AB CALCULUS (0 to 100)
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
dy
dx
>0
local maximum:
dy
dx
goes (+,0,–) or (+,und,–) or
2
2
dy
dx
<0
point of inflection: concavity changes
2
2
dy
dx
goes from (+,0,–), (–,0,+),
Differentiation Rules
Chain Rule
( ) '( )
d du dy dy du
f u f u
dx dx dx du
Rx
Od
Product Rule
() ''
d dv du
uv u v OR u v v
dx dx dx u
Quotient Rule
22
''
du dv
dx dx
vu vu uv
O
du
dv
xv vR
Numerical Methods for Integration
Trapezoidal Rule
101
2
1
( ) [ ( ) 2 ( ) ...
2 ( ) ( )]
bba
n
a
nn
f x dx f x f x
f x f x
RRAM (Right-hand Rect. Approx.)
LRAM (Left-hand Rect. Approx)
MRAM (Midpt. Rect. Approx)
Theorem of the Mean Value
i.e. AVERAGE VALUE
Basic Derivatives
1nn
dx nx
dx
sin cos '
du u u
dx
cos sin '
du u u
dx
2
tan sec '
du u u
dx
2
cot csc '
du u u
dx
sec sec tan '
du u u u
dx
csc csc cot '
du u u u
dx
1
ln
d du
u
dx u dx
uu
d du
ee
dx dx
“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
1
( ) ( )
b
a
f c f x dx
ba
This value f(c) is the “average value” of
the function on the interval [a, b].
The Fundamental Theorem of
Calculus
( ) ( ) ( )
where '( ) ( )
b
af x dx F b F a
F x f x
Corollary to FunThmCalculus
()
()
( ( )) '( ) ( ( ))
()
'( )
bx
ax
f b x b x f a x
f t dt
d
d
ax
x
Solids of Revolution and friends
Disk Method
2
()
xb
xa
V R x dx
(about x-axis)
Washer Method
22
( ) ( )
b
a
V R x r x dx
(about x-axis)
Cross Sections
()
b
a
V Area x dx to x axis
Intermediate Value Theorem
If the function f(x) is continuous on [a, b],
and k 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 k
.
More Derivatives
1
2
1
sin 1
d du
u
dx dx
u
1
2
1
cos 1'
du
du
xu
1
2
1
tan '
1
du
du
xu
1
2
1
cot 1'
du
du
xu
1
2
1
sec 1'
duu
dx uu
1
2
1
csc 1'
duu
dx uu
l'n
uu
da a a
dx u
1
lo n'gl
a
duuu
dx 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
fc ba
.
* Rolle’s Theorem: f ’(c) = 0.
Position, Velocity, and Acceleration
velocity =
d
dt
(position)
acceleration =
d
dt
(velocity)
speed =
v
displacement =
f
o
t
tvdt
final time
initial time
distance = v dt
average velocity =
s
t
final position initial position
total time
average acceleration =
v
t
final velocity initial velocity
total time
Area Formulas
Trapezoid
)(
2
121 bbhA
Circle
2
Ar
Square
2
As
Rectangle
A lw
Triangle
1
2
A bh
OR at endpoints
(+,und,–), or (–,und,+)
