Formula Quiz #1

tan x =

sin x/cos x

cot x =

cos x/sin x

sec x =

1/cos x

csc x =

1/sin x

double angle id’s

2sinxcosx = sin2x, cos²x - sin²x = cos2x

Pythagorean id’s

sin²x+cos²x=1, sec²x-tan²x = 1

Even - odd id’s

sin(-x) = -sinx, cos(-x) = cosx, tan(-x) = -tanx

sin(A+B) =

sinAcosB + cosAsinB

cos(A+B) =

cosAcosB-sinAsinB

sin(A-B) =

sinAcosB-cosAsinB

cos(A-B) =

cosAcosB+sinAsinB

|x| =

x if x>=0, -x if x<0

Law of cosines

c² = a²+ b² - 2abcosC

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(aⁿ) =

nlna

ln(1/a) =

-lna

ln(0) =

undefined

ln(1) =

0

ln(e) =

1

One sided limit, from the left

lim_x→a^-f(x)=L

One sided limit, from the right

lim_x→a⁺f(x)=L

Definition of a limit

lim_x→af(x)= L iff lim_x→a^-f(x)=L=lim_x→a⁺f(x)

sin =

y

cos =

x

lim (f+-g) =

limf +- limg

lim(c x f) =

c x limf

lim(fg) =

limf x limg

lim(f/g)

limf/limg for lim≠0

lim_x→ak =

k

lim_x→ax =

a

lim_x→asqrt(f(x)) =

sqrt(lim_x→af(x))

lim_x→af(g(x)) =

f(lim_x→ag(x)) provided that f is continuous at lim x >=a of g(x)

definition of continuity

A function f is continuous at x=c iff

1. f( c) exists

2. lim_x→cf(x) exists

3. lim_x→cf(x)=f(c )

Intermediate Value Theorem

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 lim_x→af(x) = lim_x→ah(x) then lim_x→ag(x) =L

common limits, lim_x→0

lim_x→0sinx/x = 1,lim_x→01-cosx/x = 0

Common limits, lim_x→a

lim_x→ax = a

Common limits, lim_x→0^-

lim_x→0^-1/x → -infinity

Common limits, lim_x→0⁺

lim_x→0⁺1/x→ infinity