typed by Sean Bird, Covenant Christian High School
updated April 6, 2006
Curve sketching and analysis
y = f(x) must be continuous at each:
dy
critical point:
= 0 or undefined
dx
or endpoints
local minimum:
dy goes (–,0,+) or (–,und,+) or d 2 y >0
dx 2
dx
local maximum:
2
dy goes (+,0,–) or (+,und,–) or d y <0
dx 2
dx
point of inflection: concavity changes
d 2 y goes from (+,0,–), (–,0,+),
dx 2
(+,und,–), or (–,und,+)
Basic Derivatives
 
d n
x  nx n1
dx
d
 sin x   cos x
dx
d
 cos x    sin x
dx
d
 tan x   sec2 x
dx
d
 cot x    csc2 x
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
du
e  eu
dx
dx
 
AP CALCULUS
Stuff you MUST know Cold
Differentiation Rules
Chain Rule
d
du
dy dy du
f (u)  f '(u)
OR


dx
dx
dx du dx
* means topic only on BC
Approx. Methods for Integration
Trapezoidal Rule

b
a
f ( x)dx 
d
du
dv
(uv) 
vu
OR u ' v  uv '
dx
dx
dx
 2 f ( xn 1 )  f ( xn )]

b
a
f ( x)dx 
1
3
x[ f ( x0 )  4 f ( x1 )  2 f ( x2 )  ...
2 f ( xn2 )  4 f ( xn1 )  f ( xn )]
Quotient Rule
d u

dx  v 
[ f ( x0 )  2 f ( x1 )  ...
Simpson’s Rule
Product Rule
du
dx
1 ba
2 n
v u
v2
dv
dx
OR
u ' v  uv '
v2
“PLUS A CONSTANT”
The Fundamental Theorem of
Calculus
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

f (c ) 
b
a

b
a
f ( x)dx  F (b)  F (a )
where F '( x)  f ( x)
Corollary to FTC
d b( x)
f (t )dt 
dx a ( x )
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 .
(b  a)
This value f(c) is the “average value” of
the function on the interval [a, b].
Solids of Revolution and friends
Disk Method
V 
x b
xa
 R( x) 
2
dx
Washer Method
V 
b
a
 R(x)  r(x)  dx
2
2
General volume equation (not rotated)
b
V   Area ( x) dx
a
*Arc Length L   1   f '( x) dx
b
2
a

b
a
More Derivatives


d
1
du
sin 1 u 
2 dx
dx
1 u
d
1
cos1 x 
dx
1  x2
d
1
tan 1 x 
dx
1  x2
d
1
cot 1 x 
dx
1  x2
d
1
sec1 x 
dx
x x2  1








d
1
csc1 x 
dx
x x2  1


 
d x
a  a x ln a
dx
d
1
 loga x  
dx
x ln a
Mean Value Theorem
f ( x)dx
 x '(t )   y '(t ) dt
2
2
Distance, Velocity, and Acceleration
velocity = d (position)
dt
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
acceleration =
*velocity vector =
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 .
dx dy
,
dt dt
speed = v  ( x ')2  ( y ')2 *
displacement =
Rolle’s Theorem
d (velocity)
dt
distance =
tf
t
v dt
o
final time
initial time v dt
tf
t
( x ')2  ( y ')2 dt *
o
average velocity =
final position  initial position

total time
x
=
t
BC TOPICS and important TRIG identities and values
l’Hôpital’s Rule
f (a) 0

 or  ,
If
g (b) 0

f ( x)
f '( x)
 lim
then lim
x a g ( x)
x  a g '( x )
Euler’s Method
If given that dy
dx  f ( x, y ) and that
the solution passes through (xo, yo),
y ( xo )  yo
y ( xn )  y ( xn1 )  f ( xn1 , yn1 )  x
In other words:
xnew  xold  x
ynew  yold 
Slope of a Parametric equation
Given a x(t) and a y(t) the slope is
dy dy
 dt
dx dx
dt
2

1
r   d

4
If the limit equal 1, you know nothing.
Lagrange Error Bound
If Pn ( x) is the nth degree Taylor polynomial
of f(x) about c and f ( n1) (t )  M for all t
between x and c, then
M
n1
xc
n

1
!
 
Alternating Series Error Bound
N
If S N    1 an is the Nth partial sum of a
n
k 1
convergent alternating series, then
S  S N  aN 1
Geometric Series
a  ar  ar 2  ar 3 
 ar n 1 

  ar n 1
a
if |r|<1
1 r
This is available at http://cs3.covenantchristian.org/bird/Smart/Calc1/StuffMUSTknowColdNew.htm
cos θ
1
3
2
4/5
2
2
3/5
1
2
tan θ
0
1
0
“ ”
3
3
3/4
1
4/3
3
0
1
1  cos 2 x 
2
Pythagorean
sin 2 x  cos2 x  1
(others are easily derivable by
dividing by sin2x or cos2x)
1  tan 2 x  sec2 x
sin 2 x 
cot 2 x  1  csc2 x
Reciprocal
1
sec x 
or cos x sec x  1
cos x
1
csc x 
or sin x csc x  1
sin x
Odd-Even
sin(–x) = – sin x (odd)
cos(–x) = cos x
(even)
Some more handy INTEGRALS:

tan x dx  ln sec x  C
  ln cos x  C
n 1
diverges if |r|≥1; converges to
sin θ
0
1
2
3/5
2
2
4/5
3
2
0
1
Trig Identities
Double Argument
sin 2 x  2sin x cos x
cos 2 x  cos2 x  sin 2 x  1  2sin 2 x
1
cos 2 x  1  cos 2 x 
2
ak 1
1
ak
f ( x)  Pn ( x) 
,60°
,90°
2
π,180°
k 0
u=LIPET)



k 
,45°
53°
3
The series  ak converges if
lim
,30°
37°
2
dy dy / d

dx dx / d
d
r   sin  
d 
 d 
d 
 r   cos  
Ratio Test
Use IBP and let u = ln x (Recall
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
ex  1  x   
2! 3!
2
x
x4
cos x  1   
2! 4!
x3 x5
sin x  x   
3! 5!
1
 1  x  x 2  x3 
1 x
x 2 x3 x 4
ln( x  1)  x    
2 3 4
1
2
6
where θ1 and θ2 are the “first” two times that r =
0.
The SLOPE of r(θ) at a given θ is
Integral of Log
ln x dx  x ln x  x  C

Polar Curve
Integration by Parts

θ
0°
For a polar curve r(θ), the
AREA inside a “leaf” is
dy
 x
dx  xold , yold 
 udv  uv   vdu
Values of Trigonometric
Functions for Common Angles

sec x dx  ln sec x  tan x  C