Crib Sheet AP Calculus AB and BC exams

1. typed by Sean Bird, Covenant Christian High School
updated May 2014
AP CALCULUS
Stuff you MUST know Cold * means topic only on BC
Curve sketching and analysis
y = f(x) must be continuous at each:
critical point:
dy
dx
= 0 or undefined
local minimum:
dy
dx
goes (–,0,+) or (–,und,+) or
2
2
d y
dx
>0
local maximum:
dy
dx
goes (+,0,–) or (+,und,–) or
2
2
d y
dx
<0
point of inflection: concavity changes
2
2
d y
dx
goes from (+,0,–), (–,0,+),
Differentiation Rules
Chain Rule
 ( ) '( )
d du dy dy du
f u f u
dx dx dx du
R
x
O
d
 
Product Rule
( ) ' '
d du dv
uv v u OR u
dx
v
dx dx
uv 
Quotient Rule
2 2
' 'du dv
dx dxv u u v uv
O
d u
R
dx v vv
 
 



Approx. Methods for Integration
Trapezoidal Rule
1
0 12
1
( ) [ ( ) 2 ( ) ...
2 ( ) ( )]
b
b a
na
n n
f x dx f x f x
f x f x


  
 

With data, do each trap separately
using ½ (f(x1)+f(x2)). It is an over
approximation if ( ) (concave up)
Riemann Sums are rectangles.
Left Riemann sums under approximate
when f(x) is increasing ( ( ) )…
Theorem of the Mean Value
i.e. AVERAGE VALUE
Basic Derivatives
  1n nd
x nx
dx


 sin cos
d
x x
dx

 cos sin
d
x x
dx
 
  2
tan sec
d
x x
dx

  2
cot csc
d
x x
dx
 
 sec sec tan
d
x x x
dx

 csc csc cot
d
x x x
dx
 
 
1
ln
d du
u
dx u dx

 u ud du
e e
dx dx

where u is a function of x,
and a is a constant.
“PLUS A CONSTANT” If the function f(x) is continuous on [a, b]
and the first derivative exists on the
interval (a, b), then there exists a number
x = c on (a, b) such that
( )
( )
( )
b
a
f x dx
f c
b a



This value f(c) is the “average value” of
the function on the interval [a, b].
The Fundamental Theorem of
Calculus
( ) ( ) ( )
where '( ) ( )
b
a
f x dx F b F a
F x f x
 


Corollary to FTC
( )
( )
( ( )) '( ) ( ( ))
( )
'( )
b x
a x
f b x b x f a x
f t dt
d
d
a x
x



Solids of Revolution and friends
Disk Method
 
2
( )
x b
x a
V R x dx


 
Washer Method
    2 2
( ) ( )
b
a
V R x r x dx 
General volume equation (not rotated)
( )
b
a
V Area x dx 
*Arc Length  
2
1 '( )
b
a
L f x dx 
   
2 2
'( ) '( )
b
a
x t y t dt 
Intermediate Value Theorem
If the function f(x) is continuous on [a, b],
and y is a number between f(a) and f(b),
then there exists at least one number x= c
in the open interval (a, b) such that
( )f c y .
More Derivatives
1
2 2
1
sin
d u du
dx a dxa u
 
 
  
 1
2
1
cos
1
d
x
dx x
 


1
2 2
tan
d u a du
dx a dxa u
 
    
 1
2
1
cot
1
d
x
dx x
 


1
2 2
sec
d u a du
dx a dxu u a
 
  
  
 1
2
1
csc
1
d
x
dx x x
 


 
   
ln
u x u xd du
a a a
dx dx
 
 
1
log
ln
a
d
x
dx x a

Mean Value Theorem
If the function f(x) is continuous on [a, b],
AND the first derivative exists on the
interval (a, b), then there is at least one
number x = c in (a, b) such that
( ) ( )
'( )
f b f a
f c
b a



.
Distance, Velocity, and Acceleration
velocity = d
dt
(position)
acceleration = d
dt
(velocity)
*velocity vector = ,
dx dy
dt dt
speed = 2 2
( ') ( ')v x y  *
displacement =
f
o
t
t
vdt
final time
initial time
2 2
distance =
( ') *'( )
f
o
t
t
v dt
x y dt


average velocity =
final position initial position
total time


=
x
t


Rolle’s Theorem
If the function f(x) is continuous on [a, b],
AND the first derivative exists on the
interval (a, b), AND f(a) = f(b), then there
is at least one number x = c in (a, b) such
that
'( ) 0f c  .
Derivative Formula for Inverses
df
dx df
dx
x f a
x a




1
1
( )
(+,und,–), or (–,und,+)
(and look out for endpoints)

2. BC TOPICS and important TRIG identities and values
l’Hôpital’s Rule
If
( ) 0
or
( ) 0
f a
g b

 

,
then
( ) '( )
lim lim
( ) '( )x a x a
f x f x
g x g x 

Slope of a Parametric equation
Given a x(t) and a y(t) the slope is
dy
dt
dx
dt
dy
dx

Values of Trigonometric
Functions for Common Angles
θ sin θ cos θ tan θ
0° 0 1 0
6
 1
2
3
2
3
3
4
 2
2
2
2
1
3
 3
2
1
2
3
2

1 0 “ ”
π 0 1 0
Know both the inverse trig and the trig
values. E.g. tan(π/4)=1 & tan-1
(1)= π/4
Euler’s Method
If given that ( , )dy
dx f x y and that
the solution passes through (xo, yo),
1 1 1
( )
( ) ( ) ( , )
o o
n n n n
y x y
y x y x f x y x  

  
In other words:
new oldx x x  
 old old
new old
,x y
dy
y y x
dx
  
Polar Curve
For a polar curve r(θ), the
AREA inside a “leaf” is
 
2
1
2
1
2 r d


   
where θ1 and θ2 are the “first” two times that r =
0.
The SLOPE of r(θ) at a given θ is
 
 
/
/
sin
cos
d
d
d
d
dy dy d
dx dx d
r
r




 
 

  
  
Integration by Parts
udv uv vdu  
Ratio Test
The series
0
k
k
a


 converges if
1
lim 1k
k
k
a
a



If the limit equal 1, you know nothing.
Trig Identities
Double Argument
sin2 2sin cosx x x
2 2 2
cos2 cos sin 1 2sinx x x x   
 2 1
cos 1 cos2
2
x x 
Integral of Log
Use IBP and let u = ln x (Recall
u=LIPET)
ln lnxdx x x x C  
Taylor Series
If the function f is “smooth” at x =
a, then it can be approximated by
the nth
degree polynomial
2
( )
( ) ( ) '( )( )
''( )
( )
2!
( )
( ) .
!
n
n
f x f a f a x a
f a
x a
f a
x a
n
  
  
 
Lagrange Error Bound
If ( )nP x is the nth
degree Taylor polynomial
of f(x) about c and ( 1)
( )n
f t M
 for all t
between x and c, then
 
1
( ) ( )
1 !
n
n
M
f x P x x c
n

  

 2 1
sin 1 cos2
2
x x 
Pythagorean
2 2
sin cos 1x x 
(others are easily derivable by
dividing by sin2
x or cos2
x)
2 2
2 2
1 tan sec
cot 1 csc
x x
x x
 
 
Reciprocal
1
sec cos sec 1
cos
x or x x
x
 
1
csc sin csc 1
sin
x or x x
x
 
Odd-Even
sin(–x) = – sin x (odd)
cos(–x) = cos x (even)
Some more handy INTEGRALS:
tan ln sec
ln cos
sec ln sec tan
xdx x C
x C
xdx x x C
 
  
  


Maclaurin Series
A Taylor Series about x = 0 is
called Maclaurin.
2 3
1
2! 3!
x x x
e x    
2 4
cos 1
2! 4!
x x
x    
3 5
sin
3! 5!
x x
x x   
2 31
1
1
x x x
x
    

2 3 4
ln( 1)
2 3 4
x x x
x x     
Alternating Series Error Bound
If  
1
1
N
n
N n
k
S a

  is the Nth
partial sum of a
convergent alternating series, then
1N NS S a  
Geometric Series
2 3 1 1
1
n n
n
a ar ar ar ar ar

 

       
diverges if |r|≥1; converges to
1
a
r
if |r|<1
This is available at http://covenantchristian.org/bird/Smart/Calc1/StuffMUSTknowColdNew.htm