AP Calc AB Formulas

sin²x + cos²x =

1

1 + tan²x =

sec^2 x

1 + cot²x =

csc^2 x

sin(A + B) =

sinA cosB + cosA sinB

sin(A - B) =

sinA cosB - cosA sinB

cos (A + B) =

cosA cosB - sinA sinB

cos (A - B)

cosA cosB + sinA sinB

sin(2x) =

2sinx cosx

cos(2x)

cos^2 x - sin^2 x

sin(-x) =

-sinx

cos(-x) =

cosx

tan(-x) =

-tanx

Definition of |x|

|x| = {
x if x >= 0
-x if x < 0

distance between two points

sqrt( (x2 - x1)^2 + (y2 - y1)^2 )

midpoint formula

((x1 + x2) / 2, (y1 + y2) / 2)

ln(ab) =

lna + lnb

ln(a/b) =

lna - lnb

ln(a^n) =

n * lna

ln(1/a) =

-lna

sec(x) =

1 / cos x

csc(x) =

1 / sin x

tan(x) =

sin x / cos x

cot(x) =

cos x / sin x

Definition of a limit

lim x -> a f(x) = L iff lim x -> a^- f(x) = L = lim x -> a^+ f(x)

lim x -> a f +- lim x -> a g

lim x -> a f * lim x -> a g

c

c * lim x -> a f

lim x -> a f / lim x -> a g for lim x -> a g != 0

Definition of Vertical Asymptote:

The line x = a is called a vertical asymptote iff lim x -> a^- f(x) = +-infinity or lim x -> a^+ f(x) = +-infinity

Definition of Horizontal Asymptote:

The line y = a is called a horizontal asymptote iff lim x -> -infinity f(x) = a or lim x -> infinity f(x) = a

if f grows faster than g, then

lim x -> infinity g(x) / f(x) = 0 and lim x -> infinity f(x) / g(x) = infinity

Definition of continuity

a function f is continuous at x = a iff
1. f(a) exists
2. lim x -> a f(x) exists
3. lim x -> a f(x) = f(a)

Intermediate Value Theorem (IVT): If, then

If

1. f is continuous on the closed interval [a,b]

2. f(a) != f(b)

3. k is between f(a) and f(b)

Then there exists a number c between a and b for which f(c) = k

Squeeze Theorem

if f(x) <= g(x) <= h(x) and as x -> a, f(x) -> L and h(x) -> L, then g(x) -> L

0

lim x -> infinity e^x =

infinity

-infinity

infinity

1

1

0

e^c = (2 equations)

lim x -> +- infinity (1 + c/x)^x
lim x -> 0^+ (1 + cx)^(1/x)

-pi / 2

pi / 2

0

0