math (fweak my life)

this will be $20 per slide pls

average rate of change (AROC)- slope between two points

\frac{f\left(b\right)-f\left(a\right)}{b-a}

inst. rate of change (IROC)- slope at a single point

f\text{'(c)}

mean value thm for a function: f\text{'(c)}=

\frac{f\left(b\right)-f\left(a\right)}{b-a}

rolles thm: if f(a)=f(b) then f’(c)=

0

average value of a function: f_{avg} =

\frac{\int_{a}^{b}\!f\left(x\right)\,dx}{b-a}

intermediate value thm: a function f(x) that is continuous on [a,b]

takes on every y-value between f(a) and f(b)

extreme value thm: if f(x) is continuous on [a,b]

then f(x) must have both an absolute min and absolute max on interval [a,b]

arc length (cartesian)

\int_{a}^{b}\!\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx

arc length (parametric)

\int_{t_1}^{t_2}\!\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt

speed

\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,

total distanced travelled

\int_{t_1}^{t_2}\!\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt

polar area

\frac12\int_{\theta_1}^{\theta_2}\!r^2\,d\theta

\frac{dy}{dx}=

\frac{\frac{dy}{dt}}{\frac{dx}{dt}}

\frac{d^2y}{dx^2} =

\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

r²=

x²+y²

x=

r\cos\theta

y=

r\sin\theta

\theta =

\arctan\frac{y}{x}

Area of a trapezoid: A=

\frac12h\left(b_1+b_2\right)

\frac{d}{dx}x^{n} =

nx^{n-1}

\frac{d}{dx}\ln x =

1/x

\frac{d}{dx}\log_{b}x =

\frac{1}{x\ln b}

\frac{d}{dx}e^{x} =

e^{x}

\frac{d}{dx}b^{x} =

b^{x}\cdot\ln b

\frac{d}{dx}\sin x =

cosx

\frac{d}{dx}\cos x =

-sinx

\frac{d}{dx}\tan x =

\sec^2x

\frac{d}{dx}\sec x =

\sec x\cdot\tan x

\frac{d}{dx}\arcsin x =

\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx}\arccos x =

- \frac{1}{\sqrt{1-x^2}}

\frac{d}{dx}\arctan x =

\frac{1}{1+x^2}

definition of a derivative: f’(x)=

\lim_{h\overrightarrow{}0}\frac{f\left(x+h\right)-f\left(x\right)}{h}

product rule: \frac{d}{dx}\left(f\cdot g\right) =

[f’(x)g(x)]+[g’(x)f(x)]

quotient rule: \frac{d}{dx}\left(\frac{f}{g}\right) =

\frac{\left\lbrack f^{\prime}\left(x\right)g\left(x\right)\right\rbrack-\left\lbrack g^{\prime}\left(x\right)f\left(x\right)\right\rbrack}{g^2}

chain rule: \frac{d}{dx}f\left(g\left(x\right)\right) =

f^{\prime}\left(g\left(x\right)\right)\cdot g^{\prime}\left(x\right)