Stuff you MUST know Cold for the AP Calculus Examin the morning of Wednesday, May 7, 2008.
AP Physics & CalculusCovenant Christian High School7525 West 21st StreetIndianapolis, IN 46214Phone: 317/390.0202 x104Email: [email protected] Website: http://cs3.covenantchristian.org/bird Psalm 111:2
Sean Bird
Updated April 24, 2009
Curve sketching and analysisy = f(x) must be continuous at each: critical point: = 0 or undefined. And don’t forget endpoints
local minimum: goes (–,0,+) or (–,und,+) or > 0
local maximum: goes (+,0,–) or (+,und,–) or < 0
point of inflection: concavity changes
goes from (+,0,–), (–,0,+),(+,und,–), or (–,und,+)
dydx
dydx
2
2
d ydx
2
2
d ydx
2
2
d ydx
dydx
Basic Derivatives
1n nd x nxdx
sin cosd x xdx
cos sind x xdx
2tan secd x xdx
2cot cscd x xdx
sec sec tand x x xdx
csc csc cotd x x xdx
1lnd duudx u dx
u ud due edx dx
Basic Integrals
1n nd x nxdx
sin cosd x xdx
cos sind x xdx
2tan secd x xdx
2cot cscd x xdx
sec sec tand x x xdx
csc csc cotd x x xdx
1lnd duudx u dx
u ud due edx dx
Some more handy integrals 1
1n naax dx x C
n
tan ln sec
ln cos
sec ln sec tan
x dx x C
x C
x dx x x C
More Derivatives 1
2
1sin1
d duudx dxu
1
2
1cos1
d xdx x
12
1tan1
d xdx x
12
1cot1
d xdx x
1
2
1sec1
d xdx x x
1
2
1csc1
d xdx x x
lnx xd a a adx
1loglna
d xdx x a
Recall “change of base”lnloglna
xxa
Differentiation Rules Chain Rule
( ) '( )d du dy dy duf u f udx dx dx du
Rx
Od
Product Rule
( ) ' 'd du dvuv v u OR udx
vdx dx
uv
Quotient Rule
2 2
' 'du dvdx dxv u
Od udx v v
u v uvv
R
The Fundamental Theorem of Calculus
Corollary to FTC
( )
( )
( ( )) '( ) ( ( ))
( )
'( )
b x
a x
f b x b x f a x
f t dtdd
a xx
( ) ( ) ( )
where '( ) ( )
b
af x dx F b F a
F x f x
Intermediate Value Theorem
. Mean Value Theorem
( ) ( )'( ) f b f af cb a
.
If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that
If the function f(x) is continuous on [a, b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a, b) such that f(c) = y.
( ) ( )'( ) f b f af cb a
If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that
If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), AND f(a) = f(b),then there is at least one number x = c in (a, b) such that f '(c) = 0.
Mean Value Theorem & Rolle’s Theorem
Approximation Methods for IntegrationTrapezoidal Rule
10 12
1
( ) [ ( ) 2 ( ) ...
2 ( ) ( )]
b
a
n n
b anf x dx f x f x
f x f x
Simpson’s Rule
10 1 23
2 1
( )
[ ( ) 4 ( ) 2 ( ) ...2 ( ) 4 ( ) ( )]
b
a
n n n
f x dx
f x f x f xf x f x
xf x
Simpson only works forEven sub intervals (odd data points) 1/3 (1 + 4 + 2 + 4 + 1 )
Theorem of the Mean Valuei.e. AVERAGE VALUE 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
This value f(c) is the “average value” of the function on the interval [a, b].
( )( )
( )
b
af x dx
f cb a
Solids of Revolution and friends Disk Method
2( )x b
x aV R x dx
2 2( ) ( )b
aV R x r x dx
( )b
aV Area x dx
Washer Method
General volume equation (not rotated)
21 '( )b
aL f x dx Arc Length *bc topic
Distance, Velocity, and Acceleration
velocity =
ddt
(position) ddt
(velocity)
,dx dydt dt
speed = 2 2( ') ' *( )v x y
displacement = f
o
t
tv dt
final time
initial time
2 2
distance =
( ') *'
( )f
o
t
t
v dt
x y dt
average velocity = final position initial position
total time
xt
acceleration =
*velocity vector =
*bc topic
Values of Trigonometric Functions for Common Angles
12
32
33
22
22
32
3
0–10π,180°
∞01 ,90°
,60°
4/33/54/553°
1 ,45°
3/44/53/537°
,30°
0100°
tan θcos θsin θθ
126
4
3
2
π/3 = 60° π/6 = 30°
sine
cosine
Trig IdentitiesDouble Argument
sisi n c2n 2 osx xx
2 2 2cos2 cos sin 1 2sinx x x x
Trig IdentitiesDouble Argument
sin 2 2sin cosx x x2 2 2cos2 cos sin 1 2sinx x x x
2 1sin 1 cos22
x x
2 1cos 1 cos22
x x
Pythagorean2 2sin cos 1x x
sine
cosine
Slope – Parametric & PolarParametric equation Given a x(t) and a y(t) the slope is
Polar Slope of r(θ) at a given θ is
dydtdxdt
dydx
//
sincos
dd
dd
dy dy ddx dx d
r
r
What is y equal to in terms of r and θ ? x?
Polar Curve
For a polar curve r(θ), the AREA inside a “leaf” is
(Because instead of infinitesimally small rectangles, use triangles)
where θ1 and θ2 are the “first” two times that r = 0.
2
1
212 r d
12
A bh
r d
r 21 1
2 2dA rd r r d
and
We know arc length l = r θ
l’Hôpital’s Rule
If
then
( ) 0 or( ) 0
f ag b
( ) '( )lim lim( ) '( )x a x a
f x f xg x g x
Integration by Parts
Antiderivative product rule(Use u = LIPET)e.g.
udv uv vdu
( ) ' 'd du dvuv v u OR udx
vdx dx
uv We know the product rule
( )( )
( )
d uv v du u dvu dv d uv v du
u dv d uv v du
ln x dxlnx x x C
Let u = ln x dv = dx du = dx v = x1
x
LIPET
Logarithm Inverse Polynomial ExponentialTrig
Maclaurin SeriesA Taylor Series about x = 0 is called
Maclaurin.
If the function f is “smooth” at x = a, then it can be approximated by the nth degree polynomial
2 3
12! 3!
x x xe x
2 4
cos 12! 4!x xx
3 5
sin3! 5!x xx x
2 31 11
x x xx
2 3 4
ln( 1)2 3 4x x xx x
Taylor Series
2
( )
( ) ( ) '( )( )''( ) ( )2!
( ) ( ) .!
nn
f x f a f a x af a x a
f a x an