AP Calc - Formulas

continuity

point f(c ) exists, limit exists, limit is the same as point

critical point

f’(x) = 0, DNE

POI

concavity changes - f’’(x) changes sign, f’(x) changes direction

min/max

f’(x) = 0, changes sign

d/dx sec x

sec x tan x

d/dx csc x

- csc x cot x

d/dx arcsin x

1 / sqrt(1-x²)

d/dx arccos x

-1 / sqrt(1-x²)

d/dx arctan x

1 / (1+x²)

d/dx arccot x

-1 / (1+x²)

d/dx arcsec x

1 / [|x| sqrt(x²+1)]

d/dx arccsc x

-1 / [|x| sqrt(x²+1)

d/dx log(base a) x

1 / x ln a

d/dx a^x

a^x ln a

lim x—> 0 (sin x / x)

1

lim x—> 0 (1-cosx / x)

0

limit definition of derivative

lim Δx—>0 [f(x+Δx) - f(x)] / Δx

alternate limit definition of derivative

lim x—>c [f(x)-f(c )] / (x-c)

average rate

[f(b) - f(a)] / b - a

displacement

int (a,b) f(x) dx

total distance

| int (a,b) f(x) dx |