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BC calculus memorization sheet with derivatives, integrals, trig identities, volume, differentiation rules and Taylor series.
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BC Calc Memorization Sheet
b 2
a
V r dx
d n n 1
dx
x nx
1 ln
d
dx x
x
1
ln
logb
d
dx x b
x
d x x
dx
e e
ln
d x x
dx
b b b
sin cos
d
dx
x x
cos sin
d
dx
x x
2 tan sec
d
dx
x x
sec sec tan
d
dx
x x x
2
arcsin
1
d
dx
x
x
2
arccos
1
d
dx
x
x
2
arctan 1
d
dx
x x
Integrals
, 1 1
1
C n n
x x dx
n n
1 ln x
dx x C
Trig Identities
sin
cos
tan
x
x
x
sin cos 1
2 2 x x
McLaurin Series to have memorized
2 3
1 2! 3!!
n x x^ x^ x e x n
3 5 2 1 1 sin 3! 5! 2 1!
n (^) n x x^ x x x n
2 4 2 1 cos 1 2! 4! 2!
n (^) n x x^ x x n
Taylor Series
2 3
2! 3!!
n f a f a f a n f x f a f a x a x a x a x a n
Maclaurin Series ( Taylor series with a 0 )
Logistic
dP
dt
k P M P M
kt
Ce
M = carrying capacity
Euler’s Method
(x,y) dy
dx
x dy y x dx
(x,y)
First Fundamental Theorem
( ) ( ) '( )
g x
a
d
dx
f t dt f g x g x
Alt. Series Error: error an (^) 1 (the next
term)
Lagrange Error:
1 1
error 1!
n^ n f c b a
n
where
n 1 f c
is the maximum value of f
n+ (x) on [a,b].
Volume
Disc
2
b
a
Washer
b 2 2
a
Shell
b
a
Cross Section
b
a
V A dx
Definition of Derivative
0
( ) lim h
f x h f x f x h
Second Fundamental Theorem
( )^ ( )
b
a
f 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
g g
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
u dv uv v du
Decomposing into P.F.
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 D T v t dt
speed vel
L’Hopital’s Rule
If
( ) 0 lim or x a ( ) 0
f x
g x
,
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 f ( )x
(b,a) slope =^1 m
Pt Slope Form
y - y 1 = m(x – x 1 )
Tests for Convergence/Divergence
Average Rate of Change: AROC
f b( ) f a( )
b a
(slope between two points)
Mean Value Thm Part 1:
b a
f b f a f c
( ) Rolles Thm.: if f(a) = f(b), then f’(c) = 0
Average Value of a Function:
b
a avg
f x dx
f b a
Mean Value Thm Part 2:
b
a
f x dx
f c b a
0
1
term test div. if lim 0 (cannot be used to show convergence)
a Geom. series test 1 conv. , 1 div. , S= 1
-series 1 conv. , 1 div.
Alternating s
th n n
n
n
p n
n a
ar r r r
p p p n
1 1 1 1
1 1
eries decr. terms and lim 0 conv.
Integral test ( ) conv. if ( ) conv., div. if ( ) div.
lim 1 conv. , lim 1 div. , (in Ratio test
n n
n n n n n
n n
n n n n
a
a f x a f x dx a f x dx
a a
a a
(^)
1 conclusive if lim 1)
(works well for factorials and exponentials)
a series with terms than a known convergent series also converges Direct Comparison a series with terms than a kn
n
n n
a
a
smaller
larger own divergent series also diverges
if lim is finite and positive both series converge or both diverge Limit Comparison
(use with "messy" algebraic series, usually compared to a -series)
n
n n
a
b
p
Arc Length
2
1
b
a
dy dx dx
2
1
t^2
t
dx dy
dt dt
dt
2 2 dx dy
dt dt
2
1
t^2
t
dx dy
dt dt
dt
Polar Area
2
1
(^12)
2
r d
Parametric Derivatives:
dy
dy (^) dt
dx dx
dt
2
2
d dy
d y dt dx
dx dx dt
Polar Conversions:
2 2 2
y
x
cartesian parametric
Area of Trapezoid
A = 1/2h(b 1 + b 2 )