AP CALCULUS BC
Stuff you MUST Know Cold
l’Hopital’s Rule
f (a ) 0
∞
,
If
=
or =
g (a ) 0
∞
f (x)
f '( x )
= lim
then lim
x→ a g ( x)
x → a g '( x )
Properties of Log and Ln
1. ln 1 = 0
2. ln e a = a
“PLUS A CONSTANT”
3.eln x = x
5. ln ( ab ) = ln a + ln b
The Fundamental Theorem of
Calculus
6.ln ( a b ) = ln a − ln b
∫ f ( x)dx = F (b) − F (a)
4. ln x n = n ln x
b
a
Average Rate of Change
(slope of the secant line)
If the points (a, f(a)) and (b, f(b))
are on the graph of f(x) the average
rate of change of f(x) on the interval
[a,b] is
f (b) − f (a)
b−a
Differentiation Rules
Chain Rule
d
du
[ f (u )] = f '(u )
dx
dx
Product Rule
d
dv
du
( uv ) = u + v
dx
dx
dx
Quotient Rule
Definition of Derivative
(slope of the tangent line)
 f ( x + h) − f ( x) 
f '( x) = lim 

h →0
h


Derivatives
d n
x ) = nx n −1
(
dx
d
( sin x ) = cosx
dx
d
( cos x ) = −sin x
dx
d
( tan x ) = sec2 x
dx
d
( cot x ) = −csc2 x
dx
d
( sec x ) = tanx sec x
dx
d
( csc x ) = −cotx csc x
dx
d
1
( ln u ) = du
dx
u
d u
e ) = eu du
(
dx
d
1
( log a x ) =
dx
x ln a
d u
a ) = a x ( ln a ) du
(
dx
d u
 =
dx  v 
v
du
dv
−u
dx
dx
2
v
Mean Value & 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
exists a number x = c on (a, b) such
that f '( c ) =
f (b ) − f ( a )
b−a
where F '( x) = f ( x)
2nd Fundamental Theorem of
Calculus
g ( x)
d
f ( x)dx = f ( g ( x )) ⋅ g '( x)
dx ∫#
Average Value
If the function f(x) is continuous on
[a, b] and the first derivative exist
on the interval (a, b), then there
exists a number x = c on (a, b) such
that
b
1
f (c ) =
f ( x)dx
b − a ∫a
f (c) is the average value
Euler’s Method
if f(a) = f(b), then f ’(c) = 0.
If given that
Curve sketching and analysis
y = f(x) must be continuous at each:
dy
critical point: dx = 0 or undefined.
dy
local minimum:
goes (-,0,+) or
dx
d2y
(-,und,+) or
>0
dx 2
dy
local maximum:
goes (+,0,-) or
dx
d2y
(+,und,-) or
<0
dx 2
Absolute Max/Min.: Compare local
extreme values to values
at endpoints.
pt of inflection : concavity changes.
d2y
goes (+,0,-),(-,0,+),
dx 2
(+,und,-), or (-,und,+)
dy
= f ( x, y ) and
dx
that the solution passes through
(x0, y0), then
x new = x old + ∆x
ynew = yold +
dy
⋅ ∆x
dx ( xold , yold )
Logistics Curves
L
P (t ) =
,
1 + Ce− ( Lk ) t
where L is carrying capacity
Maximum growth rate occurs when
P=½L
dP
= kP ( L − P ) or
dt
dP
P
= ( Lk ) P (1 − )
dt
L
Distance, velocity and
Acceleration
Integrals
∫ kf (u )du = k ∫ f (u)du
d
( position )
dt
d
Acceleration =
( velocity )
dt
dx dy
Velocity Vector =
,
dt dt
∫ du = u + C
u n +1
u
du
=
+ C , n ≠ −1
∫
n +1
1
∫ u du = ln | u | +C
u
u
∫ e du = e + C
Speed = |v(t)| =
 1  u
u
∫ a du =  ln a  a + C
2
2
final
time
=
1
|u |
arc sec 
+C
a
 a 
u
∫ a 2 − u 2 = arcsin  a  + C
du
1
u
∫ a 2 + u 2 = a arctan  a  + C
du
b
V = π∫ [ R ( x)] − [ r ( x) ] dx
∫ v(t ) dt = ∫ ( x ') + ( y ') dt
2
Volume of Known Cross Sections
initial
time
Perpendicular to
x-axis:
b
∫ x '(t )dt
b
V = ∫ A( x)dx
a
b
a
∫ y '(t )dt
y(b) = y(a) +
d
V = ∫ A( y )dy
c
If the function f is “smooth”at x = c,
then it can be approximated by the nth
degree polynomial
f ( x ) ≈ f (c ) + f '(c)( x − c )
Polar Curves
For a polar curve r(θ), the
θ2
1
2
∫ [ r(θ)] d θ
f "(c )
( x − c ) 2 + ...
2!
f '''(c )
+
( x − c )3 + ...
3!
f ( n ) (c )
+
( x − c) n
n!
2
+
Elementary Functions
Centered at x = 0
x 2 x3
+
+ ...
2! 3!
x2 x4 x6
cos x = 1 −
+
−
+ ...
2! 4! 6!
x3 x 5 x 7
sin x = x −
+
−
+ ...
3! 5! 7!
1
= 1 + x + x 2 + x3 + ...
1− x
x 2 x3 x 4
ln( x + 1) = x −
+
−
+ ...
2
3
4
∞
The series ∑an converges if
L = ∫ 1 + [ f '( x ) ] dx
ex = 1 + x +
n=0
CHECK
ENDPOINTS
an+1
<1
n→∞ a
n
a
lim
For a polar graph, r(θ)
θ2
2
[ r (θ)] + [ r '(θ)] d θ
Alternating Series Error Bound
θ1
If S N =
Lagrange Error Bound
(z)
x−c
∑
N
n =1
( −1) n an is the Nth
partial sum of a convergent
alternating series, then
If Pn(x) is the nth degree Taylor
polynomial of f(x) about c, then
for all z between x and c.
y-axis:
Taylor Series
a
Ratio Test
(use for interval of convergence)
( n +1) !
∫ r ( x)h( x)dx
a
2
x(b) = x(a) +
)
b
final
time
initial
time
2
Shell Method: V = 2π
dy dy d θ ddθ [ r (θ) sin θ]
=
=
dx dx d θ ddθ [ r (θ) cos θ]
2
max f
2
2
The slope of r(θ) at a given θ is
b
( n+1)
(
a
θ1
Arc Length
For a function, f(x)
f (x) − Pn (x) ≤
Washer Method:
where θ1 and θ2 are the “first” two
times that r = 0.
∫ udv = uv − ∫ vdu
2
2
2
a
( x ') + ( y ') .
Area inside a “leaf” is
Integration by Parts
L= ∫
∫
Disk Method: V = π [ R ( x) ] dx
Distance Traveled =
∫ cos udu = sin u + C
∫ sin udu = − cos u + C
∫ tan udu = − ln | cos u | +C
∫ cot udu = ln | sin u | +C
∫ sec udu = ln | sec u + tan u | +C
∫ csc udu = − ln | csc u + cot u | +C
∫u u −a
b
Velocity =
n
du
Volume
Solids of Revolution
S∞ − S N ≤ aN +1
n+1
Most Common Series
∞
∞
n
1
(−1)
diverges ∑
converges
∑
n
n=1 n
n=1
∞
A
∑ A(r) converges to 1 − r if |r|<1
n
n= 0