Stuff you MUST know Cold
for the AP Calculus Exam
in the morning of Wednesday, May 7, 2008.
Sean Bird
AP Physics & Calculus
Covenant Christian High School
7525 West 21st Street
Indianapolis, IN 46214
Phone: 317/390.0202 x104
Email: seanbird@covenantchristian.org
Website: http://cs3.covenantchristian.org/bird
Updated April 24, 2009
Psalm 111:2
Curve sketching and analysis
y = f(x) must be continuous at each:
dy
 critical point:
= 0 or undefined. And don’t forget endpoints
dx



d2y
dy
local minimum:
goes (–,0,+) or (–,und,+) or
2 > 0
dx
dx
d2y
dy
local maximum:
goes (+,0,–) or (+,und,–) or
2 < 0
dx
dx
point of inflection: concavity changes
d2y
goes from (+,0,–), (–,0,+),
2
dx
(+,und,–), or (–,und,+)
Basic Derivatives
d
 sin x   cos x
dx
d
 cos x    sin x
dx
d
2
 tan x   sec x
dx
d
2
 cot x    csc x
dx
 
d n
n 1
x  nx
dx
d
 sec x   sec x tan x
dx
d
 csc x    csc x cot x
dx
d
1 du
 ln u  
dx
u dx
d u
u du
e e
dx
dx
 
 
d
n
n 1
Basic Integrals
x  nx
dx
d
 sin x   cos x d
dx
 sec x   sec x tan x
dx
d
d
cos
x


sin
x
 
csc
x


csc
x
cot
x


dx
dx
d
2
d
1
du
tan
x

sec
x
 
ln u  

dx
dx
u
dx
d
2
d u
 cot x    csc x
u du
e e
dx
dx
dx
 
Some more handy integrals

a n 1
ax dx 
x C
n 1

tan x dx  ln sec x  C
n
  ln cos x  C

sec x dx  ln sec x  tan x  C
More Derivatives


d
1 du
1
sin u 
2 dx
dx
1 u
d
1
1
cos x 
2
dx
1 x
d
1
1
tan x 
2
dx
1 x
d
1
1
cot x 
dx
1  x2


d
1
1
sec x 
2
dx
x x 1


d
1
1
csc x 
dx
x x2  1




d x
x
a  a ln a
dx
d
1
 loga x  
dx
x ln a


 
Recall “change of base” log a x 
ln x
ln a
Differentiation Rules
Chain Rule
d
du
dy dy du
f (u)  f '(u)
OR


dx
dx
dx du dx
Product Rule
d
du
dv
(uv) 
vu
OR u ' v  uv '
dx
dx
dx
Quotient Rule
d u



dx  v 
du
dx
dv
v  u dx
2
v
u ' v  uv '
OR
2
v
The Fundamental Theorem of Calculus

b
a
f ( x)dx  F (b)  F (a )
where F '( x)  f ( x)
d b( x)
f
(
t
)
dt


dx a ( x )
Corollary to FTC
f (b( x))b '( x)  f (a( x))a '( x)
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.
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)
f (b)  f (a)
such that
f '(c) 
ba
Mean Value Theorem & Rolle’s 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
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 f '(c) = 0.
Approximation Methods for Integration
Trapezoidal Rule

b
a
f ( x)dx 
Simpson’s Rule
b

a
1 ba
2 n
[ f ( x0 )  2 f ( x1 )  ...
 2 f ( xn 1 )  f ( xn )]
f ( x)dx 
1
3
x[ f ( x0 )  4 f ( x1 )  2 f ( x2 )  ...
2 f ( xn2 )  4 f ( xn1 )  f ( xn )]
Simpson only works for
Even sub intervals (odd data points) 1/3 (1 + 4 + 2 + 4 + 1
)
Theorem of the Mean Value
i.e. AVERAGE VALUE

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

f (c ) 
a

f ( x)dx
(b  a)
This value f(c) is the “average value” of the
function on the interval [a, b].
Solids of Revolution and friends

L
Disk Method
a
x b
R( x)

xa
V 

2
1   f '( x) dx
dx
Washer Method
V 
b
a

b
 R(x)  r(x)  dx
2
2
General volume equation (not rotated)
b
V   Area ( x) dx
a
2
Distance, Velocity, and Acceleration
d
velocity =
(position)
average velocity =
dt
final position  initial position

d
total time
(velocity)
acceleration =
x
dt

t
2
2
speed = v  ( x ')  ( y ') *
dx dy
,
dt dt
*velocity vector =
displacement =
tf
t
o
v dt
distance =
final time
initial time v dt
tf
t
o
*bc topic
( x ')2  ( y ')2 dt *
Values of Trigonometric Functions for
Common Angles
π/3 = 60°
θ
sin θ
cos θ
tan θ
0°
0
1
0

,30°
6
1
2
3
2
3
3
37°
3/5
4/5
3/4

,45°
4
53°
2
2
2
2
1
4/5
3/5
4/3

,60°
3
3
2
1
2
3

,90°
2
1
0
∞
π,180°
0
–1
0
π/6 = 30°
sine
cosine
Trig Identities
Double Argument
sin 2 x  2 sin x cos x
cos 2 x  cos x  sin x  1  2sin x
2
2
2
Trig Identities
Double Argument
sin 2 x  2sin x cos x
2
2
2
cos 2 x  cos x  sin x  1  2sin x
1
cos x  1  cos 2 x 
2
1
2
sin x  1  cos 2 x 
2
2
Pythagorean
2
sin x  cos x  1
2
sine
cosine
Slope – Parametric & Polar
Parametric equation

Given a x(t) and a y(t) the slope is
Polar
dy

dx
dy
dt
dx
dt
dy
dy
/
d

 Slope of r(θ) at a given θ is

dx dx / d
What is y equal to in terms of r and θ ? x?
d
r   sin  
d 

 d
d 
 r   cos  
Polar Curve
For a polar curve r(θ), the AREA inside a “leaf” is
2

1
1
2
r   d
2
(Because instead of infinitesimally small rectangles, use triangles)
where θ1 and θ2 are the “first” two times that r = 0.
r  d
We know arc length l = r θ
and
1
A  bh
2
1
1 2
dA   rd   r  r d
2
2
r
l’Hôpital’s Rule
If
then
f (a) 0

 or 
g (b) 0

f ( x)
f '( x)
lim
 lim
xa g ( x)
x  a g '( x )
Integration by Parts
d
du
dv
(uv) 
vu
OR u ' v  uv '
dx
dx
dx
We know the product rule
L Logarithm
 d (uv)  v du  u dv
I Inverse
 u dv  d (uv)  v du
P Polynomial
  u dv   d (uv)   v du
E Exponential
T Trig
Antiderivative product rule
udv  uv  vdu
(Use u = LIPET)
Let u = ln x
dv = dx
e.g.
du = 1 dx
v=x


ln x dx
x
 x ln x  x  C

Taylor Series
If the function f is “smooth” at x = a, then
it can be approximated by the nth degree
polynomial
f ( x)  f (a )  f '(a )( x  a )
f ''(a )

( x  a)2 
2!
f ( n ) (a)

( x  a)n .
n!
Maclaurin Series
A Taylor Series about x = 0 is called Maclaurin.
x 2 x3 x 4
ln( x  1)  x    
2 3 4
x 2 x3
e  1 x   
2! 3!
x
x3 x5
sin x  x   
3! 5!
x2 x4
cos x  1   
2! 4!
1
 1  x  x 2  x3 
1 x