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BC Calculus Formula Sheet, Cheat Sheet of Calculus

BC calculus memorization sheet with derivatives, integrals, trig identities, volume, differentiation rules and Taylor series.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

ekagarh
ekagarh 🇺🇸

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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
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
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)
pf2

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BC Calc Memorization Sheet

b 2

a

V  r dx

^ Derivatives

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

M
P

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

V   r dx

Washer

 

b 2 2

a

V   R r dx

Shell

b

a

V   rh dx

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.

A B

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)

Inst. Rate of Change: IROC f  c (slope at a single point)

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

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, .

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

Arc Length    

2

1

t^2

t

dx dy

dt dt

  dt

Speed =    

2 2 dx dy

dt dt

  T.D.T. =    

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

r  x  y , x  r cos , y rsin , arctan

y

x

 

cartesian parametric

Area of Trapezoid

A = 1/2h(b 1 + b 2 )