Articles of calculus

The derivative of a function of a variable with respect to a function of the same variable

I have a function $f(x)$ defined as follows: $$ f(x)=\frac{1-g(x)}{1-xg(x)}. $$ Because $f$ contains the function $g(x)$, I guess you could say $f$ is a function of $g(x)$ and $x$. Given this, a mathematician says $$ \frac{\partial}{\partial x}f=\frac{\partial f}{\partial g}\frac{\partial g}{\partial x} + \frac{\partial f}{\partial x}. $$ I’m not following his reasoning. For one, I […]

using the same symbol for dependent variable and function?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$, $y=y(t)$ where $t$ is the parameter? Also, if someone write the following equation $y=y(t) = t^2$ where $y$ represents the dependent variable and t represents […]

How to obtain asymptotes of this curve?

I am reading a text, where the neutral curve as a result of linear stability analysis of a delayed differential equation is given. $\delta$ and $\alpha$ are parameters of this model. The curve is defined as $$ \delta=\arccos\left(\frac{3\alpha-2}{\alpha}\right)\frac{1}{\sigma_i(\alpha)},$$ where $$\sigma_i(\alpha)=\sqrt{\alpha^2-(2-3\alpha)^2}.$$ Furthermore, I know that $\delta\gt 0$ and $0\lt\alpha\leq 1$. I do understand how these equations […]

Finding the antiderivative of the product of two functions given only their derivative properties

let $\alpha'(x)=\beta(x), \beta'(x)=\alpha(x)$ and assume that $\alpha^2 – \beta^2 = 1$. how would I go about calculating the following anti derivative : $\int (\alpha (x))^5 (\beta(x))^4$d$x$. Thank you.

On the $r$-neighborhood of a set in $\mathbb{R}^2$.

Suppose $A$ is a simply connected open set in $\mathbb{R}^2$ with rectifiable Jordan boundary. Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be the parametrization of $\partial A$. Let $A^r$ denote the $r$-neighborhood of $A$, that is, $A^r:=\{x: d(x,A) \le r\}$. Let $0=t_0<t_1< \cdots <t_n=1$ be the partition of $[0,1]$ such that $P(A_n) \rightarrow P(A)$ where $P$ is the […]

Evaluating $ \int \frac{1}{5 + 3 \sin(x)} ~ \mathrm{d}{x} $.

What is the integral of: $\int \frac{1}{5+3\sin x}dx$ My attempt: Using: $\tan \frac x 2=t$, $\sin x = \frac {2t}{1+t^2}$, $dx=\frac {2dt}{1+t^2}$ we have: $\int \frac{1}{5+3\sin x}dx= 2\int \frac 1 {5t^2+6t+5}dt $ I’ll expand the denominator: $5t^2+6t+5=5((t+\frac 3 5 )^2+1-\frac 1 4 \cdot (\frac 6 5)^2)=5((t+\frac 3 5)^2+0.64)$. So: $2\int \frac 1 {5t^2+6t+5}dt = \frac […]

Find by integrating the area of ​​the triangle vertices $(5,1), (1,3)\;\text{and}\;(-1,-2)$

Find by integrating the area of ​​the triangle vertices $$(5,1), (1,3)\;\text{and}\;(-1,-2)$$ I tried to make straight and integrate, but it is very complicated, there is some better way?

Calculate the slope of a line passing through the intersection of two lines

Let say I have this figure, I know slope $m_1$, slope $m_1$, $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$. I need to calculate slope $m_3$. Note the line with $m_3$ slope will always equally bisect line with $m_1$ slope and line with $m_2$.

Showing if $n \ge 2c\log(c)$ then $n\ge c\log(n)$

Is this true that if $n \ge 2c\log(c)$ then $n\ge c\log(n)$, for any constant $c>0$? Here $n$ is a positive integer.

Calculate the surface integral $\iint_S (\nabla \times F)\cdot dS$ over a part of a sphere

How can I calculate the integral $$\iint_S (\nabla \times F)\cdot dS$$ where $S$ is the part of the surface of the sphere $x^2+y^2+z^2=1$ and $x+y+z\ge 1$, $F=(y-z, z-x, x-y)$. I calculated that $\nabla\times F=(-2 , -2 ,-2)$. It’s difficult for me to find the section between the sphere and the plane. Also, I can’t calculate […]