1. typed by Sean Bird, Covenant Christian High School
updated August 15, 2009
AP CALCULUS
Stuff you MUST know Cold * means topic only on BC
Curve sketching and analysis Differentiation Rules Approx. Methods for Integration
y = f(x) must be continuous at each: Chain Rule Trapezoidal Rule
dy d du dy dy du b
critical point: = 0 or undefined
dx and look out for endpoints dx
[ f (u )] = f '(u ) dx OR dx = du dx ∫a
f ( x)dx = 1 b− a
2 n [ f ( x0 ) + 2 f ( x1 ) + ...
local minimum: + 2 f ( xn −1 ) + f ( xn )]
dy goes (–,0,+) or (–,und,+) or d 2 y >0 Product Rule Simpson’s Rule
dx dx 2 b
local maximum:
d
dx
(uv) =
du
dx
v+u
dv
dx
OR u ' v + uv ' ∫a
f ( x)dx =
2 1
dy goes (+,0,–) or (+,und,–) or d y <0 3 ∆x[ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + ...
dx dx 2 2 f ( xn −2 ) + 4 f ( xn−1 ) + f ( xn )]
point of inflection: concavity changes
Quotient Rule
d 2 y goes from (+,0,–), (–,0,+),
d u du
dx v −u dv
dx u ' v − uv '
 =
dx  v  v2
OR
v2
dx 2 (+,und,–), or (–,und,+) Theorem of the Mean Value
i.e. AVERAGE VALUE
If the function f(x) is continuous on [a, b]
Basic Derivatives
“PLUS A CONSTANT” and the first derivative exists on the
d n interval (a, b), then there exists a number
dx
( )
x = nx n−1 The Fundamental Theorem of x = c on (a, b) such that
Calculus b
d
( sin x ) = cos x f (c ) =
∫ a
f ( x)dx
dx b
(b − a)
d
( cos x ) = − sin x
∫a
f ( x)dx = F (b) − F (a)
This value f(c) is the “average value” of
dx where F '( x) = f ( x) the function on the interval [a, b].
d
( tan x ) = sec2 x
dx
Corollary to FTC Solids of Revolution and friends
d
( cot x ) = − csc2 x Disk Method
dx d b( x) x =b
V =π∫
2
[ R ( x)]
dx ∫a ( x )
d f (t )dt = dx
( sec x ) = sec x tan x x=a
dx Washer Method
d f (b( x))b '( x) − f (a ( x))a '( x)
dx
( csc x ) = − csc x cot x V =π∫
b
a
([ R( x)] − [r ( x)] ) dx
2 2
Intermediate Value Theorem General volume equation (not rotated)
d 1 du
( ln u ) = If the function f(x) is continuous on [a, b], b
dx u dx and y is a number between f(a) and f(b), V = ∫ Area ( x) dx
a
d u du then there exists at least one number x= c
dx
( )
e = eu
dx in the open interval (a, b) such that *Arc Length L = ∫ 1 + [ f '( x)] dx
b 2
a
where u is a function of x, f (c ) = y . b
=∫
2 2
and a is a constant. a
[ x '(t )] + [ y '(t )] dt
More Derivatives Distance, Velocity, and Acceleration
d  −1 u  1 du Mean Value Theorem velocity = d (position)
sin =
dx 
 a
 2
a −u 2 dx dt
If the function f(x) is continuous on [a, b], d (velocity)
d −1 acceleration =
AND the first derivative exists on the
dx
(
cos −1 x = ) interval (a, b), then there is at least one
dt
1 − x2 dx dy
number x = c in (a, b) such that *velocity vector = ,
d  −1 u  a du dt dt
tan = ⋅ f (b) − f (a )
dx  a  a 2 + u 2 dx
 f '(c) = .
speed = v = ( x ')2 + ( y ') 2 *
b−a
d −1
dx
(
cot −1 x =
1 + x2
) displacement = ∫t
tf
v dt
o
d  −1 u  a du Rolle’s Theorem final time
dx 

sec =
a  u u 2 − a 2 dx

⋅ distance = ∫initial time v dt
If the function f(x) is continuous on [a, b], tf
d
(
csc −1 x =
−1
) AND the first derivative exists on the ∫t o
( x ') 2 + ( y ') 2 dt *
dx x x2 − 1 interval (a, b), AND f(a) = f(b), then there average velocity =
is at least one number x = c in (a, b) such final position − initial position
d u( x) du
dx
( )
a = a ( ) ln a ⋅
u x
dx
that
f '(c) = 0 .
=
total time
∆x
d 1 =
( log a x ) = ∆t
dx x ln a

2. BC TOPICS and important TRIG identities and values
l’Hôpital’s Rule Slope of a Parametric equation Values of Trigonometric
f (a) 0 ∞ Given a x(t) and a y(t) the slope is Functions for Common Angles
If = or = ,
g (b) 0 ∞ dy dy
= dt θ sin θ cos θ tan θ
f ( x) f '( x) dx dx
then lim = lim dt
x →a g ( x) x → a g '( x ) 0° 0 1 0
Euler’s Method Polar Curve π 1 3 3
If given that dy = f ( x, y ) and that For a polar curve r(θ), the 6 2 2 3
dx
AREA inside a “leaf” is
the solution passes through (xo, yo), θ2 2
π 2 2
y ( xo ) = yo ∫θ1
1
2
 r (θ )  dθ
  4 2 2
1
where θ1 and θ2 are the “first” two times that r = π 3 1
0. 3
y ( xn ) = y ( xn−1 ) + f ( xn−1 , yn−1 ) ⋅ ∆x 3 2 2
The SLOPE of r(θ) at a given θ is
In other words: dy dy / dθ π
= 1 0 “∞ ”
xnew = xold + ∆x dx dx / dθ 2
dy π 0 −1 0
dθ  r (θ ) sin θ 
d
= d  
ynew = yold + ⋅ ∆x Know both the inverse trig and the trig
dx ( xold , yold )
dθ  r (θ ) cos θ 
  values. E.g. tan(π/4)=1 & tan-1(1)= π/4
Integration by Parts Ratio Test Trig Identities
∞
Double Argument
∫ udv = uv − ∫ vdu The series ∑ ak converges if
k =0
sin 2 x = 2sin x cos x
Integral of Log cos 2 x = cos 2 x − sin 2 x = 1 − 2sin 2 x
Use IBP and let u = ln x (Recall ak +1
lim <1 1
u=LIPET) k →∞ ak cos 2 x = (1 + cos 2 x )
2
∫ ln x dx = x ln x − x + C If the limit equal 1, you know nothing.
Taylor Series 1
If the function f is “smooth” at x = Lagrange Error Bound sin 2 x = (1 − cos 2 x )
2
a, then it can be approximated by If Pn ( x) is the nth degree Taylor polynomial Pythagorean
the nth degree polynomial
of f(x) about c and f ( n+1) (t ) ≤ M for all t sin 2 x + cos 2 x = 1
f ( x) ≈ f (a ) + f '(a )( x − a) (others are easily derivable by
between x and c, then
+
f ''(a)
( x − a)2 + … dividing by sin2x or cos2x)
M n+1
2! f ( x) − Pn ( x) ≤ x−c 1 + tan 2 x = sec 2 x
f ( n ) (a)
( n + 1)!
+ ( x − a)n . cot 2 x + 1 = csc 2 x
n! Reciprocal
Maclaurin Series 1
A Taylor Series about x = 0 is Alternating Series Error Bound sec x = or cos x sec x = 1
cos x
called Maclaurin.
1
x 2 x3 N csc x = or sin x csc x = 1
If S N = ∑ ( −1) an is the Nth partial sum of a
n
ex = 1 + x + + +… sin x
2! 3! k =1 Odd-Even
2
x x4 convergent alternating series, then sin(–x) = – sin x (odd)
cos x = 1 − + − …
2! 4! S∞ − S N ≤ aN +1 cos(–x) = cos x (even)
x3 x5 Some more handy INTEGRALS:
sin x = x − + − …
1
3! 5! Geometric Series
∞ ∫ tan x dx = ln sec x + C
= 1 + x + x2 + x3 + … a + ar + ar 2 + ar 3 + … + ar n −1 + … = ∑ ar n −1 = − ln cos x + C
1− x n =1
x 2 x3 x 4 a
ln( x + 1) = x − + − + …
2 3 4
diverges if |r|≥1; converges to
1− r
if |r|<1 ∫ sec x dx = ln sec x + tan x + C
This is available at http://covenantchristian.org/bird/Smart/Calc1/StuffMUSTknowColdNew.htm