You mention mathematical writing and symbols/notation. Though I’ve never taken a linear algebra course, here are some relevant tips: Make sure every line of work has a meaningful relation (symbol) in ...

It depends on your audience and the context. In physics, $\mathcal E$ usually denotes emf, so you would say that. In set theory, $\mathcal P$ usually denotes a power set, so you would say that. ...

Historically speaking, it is not likely that beautifying the Taylor expansion of sine was “the” reason for defining the radian. After all, the radian was defined by the relationship $r\theta=s$, which ...

$$x=-1 \implies x^2 = 1$$ but $$x=-1 \not\Longleftarrow x^2=1$$ because it could be that $x=1$.

Edit: In light of Carmeister’s comment, I would like to clarify that (in my opinion) there does exist an integral analogue to the product rule, and that analogue is integration by parts. ...

Roll the dice and eliminate any that show up blank. Continue rolling until one die remains. If all of your remaining dice get eliminated in one throw, then reroll. Edit (thanks Rahul): Eliminate ...

Look at the middle line of points formed by the intersections of the line segments. Count how many segments run through each point, counting the left- and rightmost points as having one segment ...

It’s really quite simple: $\lim_{n\to\infty}\left[ (-1)^n\right]$ does not approach $0$; hence $\sum_{n=1}^\infty\left[ (-1)^n\right]$ diverges.

I agree that there is no formal difference and that either is acceptable provided you are consistent. I also think that punctuation creates unnecessary clutter. However, it is important to keep your ...

The connecting factor amongst the hyperbolic and circular functions and the unit circle and hyperbola is not the angle subtended by the curve but rather the area bounded by it. Here are two images I ...

These are often called operators. Some operators take two arguments, called binary, like $()\times()$ or $()-()$; others take only one, called unary, such as $()^2$. Some examples of operators or “...

I’m not entirely sure if this is what you’re asking, but I suppose you could use the first few terms of the Taylor series $$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} +\cdots$$ for $|x|<\pi/2$....

Perhaps you create a mapping from the length of each segment to the number of segments necessary, namely $n= (1 \ \mathrm{cm})/{\ell}$. Graph this function with $n$ on the vertical axis and $\ell / \... View answer Accepted answer 3 votes Desmos.com all the way. It does everything you want and more. They also have a mobile app that will sync with your account on desktop. Go check it out. View answer 3 votes Hint: $$e^{iat}=\cos at + i \sin at$$ View answer Accepted answer 3 votes You are correct to transform$2\log_b x + 2\log_b (1-x) = 4$into$x^2-x+b^2=0$. These are the restrictions implicated by the original equation:$b$is a positive number not equal to$0$nor$1x&...

The issue here is the difference between equality $=$ and approximation $\approx$. Here is what’s actually happening: $$\frac39 = \frac13 = 0.33333\cdots \approx 0.3$$ The approximation comes when ...

Formulaic approach via the substitution $u=t^3$: \begin{align} {d \over dt}\left(2^{t^3}\right) &= {d \over dt}\left( 2^u \right) \\ &= 2^u\ln(2){du \over dt} \\ &= \ln(2)2^{\left(t^3\... View answer Accepted answer 3 votes We can calculate the angle by using the properties of dot products. Since R is the vertex of the angle, \angle PRQ = \Theta is between \overrightarrow{RP} and \overrightarrow{RQ}.\...

When we divide by $\Delta x$, why do we put the $\Delta x$ under the other $\Delta$? Why does it not go under the constant v or u? If you’re asking why we write $u\frac{\Delta v}{\Delta x}$ ...

Say $Z_1 = a+bi$ and $Z_2 = x+yi$ where $\{a,b,x,y\}\subset\Bbb{R}$. Say $Z_1\overline{Z_2} = R+Ji$ where $\{R,J\}\subset\Bbb R$. The notation $\overline Z$ means the complex conjugate of $Z$. The ...
Two vectors $\mathbf v$ and $\mathbf w$ are in the same direction if and only if $$\frac{\mathbf{v}}{v}\cdot\frac{\mathbf{w}}{w}=1$$ One of the many ways your can rephrase this is $\mathbf{\hat v}=\... View answer 3 votes Adding up the sections of$A_h$is a lot like stacking up incrementally smaller onion rings and adding up the surface areas of their exteriors. a very dextrous sketch of a stack of onion rings in the ... View answer 3 votes Polynomials are already in this Taylor form. If you want to start with a polynomial, then convert it into this Taylor form, you can do that simply by differentiating the polynomial up to$ntimes ... View answer 3 votes \begin{align} f(x) &= \tan x \\ f’(x) &= \sec^2x \\ f’’(x) &= 2\sec(x)\tan(x) \\ \end{align} You should just memorizef’(x)$, and$f’’(x)$comes from chain rule. Assuming we expand ... View answer Accepted answer 3 votes I believe you are asking if integration can be defined in a way other than setting an identity (e.g., with a limit like differentiation). This is how the Riemann integral is defined. The integral is ... View answer 3 votes If I understand the problem correctly, these two vectors will outline two of the four sides of something that looks like a slanted rectangle. Therefore, the length of the diagonal will be the segment ... View answer Accepted answer 2 votes Call the unique base-ten expansion of a finite number$X$that doesn’t involve insignificant zeroes the series $$X = \sum_{k=a}^{b} 10^k x_k$$ Therefore, in base ten, the$k$th digit of$X$is$x_k$(... View answer Accepted answer 2 votes You can parametrise: get the equation in terms of$t$. Or you could convert to polar coordinates: use the form$r$as a function of$\theta$. However, in general, this can be complicated. Instead you ... View answer Accepted answer 2 votes Take$\newcommand{\e}{\mathrm e}\newcommand{\i}{\mathrm i} z= \sqrt{11}-\i = r\e^{\theta\i}$, where$r=\lvert z\rvert$. Then$\$\begin{align} z^{1000} &= (r\e^{\theta\i})^{1000} \\ &= r^{1000}\...