1
BC Calc Memorization Sheet
2
b
a
V r dx
Derivatives
1nn
d
dx x nx
1
ln
d
dx x
x
1
ln
logb
d
dx x b
x
xx
d
dx ee
ln
xx
d
dx b b b
sin cos
d
dx xx
cos sin
d
dx xx
2
tan sec
d
dx xx
sec sec tan
d
dx x x x
2
1
arcsin 1
d
dx xx
2
1
arccos 1
d
dx xx
2
1
arctan 1
d
dx xx
Integrals
1,
1
1nC
n
x
dxxn
n
1ln
xdx x C
.
Trig Identities
sin
cos
tan x
x
x
1cossin 22 xx
McLaurin Series to have memorized
23
12! 3! !
n
xx x x
ex n
21
35 1
sin 3! 5! 2 1 !
nn
x
xx
xx n
2
24 1
cos 1 2! 4! 2 !
nn
x
xx
xn
Taylor Series
23
2! 3! !
nn
f a f a f a
f x f a f a x a x a x a x a
n
Maclaurin Series ( Taylor series with
0a
)
Logistic
dP
dt
kP M P
M
1kt
M
PCe
M = carrying capacity
Euler’s Method
(x,y)
dy
dx
x
dy
yx
dx
(x,y)
First Fundamental Theorem
() ( ) '( )
gx
a
d
dx f t dt f g x g x
Alt. Series Error:
1
error n
a
(the next
term)
Lagrange Error:
1
1
error 1!
n
n
f c b a
n
where
1n
fc
is the maximum value of fn+1(x) on [a,b].
Volume
Disc
2
b
a
V r dx
Washer
22
b
a
V R r dx
Shell
2b
a
V rh dx
Cross Section
b
a
V A dx
Definition of Derivative
0
( ) ( )
( ) lim
h
f x h f x
fx h
Second Fundamental Theorem
( ) ( )
b
af t dt F b F a
where F’(x) = f(x)
Differentiation Rules
Prod.
''
d
dx f g f g fg
Quot.
2
''
d
dx
f f g fg
gg
Chain
( ) ( ) '( )
d
dx f g x f g x g x
Integration Rules
U-Substitution
( ( ))f g x dx
let u = g(x)
Integration by Parts
udv uv vdu
Decomposing into P.F.
1
( )( ) ( ) ( )
AB
cx d hx k cx d hx k
Position, Vel, Acc
( ) ( )
d
v t pos
dt
( ) ( ( ))
d
a t v t
dt
()
b
a
displacement v t dt
. . . ( )
b
a
T DT v t dt
speed vel
L’Hopital’s Rule
If
( ) 0
lim or
( ) 0
xa
fx
gx
,
then
( ) '( )
lim lim
( ) '( )
x a x a
f x f x
g x g x
Inv Fun Theorem
f(x) (a,b) slope = m
1()fx
(b,a) slope =
1m
Pt Slope Form
y - y1 = m(x – x1)
