Title: Math 2250 Facts to Memorize
Tags: Derivative Trig Identity Concepts Exponential Geometry Rules Algebra Polynomial

Q: [latex]$\sin(a+b)$[/latex]
A: [latex]$\sin a \cos b + \cos a \sin b$[/latex]

Q: [latex]$1 + \tan^2 \theta$[/latex]
A: [latex]$\sec^2 \theta$[/latex]

Q: [latex]$\cos(a+b)$[/latex]
A: [latex]$\cos a \cos b - \sin a \sin b$[/latex]

Q: [latex]$\sin 2\theta$[/latex]
A: [latex]$2 \sin \theta \cos \theta$[/latex]

Q: [latex]$\cos 2\theta$[/latex]
A: [latex]$\cos^2 \theta - \sin^2 \theta$[/latex]

Q: [latex]$\sin^2 \theta$[/latex]
A: [latex]$$\frac{1 - \cos2\theta}{2}$$[/latex]

Q: [latex]$\cos^2 \theta$[/latex]
A: [latex]$$\frac{1 + \cos2\theta}{2}$$[/latex]

Q: [latex]$\frac{d}{dx} \sin x$[/latex]
A: [latex]$\cos x$[/latex]

Q: [latex]$\frac{d}{dx} \cos x$[/latex]
A: [latex]$-\sin x$[/latex]

Q: [latex]$\frac{d}{dt} \tan x$[/latex]
A: [latex]$\sec^2 x$[/latex]

Q: [latex]$\frac{d}{dx} \cot x$[/latex]
A: [latex]$-\csc^2 x$[/latex]

Q: [latex]$\frac{d}{dx} \sec x$[/latex]
A: [latex]$\sec x \tan x$[/latex]

Q: [latex]$\frac{d}{dx} \csc x$[/latex]
A: [latex]$-\csc x \cot x$[/latex]

Q: [latex]$\frac{d}{dx} \arcsin x$[/latex]
A: [latex]$\frac{1}{\sqrt{1 - x^2}}$[/latex]

Q: [latex]$\frac{d}{dx} \arccos x$[/latex]
A: [latex]$\frac{-1}{\sqrt{1 - x^2}}$[/latex]

Q: [latex]$\frac{d}{dx} \arctan x$[/latex]
A: [latex]$\frac{1}{1+x^2}$[/latex]

Q: [latex]$\frac{d}{dx} \operatorname{arccot} x$[/latex]
A: [latex]$\frac{-1}{1 + x^2}$[/latex]

Q: [latex]$\frac{d}{dx} x^n$[/latex]
A: [latex]$n x^{n-1}$[/latex]

Q: [latex]$\frac{d}{dx} e^x$[/latex]
A: [latex]$e^x$[/latex]

Q: [latex]$\frac{d}{dx} a^x$[/latex]
A: [latex]$a^x \ln a$[/latex]

Q: [latex]$\frac{d}{dx} \ln x$[/latex]
A: [latex]$\frac{1}{x}$[/latex]

Q: [latex]$x^{a+b}$ = [/latex] [...]
A: [latex]$x^{a}x^{b}$[/latex]

Q: [latex]$(x^a)^b$ = [/latex] [...]
A: [latex]$x^{ab}$[/latex]

Q: [latex]$e^{\ln x}=$[/latex]
A: [latex]$x$[/latex]

Q: [latex]$\ln (e^x)$[/latex]
A: [latex]$x$[/latex]

Q: [latex]$\frac{d}{dx} (f(x) + g(x))$[/latex]
A: [latex]$f'(x) + g'(x)$[/latex]

Q: [latex]$\frac{d}{dx} k f(x)$[/latex]
A: [latex]$k f'(x)$[/latex]

Q: [latex]$\frac{d}{dx} f(x) \cdot g(x)$[/latex]
A: [latex]$f'(x) g(x) + f(x) g'(x)$[/latex] (product rule)

Q: [latex]$\frac{d}{dx} f(g(x))$[/latex]
A: [latex]$f'(g(x)) g'(x)$[/latex] (chain rule)

Q: The solutions of [latex]$a x^2 + b x + c = 0$[/latex] are given by [latex]$x = $[/latex]
A: [latex]$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$[/latex] (quadratic formula)

Q: How to complete the square: [latex]$x^2 + 2 a x + b =$[/latex]
A: [latex]$(x + a)^2 + (b-a^2)$[/latex]

Q: [latex]A derivative $\frac{d}{dt} f(t)$ with respect to a variable $t$ representing time is \dots [/latex]
A: [latex] the \emph{rate of change} of $f$ with respect to time. [/latex]

Q: [latex]The derivative of \textbf{position} $p(t)$ with respect to \textbf{time} $t$ is \dots [/latex]
A: [latex] \textbf{velocity} $v(t)$.[/latex]

Q: [latex] The derivative of \textbf{velocity} $v(t)$ with respect to \textbf{time} $t$ is \dots [/latex]
A: [latex] \textbf{acceleration} $a(t)$.[/latex]

Q: [latex] A \textbf{critical point} of a function $f(x)$ is a point where \dots [/latex]
A: [latex]$f'(x) = 0$ OR $f'(x)$ does not exist (a corner or asymptote).[/latex]

Q: [latex] The maximum and minimum values of $f(x)$ on a \textbf{closed} interval $[a,b]$ are located at \dots [/latex]
A: [latex]$a$, $b$ \textbf{or} a critical point of $f(x)$.[/latex]

Q: [latex] $$\frac{d}{dx} \frac{f(x)}{g(x)} = $$[/latex]
A: [latex]$$\frac{f'(x) g(x) - f(x) g'(x)}{g^2(x)}$$[/latex]

Q: [latex] The \textbf{definition} of the derivative $\frac{d}{dx} f(x)$ is [/latex]
A: [latex] $$\frac{d}{dx} f(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$ [/latex]

Q: [latex] The \textbf{linear} approximation to $f(x+\delta x)$ is [/latex]
A: [latex]$f(x + \delta x) \simeq f(x) + f'(x) \delta x$[/latex].

Q: [latex] The \textbf{quadratic} approximation to $f(x+\delta x)$ is [/latex]
A: [latex] $f(x + \delta x) \simeq f(x) + f'(x) \delta x + \frac{f''(x)}{2} (\delta x)^2$ [/latex]

Q: [latex] The \textbf{slope of the tangent line} to $f(x)$ at $x=a$ is [/latex]
A: [latex] $f'(a)$ [/latex]

Q: [latex] The distance between $(x_1,y_1)$ and $(x_2,y_2)$ in the plane is [/latex]
A: [latex]$\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$[/latex].

Q: [latex] The \textbf{area} of a circle of radius $r$ is [/latex]
A: [latex] $\pi r^2$ [/latex]

Q: [latex] The \textbf{perimeter} of a circle of radius $r$ is [/latex]
A: [latex] $2 \pi r$ [/latex]

Q: [latex] The \emph{volume} of a sphere of radius $r$ is [/latex]
A: [latex]$\frac{4}{3} \pi r^3$[/latex]

Q: [latex] The \textbf{area} of a sphere of radius $r$ is [/latex]
A: [latex] $4 \pi r^2$ [/latex]

Q: [latex] The volume of a cylinder of height $h$ and radius $r$ is [/latex]
A: [latex] $\pi r^2 h$ [/latex]