AP CALCULUS

Stuff You MUST Know Cold

BASIC DERIVATIVES Extreme Value Theorem The Fundamental Theorem of

If the function f is continuous on
( )
d n Calculus
x = nx n−1 [a, b], then there are points c and d in b

d
dx [a,b] with f(c) ≤ f(x) ≤ f(d) for all x in ∫a f ( x)dx = F(b) − F(a)
dx
( sin( x )) = cos( x ) [a,b].
where F ′(x ) = f ( x).
d
dx
( cos( x )) = − sin( x ) Differentiation Rules
Chain Rule
Alternatively,
b

d du F ( b ) = F ( a ) + ∫ f ( x ) dx
d
( tan( x ) ) = sec2 ( x ) [ f (u)] = f ′(u) a
dx dx dx
d
dx
( cot( x ) ) = − csc2 ( x ) dy dy du
=
Corollary to FTC
dx du dx
d b(x)
∫
d
dx
( sec( x ) ) = sec( x ) tan( x ) Product Rule dx a(x )
f (t)dt =
d dv du
d (uv) = u + f ( b(x )) b′( x) − f (a(x))a ′(x )
dx
( csc( x )) = − csc( x ) cot ( x ) dx dx dx
v

d (ln( x ) ) 1 Quotient Rule Average Value of a Function

= du dv
dx x d ⎛ u ⎞ dx v − u dx
b
⎜ ⎟ =
dx ⎝ v ⎠
∫ f (x)dx
(e ) = e x
d x v2 The integral a

dx (b − a)
is the "average value" of the function f
Curve Sketching and Analysis on the interval [a, b].
MORE DERIVATIVES y = f(x) must be continuous at each:
dy
= 0 or undefined.
( )
d 1 critical point: Trapezoidal Sum
sin −1 ( x ) = dx
f (x0 ) + f (x1 )
b
dx 1− x 2
∫ f ( x ) dx ≈
local minimum:
Δx1 +
d2y
( −1
)
d dy 2
cos −1( x ) = (–,0,+) or (–,und,+) or > 0. a
dx 1 − x2
dx dx 2 f (xn −1 ) + f (xn )
local maximum: …+ Δxn
d
(
tan −1 ( x ) =
1
) dy
2
d y 2
1+ x 2 2 < 0.
dx (+,0,–) or (+,und,–) or
dx dx
d
dx
−1
(
cot ( x ) =
−1
1+ x
2 ) pt of inflection: concavity changes.
Solids of Revolution and friends
d2y Volume
( )
d 1 (+,0,–), (–,0,+),(+,und,–), or (–
−1 2
sec ( x ) = dx V = ∫ cross-sectional area
dx x x2 − 1 ,und,+)
Disk Method
d
dx
(
csc−1 ( x ) =
−1
)
x x2 −1
Mean Value Theorem
b
V = π ∫a [ R(x ) ] dx
2

If the function f is continuous on [a, Washer Method

d x
dx ( )
a = a x ln( a)
b] and the first derivative exists on the
interval (a, b), then there exists a
b
(
V = π ∫ [ R(x ) ]2 − [ r(x)] 2 dx
a )
number c in (a, b) such that
d
( loga ( x )) = x ln1 a f ′(c ) =
f (b) − f (a) Shell Method (no longer in AP)
dx ( ) b −a
. b
V = 2π ∫a r(x)h( x)dx

Intermediate Value Theorem Derivative of an Inverse ArcLength

If the function f is continuous on [a, Function + L=∫
b
1+ [ f ′ (x)] 2 dx
b], then for any number c between f(a) a
and f(b),there exists a number d in the
d
dx
( f −1 ( x )) =
1 where
f ′ ( y)
open interval (a, b) such that f(d) = c. Surface of revolution
f ( y) = x b
+ S = 2π ∫ r (x) 1+ [ f ′ (x)] dx
2
+ = BC Topic Only a
 AP CALCULUS
Stuff You MUST Know Cold

Distance, velocity, and acceleration + Taylor Polynomial Approximation + Euler's Method

d
Velocity = (position). dy
dt If Pn (x) is the nth degree Taylor
If given that = f (x , y) and that the
d polynomial of f(x) about a then dx
Acceleration = (velocity).
f (n +1) (t)
dt solution passes through (x 0 , y0 ) ,
f ( x) = Pn (x) + (x − a) (n+1)
(n + 1)!
+ Velocity vector =
dx dy
, .
y(x 0 ) = y 0
for some t between x and a.
dt dt
So if f
(n+1)
(t) ≤ M for all t between 
2 2 y(x n ) = y(x n−1 ) +
+ Speed = v = ( x ′ ) + ( y ′ ) . x and a, then
f (x) − Pn (x) ≤
M
x −a
n+1 f (x n−1, y n−1 ) ⋅ ( x n − x n−1 )
(n + 1)!
v ( t ) dt
final time
Distance = ∫ initial time In other words:
+ Maclaurin Series
tf x new = x old + Δx
+ = ∫t ( x ′) 2 + ( y ′ )2 dt A Taylor Series about x = 0 is called a
0
Maclaurin Series.
dy
Average velocity ynew = yold + ⋅ Δx
dx (x ,y )
final position - initial position x 2 x3
e x = 1+ x +
old old
= + +
total time 2 3!
x2 x 4
cos( x) = 1− + − + Polar Curves
2 4!
+ Integration by Parts
x3 x5 For a polar curve r(θ), the area inside
sin(x) = x − + −
∫ udv = uv −∫ vdu 3! 5! a "leaf" is
1
= 1+ x + x 2 + x 3 + θ 1
∫θ1 2 [r(θ)] dθ ,
2 2
1−x
+ Integral of Log x2 x3 x 4
ln(x + 1) = x − + − +
2 3 4 where θ1 and θ2 are the "first" two
∫ ln( x ) dx = x ln( x ) − x + C times that r = 0.

+ Alternating Series Error Bound The slope of r(θ) at a given θ is

+ l'Hôpital's Rule
dy (dy / dθ )
∑k =1 (−1) nan
N
If S N = is the Nth =
f (a) 0 ∞ dx (dx / dθ )
If = or = , partial sum of a convergent alternating
g(a) 0 ∞
series with "decreasing" terms, then d [ r(θ)sin θ ]
f (x) f ′ (x)
then lim = lim . dθ
= d
g ′( x)
dθ [
g(x )
x→ a x→a S∞ − SN ≤ aN +1 r(θ) cosθ ]

+ Ratio Test
+ Taylor Polynomial
∞
If the function f is "smooth" at x = a,
then it can be approximated by the n
th
The series ∑ ak converges if
k=0
degree polynomial a
k +1
lim a < 1.
f ( x) ≈ f (a) + f ′(a)( x − a) k →∞ k

f ′′ (a) 2
+ (x − a) + … If limit equals 1, you know nothing.
2!
( n)
f (a)
+ (x − a) n .
n!