x)\, \mathrm dx:$$ Make substitution $ u = x(x-a-b)$. Now this changes the limit to $$\int _{-ab}^{-ab} \text {(something)}\,\...
1
vote
0answers
31 views

A generalized differential equation for a convolved differential operator.

The solution to perhaps the world's very first differential equation $$f'(x) = f(x)$$ is the well known exponential functions $$f(x) = k\exp\left[x\right]$$ But what if we consider another ...
-1
votes
0answers
26 views

What are some financial or economic applications of the improper integral? [closed]

So basically I'm really interested in improper integration and also I am very passionate about finance. I just need some pointers, hints or references to where I can apply improper integration in the ...
1
vote
1answer
65 views

How to solve differential equation - $x^2f''(x)+xf'(x)+f(x)=0\ , f(x_0)=x_0$

Solve$$x^2f''(x)+xf'(x)+f(x)=0\ , ~~~~~~f(x_0)=x_0$$ I didn't even know what method to be used to solve this problem. How to find solution for this problem. Please help me.
3
votes
2answers
54 views

Uniform convergence of $f_n(x) = \cos^n(x/ \sqrt{n})$ .

I'm studying the uniform convergence of the following sequence : \begin{equation*} f_n(x) = \left\{ \begin{split} & \ \cos^n\left(\frac{x}{\sqrt{n}}\right) \ \ \textrm{if} \ x \in \left[0, \...
1
vote
1answer
94 views

I need help with this integration: $\int_{0}^{\infty}x^{a}e^{x^{b}}e^{-(e^{x^{b}}-1)^c}dx$

I am trying to fined a closed form for this integration $$\int_{0}^{\infty}x^{a}e^{x^{b}}e^{-(e^{x^{b}}-1)^c}dx,$$ where $a,b,c>0$ I think the generalized integro-exponential ($E_{s}^{r}(z)=\...
0
votes
2answers
58 views

Question based on $\epsilon - \delta$ definition of limits.

Question: Let $f :(a,b)\rightarrow \mathbb{R}$ be such that $\lim_{x\to c} f(x)> \alpha$, where $c\in(a,b)$ and $\alpha\in\mathbb{R}$. Prove that there exists some $\delta > 0$ such that $$f(c+...
1
vote
0answers
35 views

Calculus system expansion to complex numbers

Does calculus for complex numbers exists? did any one ever tried to make research on this? What would it mean to have complex numbers calculus? for instance: What is the meaning of $$\lim_{x\to z} f(...
4
votes
0answers
68 views

Volume of solid revolution of region bounded by $y = x^{2} - 4, y = 8 - 2x^{2}, x=0, x=3$

I have two curves: $y_{1} = x^{2} - 4$, $y_{2} = 8 - 2x^{2}$. I have a region bounded by $y_{1}, y_{2}$, $y$-axis, and $x=3$. What is the volume of solid revolve around the $x$-axis? Attempt: In the ...
1
vote
1answer
18 views

Complex Amplitudes and their Exp'l Increase/Decrease

The following are quotes from a textbook, If S(t) is represented as a rotating phasor, the angular frequency of the phasor can be thought of as velocity at the end of the phasor. In particular ...
3
votes
1answer
68 views

Computer the flux of $\nabla \ln \sqrt{x^2 + y^2 + z^2}$ across an icosahedron centered at the origin

Let $S$ be the surface of an icosahedron centered at origin) and let $$f(x,y,z)=\ln \sqrt{x^2+y^2+z^2} .$$ Calculate the flux $$\iint_S (\nabla f \cdot n) d\sigma,$$ where $n$ is the outward unit ...
4
votes
1answer
61 views

Find $\frac{d \rho}{d x}$ for $\rho = \rho(t,x(t),p(t))$

I got a question relating to this thread difference between implicit, explicit, and total time dependence Considering the reply in the top by Kostya, I konw what is the difference between $\frac{\...
-1
votes
1answer
44 views

What is the taylor series of this function at$ x =0$? [duplicate]

Let the $f(x) =e^{-1 \over x^2}$ for $x \neq 0$ Plus Define $f(0) = 0$ By definition of the differentiation,$ f'(0) =0$ But can't figure out the case of the $f^n(0)$
0
votes
0answers
25 views

Is it possible to show when the area of a polygon equals the area under its connected points?

First, let me preface w/ what my own understanding consists of. I've only ever taken classes up to Discrete Math & Differential Equations, and it has been a while since these topics have been ...
3
votes
2answers
27 views

Solution verification on homework problem. Separable first order ODE IVP.

The answer is supposedly $y^2 = 1 + \sqrt{x^2 - 16}$ I don't know where I went wrong cause I know for a fact that my substitution of $x = 4 \sec(\theta)$ is correct. I know for a fact that after ...
2
votes
1answer
53 views

Book of calculus [duplicate]

Sometimes I want to test the concepts of calculus in practice, but the most popular books like Thomas/Maurice is a little approach to me.Does someone know a book that works with calculus in ...
0
votes
3answers
51 views

Limit of exponential function using natural log and change of variable without L'Hopital

$$ \lim_{x\to \:0}\left(\frac{a^x-1}{x}\right) $$ I know the answer to this is $$\ln \left(a\right)$$ but I don't know how to reach that answer without L'Hopital. I just know the first step (given ...
0
votes
3answers
70 views

Calculating the area between two curves

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves. $$ y^2 = 9x, y = \frac{3x^2}{8} $$ ...
1
vote
2answers
49 views

What is the limit $\lim_{x \to \infty} \frac{x^{\log x}}{c^x}$ where $c > 1$?

My intuition tells me the answer is 0, but I can't figure out how to prove it. I've tried using L'Hopital's rule $k$ times in a row, but since $c^x$ doesn't change when being derived, this doesn't ...
-1
votes
0answers
42 views

$ \sum_1^\infty \cos\left(\frac{\pi}{4k}\right)e^{-k} $ [closed]

what is the closed form of $ \sum_1^\infty \cos(\frac{\pi}{4k})e^{-k} $ I don't have any idea how to start with it
1
vote
2answers
30 views

Order of factors in partial decomposition

Is there a protocol for deciding which denominator fraction goes under A and which goes under B during partial decomposition? Doing this question: integral $(5x-5)/(3x^2-8x-3)$ I factored the ...
0
votes
2answers
38 views

Implicit differentiation classic [closed]

$ x^t y^m = (x+y)^{m+x} $ prove that $\frac{dy}{dx} = \frac{y}{x} $ I tried and finally got this $\frac{dy}{dx} = \frac{y-1}{1-x} $ Update: I found a solution without using logarithms, $ tx^{t-1}y^...
1
vote
1answer
54 views

Asymptote when $x\to-\infty$

I have function $f(x)=\sqrt{4x^2+5x}$ and need asymptote when $x\to-\infty$. I know that $$\;\sqrt{4x^2+5x}=\sqrt{x^2\left(4+\frac5x\right)}=|x|\sqrt{4+\frac5x}=-x\sqrt{4+\frac5x}$$ since I assume $...
1
vote
2answers
38 views

Integral of positive Part

Is $$\int_a^b \big(f(x)\big)^+\mathrm{d}x = \left( \int_a^b f(x) \mathrm{d}x \right)^+$$ provided that $f:[a,b]\to\mathbb{R}$ is integrable? This means, can taking positive part and integration be ...
0
votes
1answer
58 views

Find the convergent interval of $\sum_{n = 0}^\infty {(3n)!\over (n!)(2n!)}x^n$

I am trying to find the convergent interval of this power series and I got the absolutely convergent interval to be $(-{4\over 27},{4 \over 27})$ by applying ratio test. But how can I verify the ...
1
vote
1answer
41 views

Differential equation related to energy conservation and Newton's law of gravitation

I have been trying to determine, given the position of a point mass an initial distance $x_0$ from the surface of a spherically symmetric body with mass $M$ and radius $R$, the position of the point ...
3
votes
3answers
41 views

How to check if a function is convex

According to a calculus book I have been reading, we call a function $g(x)$ a convex function if $$g(\lambda x +(1-\lambda)y) \leq \lambda g(x) +(1-\lambda)g(y)$$, for all $x,y$ and $0<\lambda&...
4
votes
4answers
87 views

Verify the following limit using epsilon-delta definition: $ \lim_{(x,y)\to(0,0)}\frac{x^2y^2}{x^2+y^2}=0$

Show that $$ \lim\limits_{(x,y)\to(0,0)}\dfrac{x^2y^2}{x^2+y^2}=0$$ My try: We know that, $$ x^2\leq x^2+y^2 \implies x^2y^2\leq (x^2+y^2)y^2 \implies x^2y^2\leq (x^2+y^2)^2$$ Then, $$\dfrac{x^2y^2}{x^...
2
votes
0answers
68 views

A generalization involving a logarithmic integral

In the preprint, A note presenting the generalization of a special logarithmic integral by Cornel Ioan Valean, it is given the following generalization, Let $n\ge1$ be a positive integer. Then, \...
1
vote
4answers
63 views

a hint for this Taylor series$ \frac{\cos\left(2x\right)-1}{x^2}$

Compute the first three terms (nonzero) $\frac{\cos\left(2x\right)-1}{x^2}$ the first term is $\cos \left(2\right)-1$ but in the answer, the first term that I have to choose is... $-2$ or $2$ or $-1/...
6
votes
2answers
132 views

Is this true for all polynomials

I took $3$ random polynomials with non zero roots one having even degree and two having odd degrees $f(x)=\color{red}{4}x^2-(4\sqrt3+12)x+12\sqrt3$ having roots $\color{blue}{3,\sqrt3}$ and leading ...
-1
votes
3answers
47 views

What is Jacobian in Vector Calculus? [closed]

What is A Jacobian? How will you explain it in layman language? What are it's applications?