BC Calc Memorization Sheet
b
V   r 2 Derivatives
dx
Integrals
x n1
n
 x dx  n  1  C , n  1
a
d n
x  nx n 1
dx
d
1
ln x 
dx
x
d
1
logb x 
dx
x ln b
a
Shell
f ( x) 0

 or  ,
If lim
x  a g ( x)
0

f ( x)
f '( x)
then lim
 lim
x a g ( x )
x a g '( x)
b
V  2  rh dx
a
Cross Section
b
V   A dx
a
First Fundamental Theorem
d
dx
Decomposing into P.F.
1
A
B


(cx  d )(hx  k ) (cx  d ) (hx  k )
1

f  t  dt  f  g ( x)   g '( x)
g ( x)
a
Second Fundamental Theorem
 f t  dt  F (b)  F (a) where F’(x) = f(x)
b
a
Position, Vel, Acc
d
d
v(t )  ( pos) a(t )  (v(t ))
dt
dt
b
Alt. Series Error: error  an1 (the next
term)
Lagrange Error:
displacement   v(t )dt
error 
a
f
n 1
b
T .D.T .   v(t ) dt
f
a
Inv Fun Theorem
(a,b) slope = m
cos x  1 
m
2!
 x  a 
2
f   a 
3!
where
Logistic
k
dP

P
dt
M
P
n
4
 x  a    
3
Maclaurin Series ( Taylor series with a  0 )
1
f
 M  P
M
1  Ce kt
M = carrying capacity
 1 x  
x
x
   
2! 4!
 2n  !
2
Taylor Series
f   a 
n 1
c
McLaurin Series to have memorized
x 2 x3
xn
x
e  1  x       
2! 3!
n!
n
3
5
1 x 2 n 1

x
x
sin x  x     
 
3! 5!
 2n  1!
Definition of Derivative
f ( x  h)  f ( x )
f ( x)  lim
h 0
h
f  x   f  a   f   a  x  a  
n 1
 c  b  a 
 n  1!
is the maximum value of fn+1(x) on [a,b].
speed  vel
dx
slope = 1
V     R 2  r 2  dx
L’Hopital’s Rule
Integration by Parts
 u dv  uv   v du
Quot.
f ' g  fg '
d  f 



dx g
g2
 
Chain
d
f  g ( x)   f   g ( x)   g '( x)
(b,a)
a
b
U-Substitution
.
 f ( g ( x))dx let u = g(x)
 f  g   f ' g  fg '
f 1 ( x)
2
Washer
Integration Rules
Differentiation Rules
Prod.
f(x)
b
V    r 2 dx
sin x  cos x  1
2
1
1  x2
1
d
arccos x  
dx
1  x2
1
d
arctan x 
dx
1  x2
d
dx
Volume
Disc
sin x
tan x 
cos x
 x dx  ln x  C
d x
e  ex
dx
d x
b  b x  ln b
dx
d
sin x  cos x
dx
d
cos x   sin x
dx
d
tan x  sec2 x
dx
d
sec x  sec x  tan x
dx
d
arcsin x 
dx
Trig Identities
2n
 n
 a  x  a n  


n!
Pt Slope Form
y - y1 = m(x – x1)
(x,y)
dy
dx
Euler’s Method
x
dy
y  x
dx
(x,y)
f (b)  f (a)
(slope between two points)
ba
Inst. Rate of Change: IROC  f   c  (slope at a single point)
Average Rate of Change: AROC 
Mean Value Thm Part 1: f (c) 
f (b)  f (a)
ba
Rolles Thm.: if f(a) = f(b), then f’(c) = 0
 f  x  dx

 f  x  dx
Mean Value Thm Part 2: f (c) 
b
Average Value of a Function:
b
a
f avg
a
ba
ba
Intermediate Value Thm. A function f(x) that is continuous on  a, b takes on every y-value between
f  a  and f  b  .
Extreme Value Thm: If f(x) is continuous on  a, b , then f(x) must have both an absolute min and absolute
max on the interval  a, b .
Arc Length  
2
 dy 
1    dx
 dx 
b
a
Arc Length  
     dt
t2
dx
dt
t1
   
dx
dt
2
 dy
dy
dt
2
parametric
cartesian
Speed = 
2
2
dt
T.D.T. = 
    dt
t2
dx
dt
t1
2
 dy
2
dt
d  dy 
dy
 
2
d
y
dy
dt
 dx 
dt ,
Parametric Derivatives:

 dx
2
dx
dx
dx
dt
dt
y
2
2
2
Polar Conversions: r  x  y , x  r cos , y  r sin  ,   arctan
x
1 2 2
Polar Area 
r d
2 1

Area of Trapezoid
A = 1/2h(b1 + b2)
Tests for Convergence/Divergence
div. if lim an  0 (cannot be used to show convergence)
th
n term test
n 

 ar
Geom. series test
r  1  conv. , r  1  div. , S=
n
n 0

n 1
Ratio test
p
p  1  conv. , p  1  div.
decr. terms and lim an  0  conv.
Alternating series
Integral test
1
n
p -series
a
1 r
n 
an  f ( x)




n 1
1
n 1
1
 an conv. if  f ( x) dx conv.,  an div. if  f ( x) dx div.
an 1
a
a
 1  conv. , lim n 1  1  div. , (inconclusive if lim n 1  1)
n  a
n  a
n  a
n
n
n
lim
(works well for factorials and exponentials)
Direct Comparison
Limit Comparison
a series with terms smaller than a known convergent series also converges
a series with terms larger than a known divergent series also diverges
a
if lim n is finite and positive both series converge or both diverge
n  b
n
(use with "messy" algebraic series, usually compared to a p-series)
2