AP CALCULUS BC
Stuff you MUST Know Cold
l’Hopital’s Rule Properties of Log and Ln “PLUS A CONSTANT”
If
f (a ) 0
= or =
∞
, 1. ln 1 = 0 2. ln e a = a
g (a ) 0 ∞
3.eln x = x 4. ln x n = n ln x The Fundamental Theorem of
f (x) f '( x ) Calculus
then lim = lim 5. ln ( ab ) = ln a + ln b
x→ a g ( x) x → a g '( x ) b
6.ln ( a b ) = ln a − ln b ∫ f ( x)dx = F (b) − F (a)
a
Average Rate of Change Differentiation Rules where F '( x) = f ( x)
(slope of the secant line) Chain Rule
If the points (a, f(a)) and (b, f(b)) d du 2nd Fundamental Theorem of
[ f (u )] = f '(u )
are on the graph of f(x) the average dx dx Calculus
rate of change of f(x) on the interval Product Rule d
g ( x)
dx ∫#
[a,b] is f ( x)dx = f ( g ( x )) ⋅ g '( x)
d dv du
f (b) − f (a) ( uv ) = u + v
dx dx dx
b−a
Quotient Rule Average Value
du dv If the function f(x) is continuous on
Definition of Derivative v −u [a, b] and the first derivative exist
d u dx dx
(slope of the tangent line) = on the interval (a, b), then there
dx v v 2
exists a number x = c on (a, b) such
f ( x + h) − f ( x) that
f '( x) = lim
h →0
h Mean Value & Rolle’s Theorem 1
b
b − a ∫a
If the function f(x) is continuous on f (c ) = f ( x)dx
Derivatives [a, b] and the first derivative exists
on the interval (a, b), then there f (c) is the average value
d n
dx
( x ) = nx n −1 exists a number x = c on (a, b) such
f (b ) − f ( a )
that f '( c ) = Euler’s Method
d b−a
( sin x ) = cosx if f(a) = f(b), then f ’(c) = 0.
dx dy
If given that = f ( x, y ) and
d dx
( cos x ) = −sin x
dx Curve sketching and analysis that the solution passes through
d
( tan x ) = sec2 x y = f(x) must be continuous at each: (x0, y0), then
dx critical point:
dy
= 0 or undefined.
d
dx x new = x old + ∆x
( cot x ) = −csc2 x local minimum:
dy
goes (-,0,+) or dy
dx
dx ynew = yold + ⋅ ∆x
d dx ( xold , yold )
( sec x ) = tanx sec x d2y
dx (-,und,+) or >0
dx 2
d dy
( csc x ) = −cotx csc x local maximum: goes (+,0,-) or
Logistics Curves
dx dx L
d 1 P (t ) = ,
( ln u ) = du d2y 1 + Ce− ( Lk ) t
dx u (+,und,-) or <0 where L is carrying capacity
dx 2
d u Absolute Max/Min.: Compare local Maximum growth rate occurs when
dx
( e ) = eu du
extreme values to values P=½L
at endpoints. dP
d 1 = kP ( L − P ) or
( log a x ) = pt of inflection : concavity changes. dt
dx x ln a
d2y dP P
d u goes (+,0,-),(-,0,+), = ( Lk ) P (1 − )
dx
( a ) = a x ( ln a ) du dx 2 dt L
(+,und,-), or (-,und,+)
Stuff you MUST Know Cold
l’Hopital’s Rule Properties of Log and Ln “PLUS A CONSTANT”
If
f (a ) 0
= or =
∞
, 1. ln 1 = 0 2. ln e a = a
g (a ) 0 ∞
3.eln x = x 4. ln x n = n ln x The Fundamental Theorem of
f (x) f '( x ) Calculus
then lim = lim 5. ln ( ab ) = ln a + ln b
x→ a g ( x) x → a g '( x ) b
6.ln ( a b ) = ln a − ln b ∫ f ( x)dx = F (b) − F (a)
a
Average Rate of Change Differentiation Rules where F '( x) = f ( x)
(slope of the secant line) Chain Rule
If the points (a, f(a)) and (b, f(b)) d du 2nd Fundamental Theorem of
[ f (u )] = f '(u )
are on the graph of f(x) the average dx dx Calculus
rate of change of f(x) on the interval Product Rule d
g ( x)
dx ∫#
[a,b] is f ( x)dx = f ( g ( x )) ⋅ g '( x)
d dv du
f (b) − f (a) ( uv ) = u + v
dx dx dx
b−a
Quotient Rule Average Value
du dv If the function f(x) is continuous on
Definition of Derivative v −u [a, b] and the first derivative exist
d u dx dx
(slope of the tangent line) = on the interval (a, b), then there
dx v v 2
exists a number x = c on (a, b) such
f ( x + h) − f ( x) that
f '( x) = lim
h →0
h Mean Value & Rolle’s Theorem 1
b
b − a ∫a
If the function f(x) is continuous on f (c ) = f ( x)dx
Derivatives [a, b] and the first derivative exists
on the interval (a, b), then there f (c) is the average value
d n
dx
( x ) = nx n −1 exists a number x = c on (a, b) such
f (b ) − f ( a )
that f '( c ) = Euler’s Method
d b−a
( sin x ) = cosx if f(a) = f(b), then f ’(c) = 0.
dx dy
If given that = f ( x, y ) and
d dx
( cos x ) = −sin x
dx Curve sketching and analysis that the solution passes through
d
( tan x ) = sec2 x y = f(x) must be continuous at each: (x0, y0), then
dx critical point:
dy
= 0 or undefined.
d
dx x new = x old + ∆x
( cot x ) = −csc2 x local minimum:
dy
goes (-,0,+) or dy
dx
dx ynew = yold + ⋅ ∆x
d dx ( xold , yold )
( sec x ) = tanx sec x d2y
dx (-,und,+) or >0
dx 2
d dy
( csc x ) = −cotx csc x local maximum: goes (+,0,-) or
Logistics Curves
dx dx L
d 1 P (t ) = ,
( ln u ) = du d2y 1 + Ce− ( Lk ) t
dx u (+,und,-) or <0 where L is carrying capacity
dx 2
d u Absolute Max/Min.: Compare local Maximum growth rate occurs when
dx
( e ) = eu du
extreme values to values P=½L
at endpoints. dP
d 1 = kP ( L − P ) or
( log a x ) = pt of inflection : concavity changes. dt
dx x ln a
d2y dP P
d u goes (+,0,-),(-,0,+), = ( Lk ) P (1 − )
dx
( a ) = a x ( ln a ) du dx 2 dt L
(+,und,-), or (-,und,+)
