Stuff You Must Know Cold for the AP Calculus BC Exam!
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