Stuff you MUST know Cold for the AP Calculus Exam
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Transcript of Stuff you MUST know Cold for the AP Calculus Exam
Stuff you MUST know Cold for the AP Calculus Exam
Curve sketching and analysisy = f(x) must be continuous at each: critical point: = 0 or undefined.
local minimum: goes (–,0,+) or (–,und,+) or > 0 at stationary pt
local maximum: goes (+,0,–) or (+,und,–) or < 0 at stationary pt
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
x xe dx e
Basic Integralssin cosx dx xcos sinx dx x 2sec tanx dx x
sec tan secx x dx x
1lndx x
x
ln
xx a
a dxa
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
Other part ofthe 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
If the function f(x) is continuous on [a, b], then f has both an absolute maximum and an absolute minimum on [a,b]
The absolute extremes occur eitherat the critical pointsor at the endpoints.
Extreme Value Theorem
Approximation Methods for Integration
Trapezoidal Rule
10 12
1
( ) [ ( ) 2 ( ) ...
2 ( ) ( )]
b
a
n n
b anf x dx f x f x
f x f x
Riemann Sum 1
n
kk
f c x
LRAM when ck is a LEFT endpointRRAM when ck is a RIGHT endpointMRAM when ck is a MIDPOINT
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
2 2b
aL x t y t dt
2
2b
a
drL r d
d
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 =
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°
1 ,45°
,30°
0100°
tan θcos θsin θθ
1
26
4
3
2
π/3 = 60° π/6 = 30°
sine
cosine
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
Other Indeterminate forms:
0
0 01 0
Write as a ratio
Use Logs
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