Stuff you MUST know Cold for the AP Calculus Exam In preparation for Wednesday May 9, 2012. AP...
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Transcript of Stuff you MUST know Cold for the AP Calculus Exam In preparation for Wednesday May 9, 2012. AP...
Stuff you MUST know Cold for the AP Calculus ExamIn preparation for Wednesday May 9, 2012.
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 BirdUpdated by Mrs. ShakMay 2012
Curve sketching and analysis
y = f(x) must be continuous at each: critical point: = 0 or undefined. And don’t forget endpoints for
absolute min/max
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,+) or (+,und,–), or (–,und,+)
goes from incr to decr or decr to incr
dy
dx
dy
dx
2
2
d y
dx2
2
d y
dx
2
2
d y
dx
dy
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 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
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
Non-Equi-Width Trapezoids
11 0 0 1 2 1 1 22
1 1
( )
[( )( ( ) ( )) ( )( ( ) ( ))
... ( )( ( ) ( ))]
b
a
n n n n
f x dx
x x f x f x x x f x f x
x 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
af c f x dx
b a
AVERAGE RATE OF CHANGEof f(x) on [a, b]
This value is the “average rate of change” of the function on the interval [a, b]. We use the difference quotient to approximate the derivative in the absence of a function
( ) ( )'( )
( )
f b f af c
b 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 =
d
dt
(position) d
dt
(velocity)
,dx dy
dt 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
x
t
acceleration =
*velocity vector =
*bc topic
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
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
dydt
dxdt
dy
dx
/
/
sin
cos
dd
dd
dy dy d
dx 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
1
2A 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
( ) 0or
( ) 0
f a
g b
( ) '( )lim lim
( ) '( )x a x a
f x f x
g x g x
Integration by Parts
Antiderivative product rule(Use u = LIPET)e.g.
udv uv vdu
( ) ' 'd du dv
uv v u OR udx
vdx dx
uv We know the product rule
( )
( )
( )
d uv v du u dv
u 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
2 311
1x x x
x
2 3 4
ln( 1)2 3 4
x x xx x
Taylor Series
2
( )
( ) ( ) '( )( )
''( )( )
2!
( )( ) .
!
nn
f x f a f a x a
f ax a
f ax a
n