Trig / Calc A Spring Final Cheat SheetRational Functions

• y-intercept: plug in x=0 and solve

• x-intercept: set numerator=0 and solve

• Holes: common factors in n and d

• Vertical Asymptotes: denominator=0 (after holes

• !nd "ehavior Asymptoteo #$%& y=0o #=%& y=ratio of highest 'so #%& y=synthetic division

• %omain: denominator)0

• For ine*ualities& ans+er is a domaino Factor ,rsto ove everything to one sideo Find critical num.ers using convenient values

(+hatever ma/es  x=0

o a/e sign chart

imits

•

0

0 is #12 %#!

• l3H4pital3s Rule: lim x →h( x)

f ( x)g( x )

 is the same as

lim x →h( x)

f '  ( x )

g'  ( x )

• imits to 5 or -5 are end .ehavior *uestions

• 6ontinuityo Holes are remova.leo V7A7s are remova.le (den=0o "rea/s are in piece+ise (including a.solute values

are non-remova.le (chec/ given x values in pieces

• imit %#! for polynomials at 5

• og limits at 5& 8ust solve log%erivatives

•   f '  ( x )=lim

h→ 0

f  ( x+h )−f  ( x )h

• All %erivative Rules

o c=0

o  x=1

o c∗f  ( x )=c∗f ' ( x )

o [ f  ( x ) ]n=n [ f  ( x ) ]

n−1

∗f ' ( x )

o f  ( x )± g ( x )=f  '  ( x ) ± g

' ( x )o f  (g ( x ) )= f 

' (g ( x ) )∗g'  ( x )

o f  ( x )∗g ( x )=f  ( x ) g' ( x )+g ( x ) f '  ( x )

o

f  ( x )

g ( x )=

g ( x )∗f ' ( x )−f  ( x)∗g

' ( x )

g2 ( x )

o

a

ln¿¿

af  ( x )=af  ( x )∗¿

o loga f  ( x )=  f 

'  ( x )

f  ( x )∗lnao sin f  ( x )=[cos f  ( x ) ]∗f '  ( x )o cos f  ( x )=−[sin f  ( x ) ]∗f 

' ( x )o tan f  ( x )= [sec

2f  ( x ) ]∗f 

' ( x )o cot f  ( x )=−[csc

2f  ( x ) ]∗f 

' ( x )o sec f  ( x )= [ secf  ( x ) tan f  ( x ) ]∗f 

 '  ( x )o cscf  ( x )=−[ cscf  ( x )cot f  ( x ) ]∗f 

'  ( x )

!xamples

•

 x−2

| x−2|¿

={  1, x>2−1, x<2

 x →2−¿¿

lim¿

¿