AP Calc AB

THINGS TO KNOW BY HEART

MVT - Mean Value Theorem (formula)

Pre calc slope = calc slope

\frac{f\left(b\right)-f\left(a\right)}{b-a}=f^{\prime}\left(c\right)

MVT - Mean Value Theorem (description)

Diff (a,b) and continuous [a,b]
There exists at least one point c in the interval (a, b) such that the derivative at that point equals the average rate of change of the function over the interval.

Rolle’s theorem (description)

Diff (a,b) and continuous [a,b]
There exists at least one point c in the interval (a,b) where the derivative of the function is zero.

Rolle’s theorem (formula)

f\left(b\right)=f\left(a\right)

f^{\prime}\left(b\right)=0

IVT - Intermediate Value Theroem

Continuous [a,b]
Implies that for any value between f(a) and f(b), there exists at least one point c in the interval (a, b) such that f(c) equals that value.

EVT - Extreme Value Theorem

Continuous [a,b]
There’s an asbolute max and absolute min.

Average of f on a to b

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

Rate of change of f

f^{\prime}

Average rate of change of f on a to b (formula)

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

Average rate of change of f on a to b (in words)

Pre calc slope

Average rate of change of f on a to b (complicated ahh formula)

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

Average rate of change of f on a to b

if f is increasing? f ‘ =

positive

if f is increasing? graph of f ‘ is

above the x axis

if f is decreasing? f ‘ =

negative

if f is decreasing? graph of f ‘ is

below the x axis

f has a max or min, graph of f’ ?

crosses the x axis

f has a max or min, f’ ?

equals zero or undefined

f has a max, f’ ?

goes from positive to negative

f has a min, f’ ?

goes from negative to positive

f is concave up, f’’ ?

is positive

f is concave down, f’’ ?

is negative

f has a point of inflection (formula)

f’’ = 0 , undefined

f has a point of inflection (graphically)

f’’ crosses the x axis

velocity

acceleration

position + C

pos(b) - pos(a)

change in position

distance

distance

integral of the absolute value of velocity

graph

crosses at x = 1

graph

crosses at y = 1