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A. P. CALCULUS BC FORMULA BOOKLET
GRAPHING CALCULATORS
Each student will be expected to bring to the examination a graphing calculator on which the student
can:
1. produce the graph of a function within an arbitrary viewing window;
2. find the zeros of a function;
3. compute the derivative of a function numerically, and
4. compute definite integrals numerically.
Pay special attention to calculator syntax; i.e., placement of parentheses, commas, variables, and order of
operations. Important functions include graph, root, solve, nDeriv, andfnInt.
CALCULATORS should be in RADIAN MODE.
CONTINUITY: The function f(x) is said to be continuous at x = c if
1) f(c) is a finite number;
2) limx c
f(x) exists;
3) limx c
f(x) = f(c) .
DIFFERENTIABILITY: The function is continuous at x = c .
DIFFERENTIABILITY IMPLIES CONTINUITY,
BUTCONTINUITY DOES NOT IMPLY DIFFERENTIABILITY.
LIMITS: ZEROS IN NUMERATOR/DENOMINATOR OF A FRACTION
("c" is a constant.)
Zero (Root)0
c= 0
Vertical Asymptote
c
0=
= D.N.E.
( )Point of Exclusion (Removable Discontiuity)
0
0 = undefined( )
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DERIVATIVE OF A FUNCTION: f ' x( ) = limh0
f x + h( ) f x( )h
or f'
a( ) = limxa
f x( ) f a( )x a
DIFFERENTIATION RULES:
(Where "u" and "v" are differentiable functions of x, and "a" is a constant.)
d
dxau = a
du
dx
d
dxu + v( )=
du
dx+
dv
dx
d
dxu
n= n u
n1
du
dx
d
dxa = 0
d
dxuv( ) = u
dv
dx+ v
du
dxd
dx
u
v
=
v du
dx u
dv
dx
v2
CHAIN RULE:dy
dx=
dy
du
du
dx
d
dxsinu = cos u
du
dx
d
dxcosu = sinu
du
dx
d
dxtan u = sec2 u
du
dx
d
dxcotu = csc2 u
du
dx
ddx
sec u = secu tanu dudx
ddx
csc u = cscu cotu dudx
d
dxln u =
1
u
du
dx
d
dxe
u = eudu
dx
d
dxa
u = a u ln adu
dx
d
dxsin
1u =
du
dx
1 u2d
dxcos
1u =
du
dx
1 u 2
ddx
tan 1 u =
du
dx1+ u 2
ddx
cot 1 u =
du
dx1+ u 2
d
dxsec
1u =
du
dx
u u2 1
d
dxcsc
1u =
du
dx
u u2 1
2
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Decreasing
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VELOCITY: ACCELERATION:
V=ds
dt
a =dv
dt=
dv
ds
ds
dt= v
dv
ds
a =
d2s
dt2
i) If v > 0 and a > 0, the speed is increasing.
ii) If v > 0 and a < 0, the speed is decreasing.
iii) If v < 0 and a > 0, the speed is decreasing. ( Note: speed = v t( ) )iv) If v < 0 and a < 0, the speed is increasing.
DISTANCE: Ifv = f t( ) , the distance traveled by a body between t= a and t= b is given by
f(t)a
b
dt
(Be careful. Does the object change directions between a and b?)
EQUATION OF A TANGENT LINE:
y y1 = f ' (x1 ) x x1( )
EQUATION OF A NORMAL LINE:
y y1 = 1
f ' (x1)
x x1( )
TANGENTS (function must exist at xi )
Vertical tangents: f ' xi( )does not existHorizontal tangents: f ' xi( )= 0
LINEAR APPROXIMATION The linear approximation to f x( ) near x = xo is given byy = yo + f ' xo( ) x xo( ) for x sufficiently close to xo .
EULER'S METHOD ("Numerical Solutions to a Differential Equation")
Iterative use of the Linear Approximation with a given step value.
y1 = y0 + f ' xo( ) x1 xo( )
y2 = y1 + f ' x1( ) x2 x1( )y3 = y2 + f ' x2( ) x3 x2( )etc.
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INVERSES: To find the inverse of y = f(x), solve for x in terms of y, then interchange x and y.
f[ f1
x( )] = x and f1[f x( )] = x
f1( )' d( ) = 1
f' c( )or
dx
dy=
1dy
dx
MEAN-VALUE THEOREM (SPECIAL CASE -- ROLLE'S THEOREM): If the function f(x) is
continuous at each point on the closed interval a < x < b and has a derivative at each point on the open
interval a < x < b, then there is at least one number c, a < c < b, such that
f ' ( c) =f(b) f(a)
b a
a bc
MEAN VALUE THEOREM
a b
y'=0
c
ROLLE'S THEOREM
"Where average velocityf(b) f(a)
b a
meets instantaneous velocity f ' ( c)( )."
ABSOLUTE-VALUE THEOREM:
f(x)= x =x, if x 0
x, if x < 0
GREATEST-INTEGER THEOREM:
g(x) = [x] is the greatest integer not greater than x.
e.g. g(5.2) = 5, g(-1.5) = -2, g(1) = 1
DIRECT VARIATION: y = kx ("y " is directly proportional to "x ")
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INVERSE VARIATION: y =k
xor xy = k ("y " is inversely proportional to "x ")
REFLECTIONS:
The graph ofy = f x( ) is the reflection ofy = f x( ) in the x-axis;eg. y = x
2; y = x2
whereas the graph ofy = f x( ) is the reflection of the graph ofy = f x( ) in the y-axis.eg. y = x ; y = x
ODD/EVEN FUNCTIONS:
EVEN: f x( ) = f x( )ODD: f x( ) = f x( )
e. g. Even function: y = x2
or y = cosx
Odd function: y = x3 or y = sinx
SYMMETRY:
w.r.t. x-axis .... equivalent equations when y replaced by -y
w.r.t. y-axis .... equivalent equations when x replaced by -x
w.r.t. origin .... equivalent equations when x replaced by -x
and y replaced by -y
RELATIONSHIPS between the graphs of and the graphs ofy = f x( ) and the graphs ofy = kf x( ), y = f kx( ), y k= f x h( ), y = f x( ), and y = f x( ).
LOGARITHMIC FUNCTIONS:
y = loga x iff ay= x
y = ln x iff ey= x
PROPERTIES:
ln ab( )= ln a + ln b
lna
b
= ln a ln b
ln ar = r ln a
loga x =lnx
ln a
ln1 = 0lne =1
ln ex = x
e ln x = x
x
a
xb=
x
a +b
x
ab=
x
ab
xa
xb = x
a bx
o=1
xa( )b = xab x a = 1
xa
NATURAL LOGARITHM: lnx =dt
t1
x
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EQUATIONS FOR EXPONENTIAL GROWTH AND DECAY: Equations of the form y ' = ky aresolved as.
A = Aoekt
or A = Aoert
LAWS OF LOGISTIC GROWTH : Equations of the form y ' = ky A y( ).
y =A
1+ Bekt
NB. A=the Maximum Capacity and the POI x,A
2
is the moment of maximum growth.
SLOPE FIELDS
Tips associating the slope field to a particular Differential Equation:
1) Horizantal Dashes dy
dx= 0
2) Dashes\ dy
dx< 0
3) Dashes / dy
dx> 0
4) All Dashes in any column // to each other dy
dxhas no y
5) All Dashes in any row // to each other dy
dxhas no x
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INTEGRATION FORMULAS:
f(x) dx = F(x) + C, whereF' (x) = f(x)
d
dx
f(t) dt= f(x)a
x
[First Fundamental Theorem]
Remember the Chain Rule!!!:d
dxf(t) dt= f(u)
a
u
Du
f(x) dx = F(b) F(a), where F' (x)= f(x)a
b
[Second Fundamental Theorem]
un
du =un+1
n +1+ C, n 1
du
u = ln u + C
eu
du = eu+ C a
udu =
au
ln a+ C, a > 0, a 1( )
sinu du = cos u + C cosu d u = sinu + C
sec2u du = tan u + C csc
2u d u = cotu + C
sec u tan u du = secu + C csc u cot u du = cscu + C
secu du = ln sec u + tanu + C
sin2u du =
1
2u
1
4sin 2u + C cos
2u du =
1
2u +
1
4sin 2u + C
du
a2
u2= sin
1 u
a
+ C
du
a2
+ u2 =
1
a
tan 1u
a
+ C
Integration by Parts: u dv = uv v du
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Integral Boundary Rules
f x( )a
a
dx = 0
f x( )a
b
dx = f x( )b
a
dx
f x( )a
b
dx + f x( )b
c
dx = f x( )ac
dx
If f x( ) g x( ) on a,b[ ], then f x( )a
b
dx g x( )ab
dx
AVERAGE (MEAN) VALUE: If the function y = f x( ) is continuous on the interval a < x < b, then theaverage or mean value ofy with respect to x over the interval [a,b] is
yav( )x
=1
b af(x) dx
a
b
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AREA APPROXIMATIONS
RIEMANN SUMS
A = limn +
f ci( )i=1
n
x
A = f x( )a
b
dx
Left-Hand Rectangles Midpoint Rectangles Right-Hand Rectangles
TRAPEZOIDAL RULE:
f(x) dxa
b
b a
2nf(x0 )+ 2 f(x1 ) + 2f(x2 )+....+ 2f(xn 1 )+ f(xn )[ ]
AREA FORMULAS
Function: A = f(x) g(x)[ ]a
b
dx or A = f(y) g(y)[ ]cd
dy
Polar: A =1
2r ( )[ ]
2
d
Parametric: Eliminate the parameter:
i) isolate the parameter in one equation, and
ii) substitute into the other equation
and then use the Function formula
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ARC LENGTH:
Function: L = 1+dy
dx
2
a
b
dx
Polar: L = r2 + drd
2
d
Parametric: L =dx
dt
2
+
dy
dt
2
a
b
dt
PARAMETRIC, POLAR AND VECTOR FORMS
Parametric: dydx
=
dy
dtdxdt
a function in t( ) and d2
ydx
2 =
dy'
dtdxdt
Vertical Tangent:dx
dt= 0
Horizontal Tangent:dy
dt= 0
Area: Eliminate the parameter and use the Function formula
Arc length: L = dx
dt
2
+dy
dt
2
a
b
dt
Polar: Area: A =1
2r ( )[ ]
2
d
Arc Length: L = r2 +dr
d
2
d
Parametric Polar: x ( )= r cos and y ( ) = r sin
Vector: Velocity v = x' t( ) i + y' t( ) j
Speed= v = x' t( )[ ]2
+ y' t( )[ ]2
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HORIZONTAL ASYMPTOTES (Maximum Capacity) and LIMITS AT INFINITY
xLim f x( ) or
xLim f x( )
= (
L'Hpital's Rule: If limxx 0
f(x)
g(x)
is indeterminate of the form
0
0or
, and if
limxx 0
f ' (x)
g ' (x)
exists, then lim
xx 0
f(x)
g(x)
= lim
xx 0
f ' (x)
g ' (x)
.
IMPROPER INTEGRALS
1. Boundary at infinity: f x( )dxa
= limb
F b( ) F a( )[ ]
2. Boundary is a Veritical Asymptote:
f x( )dxa
b
= limcb
f x( )dxa
c
or = limc a+
f x( )dxc
b
3. Region includes a Vertical Asymptote at x=c: f x( )dxa
b
= f x( )dxa
c
+ f x( )dxcb
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TAYLOR POLYNOMIALS
f x( )= f a( )+ f ' a( ) x a( )+f' ' a( ) x a( )2
2!+
f'' ' a( ) x a( )3
3!+ +
fn a( ) x a( )n
n!+ R
x( )
where R x( )=fn+1 c
( )x
c
( )
n+1
n+ 1( )! for some c x, a( )
McLauren series=Taylor Series where a=0
SERIES OF KNOWN FUNCTIONS
y = sin x =
**y = cos x = 1 x2
2!+
x4
4!
x6
6!+ +
1( )n x2n
2n( ) !+ =
1( )n x2n
2n( ) !0
y = ex =
y =1
1 x= 1+ x + x
2+ x
3+ + xn + = xn
0
on 1< x < 1
**y =1
1+ x= 1 x + x
2 x3 + + x( )n+ = 1( )
nx
n
0
on 1< x < 1
**y =1
1+ x2= 1 x
2+ x
4 x6 + + x( )2n
+ = 1( )nx
2n
0
on 1< x < 1
**y = tan 1 x = x x3
3+
x5
5
x7
7+ +
1( )n x2n+1
2n+1+ =
1( )n x2n +1
2n +10
on 1 x 1
**y = ln 1+ x( ) = x x2
2+
x3
3
x4
4+ +
1( )n xn
n+ =
1( )n xn
n1
on 1< x 1**These can be derived from the unmarked series.
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If a n1
IS an Alternating Series:
Alternating Series Definition: 1( )n+1
an1
or a n cosn0
Liebnitz Alternating Series Test: 1( )n+1
an1
converges if
1. all an are positive,
2. an an +1,
and 3.nLim an = 0
Absolute Convergence vs Conditional Convergence (only applies to Alternating Series)
a n is absolutely convergent if an converges.
a n is conditionally convergent if a n converges but an diverges.
RADIUS OF CONVERGENCE
R is the radius of convergence whennLim
an +1 x a( )n+1
an x a( )n < 1 leads to x a < R
INTERVAL OF CONVERGENCE
Solve x a < R (from the Radius of Convergence) and test convergence at the endpoints
SPECIAL LIMITS (for comparison)
limx0
sinx
x= 1 lim
x01 cos x
x= 0 lim
x0
ex 1
x=1
limn
ln n
n= 0 lim
nnn = 1 lim
nx
1n= 1
limn
xn = 0, if x
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KNOWN SERIES (for comparison)
Geometric Series: arn
1
--converges toa
1 rfor r1--diverges for p1
Harmonic Series:1
n1
= 1+1
2+
1
3+
1
4+ Diverges
Alternating Harmonic Series:1( )n+1
n1
= 11
2+
1
3
1
4+ Converges conditionally
Telescoping Series: Any series that can be simplified by Partial Fractions such that consecutive
terms add to 0, leaving only the first and last terms e.g.,1
n n+ 1( )1
It will generally converge, by the integral Test and partial fractions.
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absolute convergence
absolute minimum
absolute maximum
acceleration
acceleration vector
algebraic functionalternating series
amplitude
antiderivative
antidifferentiation
arc length
arccosine
arcsine
arctangent
asymptote
average rate of changeaxis of rotation
axis of symmetry
base (exponential and log)
bounded above
bounded below
bounded
cartioid
Cartesian Coordinate System
Chain Rule
circlecircular functions
closed interval [a,b]
coefficient
Comparison Test
complex number
components of a vector
composition f g
concave down
concave up
conditional convergenceconic section
constant function
constant of integration
continuity at a point
continuity on an interval
continuous function
convergent improper integral
convergent sequence
convergent series
coordinate axes
cosecant function
cosine function
cotangent functioncritical point
critical value
cross-sectional area
decay model
decreasing function
decreasing on an interval
definite integral
degree
delta notation
derivativedifference quotient
differentiability
differential
differential equation
differentiation
discontinuity
disk method
distance (from velocity)
distance formula
divergent improper integraldivergent sequence
divergent series
domain
dummy variable of integration
dy/dx (leitniz notation)
e
ellipse
end behavior
endpoint extrema
essential discontinuityEuler's Method
even function
exponential function
exponential growth and decay
exponential laws
extremum
factorial
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First Derivative Test
Frequency of a periodic function
function
Fundemental Theorem of Calculus
geometric sequence
geometric series
graphgrowth models
growth rate
half-life
harmonic series
hyperbola
imaginary number
implicit differentiation
improper integral
increasing function
increasing on an intervalincrement
indefinite integral
indterminate form
infinite limit
inflection point
initial condition
initial value problem
inscribed rectangle
instantaneous rate of change
instantaneous velocityinteger
integrable function
integrand
integration by partial fractions
integration by parts
integration by substitution
Intermediate Value Theorem
interval
interval of convergence
inverse functionirrational number
Lagrange Error Bound
Law of Cosines
Law of Sines
left-hand limit
left-hand sum
Leibniz, Gottfried
L'Hopital's Rule
limit
limt at infinity
limit of integration
linear approximation
linear function
local extremalocal linearity
local linearization
logarithmic function
logarithmic laws
logistic equation
logistic growth
lower bound
Maclaurin series
maximum
mean valueMean Value Theorem
midpopint formula
minimum
monotonic
motion
natural log
Newton, Isaac
non-removable discontinuity
normal line
numerical derivativenumerical integration
odd function
one-to-one function
open interval (a,b)
optimization
order of a derivative
origin
parabola
parallel curves
parameterparametric curve
partial fractions
partial sum of a series
partition of an interval
percentage error
period
periodic function
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perpendicular curves
piece-wise defined functions
polar coordinates
polynomial
position function
position vector
power seriesprime notation f'(x)
Product Rule
proportionality
p-series
quadrant
quadratic formula
Quotient Rule
radian
radius of a circle
radius of convergencerange
rate of change
rational function
Ratio Test
real number
rectangular coordinates
region (in a plane)
related rates
relative error
relative maximumrelative minimum
removable discontinuity
Rhiemann sum
right-hand limit
right-hand sum
root of an equation
roundoff error
scalar
secant function
secant linesecond derivative
Second Derivative Test
separable differential equation
sequence
series
set
sigma notation
sine function
slope
slope field
solid (in 3-space)
solid of revolution
speed
spheresubset
symmetry
tangent function
tangent line
tangent vector
Taylor polynomial
Taylor series
term of a sequence or series
transcendental function
Trapezoidal Ruletruncation error for power series
trigonometric functions
unit circle
unit vector
upper bound
u-substitution
vector
vertex
viewing window
volume by slicingx-axis
x-intercept
y-axis
y-intercept
zero of a function
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