Stuff you MUST know Cold for the AP Calculus Exam In preparation for Wednesday May 9, 2012. AP...
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![Page 1: Stuff you MUST know Cold for the AP Calculus Exam In preparation for Wednesday May 9, 2012. AP Physics & Calculus Covenant Christian High School 7525.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649e495503460f94b3c8f0/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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.
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( ) ( )'( )
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
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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
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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
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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
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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
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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
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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
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Trig IdentitiesDouble Argument
sisi n c2n 2 osx xx
2 2 2cos2 cos sin 1 2sinx x x x
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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
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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?
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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
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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
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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
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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
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