stuff you must know cold for the ap calculus exam in the morning of wednesday, may 7, 2008. ap...
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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,+)
dy
dx
dy
dx
2
2
d y
dx2
2
d y
dx
2
2
d y
dx
dy
dx
Basic Derivatives
1n ndx nx
dx
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
1ln
d duu
dx u dx
u ud due e
dx dx
Basic Integrals
1n nx dx nx sin cosx dx x
cos sinx dx x 2tan secx dx x 2cot cscx dx x
sec sec tanx dx x x csc csc cotx dx x x
1lndu u
u
u ue du e
Some more handy integrals 1
1n na
ax dx x Cn
tan ln sec
ln cos
sec ln sec tan
x dx x C
x C
x dx x x C
More Derivatives 1
2
1sin
1
d duu
dx dxu
1
2
1cos
1
dx
dx x
12
1tan
1
dx
dx x
12
1cot
1
dx
dx x
1
2
1sec
1
dx
dx x x
1
2
1csc
1
dx
dx x x
lnx xda a a
dx
1log
lna
dx
dx x a
Recall “change of base”ln
loglna
xx
a
Differentiation Rules Chain Rule
( ) '( )d du dy dy du
f u f udx dx dx du
Rx
Od
Product Rule
( ) ' 'd du dv
uv v u OR udx
vdx dx
uv
Quotient Rule
2 2
' 'du dvdx dxv u
Od u
dx v v
u v uv
vR
The Fundamental Theorem of Calculus
Corollary to FTC
( )
( )
( ( )) '( ) ( ( ))
( )
'( )
b x
a x
f b x b x f a x
f t dtd
da x
x
( ) ( ) ( )
where '( ) ( )
b
af x dx F b F a
F x f x
Intermediate Value Theorem
. Mean Value Theorem
( ) ( )'( )
f b f af c
b 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 c
b 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
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].
( ) 1( ) ( )
( )
b
ba
a
f x dxf c OR f x dx
b a b a
Area “Under” and Between Curves Sketch the Area Find Points of Intersection Integrate “top” curve – “bottom” curve (or
right curve – left curve) Use Geometry as appropriate
( ) OR A = ( ) ( )x b x b
x a x aA R x dx f x g x dx
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- cross sections)
Distance, Velocity, and Acceleration
velocity =
d
dt
(position) d
dt
(velocity)
speed = v
displacement = f
o
t
tv dt final time
initial time distance = v dt
average velocity = final position initial position
total time
x
t
acceleration =
Values of Trigonometric Functions for Common Angles
1
23
2
3
3
2
2
2
2
3
2
3
0–10π,180°
∞01 ,90°
,60°
4/33/54/553°
1 ,45°
3/44/53/537°
,30°
0100°
tan θcos θsin θθ
1
26
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 cos2
2x x
2 1cos 1 cos2
2x x
Pythagorean2 2sin cos 1x x
sine
cosine
l’Hôpital’s Rule (BC but nice to know)If
then
( ) 0or
( ) 0
f a
g b
( ) '( )lim lim
( ) '( )x a x a
f x f x
g x g x