ALGEBRA
Properties of Absolute Value
Distance Formula
The distance d between two points ( x1 , y1 ) and ( x2 , y2 ) is:
For all real numbers a and b:
a ≥0
−a = a
a≤ a
ab = a b
d=
Midpoint Formula
a
a
= , b≠0
b
b
a+b ≤ a + b
 x1 + x2 y1 + y2 
,


2
2 
Triangle Inequalityy
Properties of Integer Exponents and Radicals
Assume that n and m are positive integers, that a and b are
nonnegative, and that all denominators are nonzero. See
Appendices B and D for graphs and further discussion.
an ⋅ am = an+ m
(a )
an
= an−m
am
( ab )
a−n =
n m
n
=a b
am n = n am =
n
a
=
b
n
a
n
b
( a)
n
m
( A − B ) ( A + B ) = A2 − B 2
( A + B )2 = A2 + 2 AB + B 2
( A − B )2 = A2 − 2 AB + B 2
( A + B )3 = A3 + 3 A2 B + 3 AB 2 + B3
( A − B )3 = A3 − 3 A2 B + 3 AB 2 − B3
Factoring Special Binomials
(
= ( A + B)( A
A3 + B 3
2
− AB + B 2
)
)
Quadratic Formula
The solutions of the equation ax2 + bx + c = 0 are:
x=
−b ± b 2 − 4ac
2a
Horizontal lines y = c have slope 0.
Vertical lines x = c have undefined slope.
slope of perpendicular line =  −1 m
Special Product Formulas
A3 − B 3 = ( A − B ) A2 + AB + B 2
y2 − y1
x2 − x1
slope of parallel line = m
a = mn a
A2 − B 2 = ( A − B ) ( A + B )
m=
Given a line with slope m:
n
n
ab = n a n b
m n
n
Slope of a Line
Parallel and Perpendicular Lines
= a nm
an
 a
  = n
b
b
1
an
a1 n = n a
n
( x2 − x1 )2 + ( y2 − y1 )2
Forms of Equations of a Line
Standard Form: ax + by = c
Slope-Intercept Form: y = mx + b, where m is the
slope and b is the y-intercept
Point-Slope Form: y − y1 = m ( x − x1 ) , where m is the
slope and ( x1 , y1 ) is a point on the line
Properties of Logarithms
Let a, b, x, and y be positive real numbers with a ≠ 1
and b ≠ 1, and let r be any real number. See Appendix B
for graphs and further discussion.
log a x = y and x = a y are equivalent
log a 1 = 0
log a a = 1
( )
a loga x = x
log a a x = x
log a ( xy ) = log a x + log a y
 x
log a   = log a x − log a y
 y
( )
log a x r = r log a x
log b x =
log a x
log a b
Change of base formula
GEOMETRY
A = area, C = circumference, SA = surface area or lateral area, V = volume
Rectangle
Circle
Triangle
A = lw
A = pr2 C = 2pr
r
w
l
Parallelogram
Trapezoid
A = lh
A=
h
1
h (b + c )
2
b
c
1
bh
2
a
h
b
,
Heron s Formula:
A = s ( s − a )( s − b)( s − c)
where s =
a+b+c
2
h
l
c
Rectangular Prism
Sphere
V = lwh A=
SA = 2lh + 2wh + 2lw
V=
Rectangular Pyramid
4 3
SA = 4pr2
π r 3
h
1
V = lwh
3
r
w
l
h
w
l
Right Cylinder
Right Circular Cylinder
V = ( Area of Base ) h
V = pr2h SA = 2pr2 + 2prh
Cone
1
V = π r 2 h SA = π r 2 + π r r 2 + h 2
3
r
h
h
h
r
Trigonometric and Hyperbolic Functions: Definitions, Graphs, and Identities
See Appendix C.
CONIC SECTIONS
Ellipse
Parabola
( h, k )
Focus
Focus
Focus
( x − h)
1. a2
2
(y − k)
+
Let p ≠ 0.
Vertex: ( h, k )
Standard form of equation:
2
vertically oriented
=1
major axis is horizontal
( x − h )2 + ( y − k )2
2. b2
a2
Focus: ( h, k + p )
Directrix: y = k − p
( y − k )2 − ( x − h )2
Foci: on major axis, c units away
from the center, where
c2 = a2 - b2
Focus: ( h + p, k )
Directrix: x = h − p
a2
b2
b
( x − h)
a
=1
foci are aligned vertically
horizontally oriented
Asymptotes: y − k = ±
2. 2. ( y − k ) = 4 p ( x − h )
major axis is vertical
Focus
( h, k )
foci are aligned horizontally
2
=1
Vertex Vertex
Let a, b > 0.
Center: ( h, k )
Standard form of equation:
( x − h )2 − ( y − k )2 = 1
1. a2
b2
1. ( x − h ) = 4 p ( y − k )
2
b2
Focus
( h, k )
Directrix
Let a, b > 0 with a ≥ b.
Center: ( h, k )
Major axis length: 2a
Minor axis length: 2b
Standard form of equation:
Hyperbola
a
( x − h)
b
Foci: c units away from the center,
where c2 = a2 + b2
Vertices: a units away from the center
Asymptotes: y − k = ±
LIMITS
Definition of Limit
The Squeeze Theorem
Let f be a function defined on an open interval
containing c, except possibly at c itself. We say that
the limit of f ( x ) as x approaches c is L, and write
lim f ( x ) = L, if for every number e > 0 there is a
x→c
number d > 0 such that f ( x ) − L < ε whenever x
satisfies 0 < x − c < δ.
If g ( x ) ≤ f ( x ) ≤ h ( x ) for all x in some open interval
containing c, except possibly at c itself, and if
lim g ( x ) = lim h ( x ) = L, then lim f ( x ) = L as well.
x→c
x→c
x→c
Continuity at a Point
Given a function f defined on an open interval
containing c, we say f is continuous at c if
Basic Limit Laws
lim f ( x ) = f ( c ) .
x→c
Sum Law:
lim  f ( x ) + g ( x ) = lim f ( x ) + lim g ( x )
x→c
x→c
x→c
Difference Law:
lim  f ( x ) − g ( x ) = lim f ( x ) − lim g ( x )
x→c
x→c
x→c
L’Hôpital’s Rule
Suppose f and g are differentiable at all points of an open
interval I containing c, and that g ′ ( x ) ≠ 0 for all x ∈ I
except possibly at x = c. Suppose further that either
lim f ( x ) = 0 and lim g ( x ) = 0
Constant Multiple Law:
x→c
lim  kf ( x ) = k lim f ( x )
x→c
x→c
or
Product Law:
x→c
Quotient Law:
lim
x→c
x→c
f ( x)
f ( x ) lim
= x→c
, provided lim g ( x ) ≠ 0
x→c
g ( x ) lim g ( x )
x→c
lim f ( x ) = ±∞ and lim g ( x ) = ±∞.
x→c
lim  f ( x ) g ( x ) = lim f ( x ) ⋅ lim g ( x )
x→c
x→c
x→c
Then
lim
x→c
f ( x)
f ′ ( x)
= lim
,
g ( x) x→c g ′ ( x)
assuming the limit on the right is a real number or ∞ or -∞.
DERIVATIVES
The Derivative of a Function
Derivatives of Exponential and Logarithmic
Functions
The derivative of f , denoted f ′, is the function whose
value at the point x is
d x
e = ex
dx
d x
a = a x ln a
dx
1
d
(ln x ) =
dx
x
d
1 1
(log a x ) = ln a ⋅ x
dx
( )
f ( x + h) − f ( x)
f ′ ( x ) = lim
,
h→ 0
h
provided the limit exists.
( )
Elementary Differentiation Rules
Constant Rule:
Product Rule:
d
(k ) = 0
dx
d


d
d
 f ( x ) g ( x ) =  f ( x ) g ( x ) + f ( x )  g ( x )
dx


 dx
 dx
Constant Multiple Rule:
Quotient Rule:
d
d
f ( x)
 k f ( x ) = k
dx 
dx
d  f ( x)  f ′ ( x) g ( x) − f ( x) g ′ ( x)
=

2
dx  g ( x ) 
 g ( x )
Sum Rule:
Power Rule:
d
d
d
f ( x) +
g ( x)
 f ( x ) + g ( x ) =
dx 
dx
dx
d r
x = rx r −1
dx
( )
Difference Rule:
Chain Rule:
d
d
d
 f ( x ) − g ( x ) =
f ( x) − g ( x)
dx 
dx
dx
d
 f ( g ( x )) = f ′ ( g ( x )) ⋅ g ′ ( x )
dx 
Derivatives of Trigonometric Functions
d
(sin x ) = cos x
dx
d
(cos x ) = − sin x
dx
d
( tan x ) = sec2 x
dx
d
(csc x ) = − csc x cot x
dx
d
(sec x ) = sec x tan x
dx
d
(cot x ) = − csc2 x
dx
Derivatives of Inverse Trigonometric Functions
d
1
sin −1 x =
dx
1 − x2
(
)
d
csc −1 x = −
dx
x
(
)
d
1
cos −1 x = −
dx
1 − x2
d
1
tan −1 x =
dx
1 + x2
d
sec −1 x =
dx
x
d
1
cot −1 x = −
dx
1 + x2
(
1
2
x −1
(
)
)
1
2
x −1
(
(
)
)
Derivatives of Hyperbolic Functions
d
(sinh x ) = cosh x
dx
d
(cosh x ) = sinh x
dx
d
( tanh x ) = sech 2 x
dx
d
(csch x ) = − csch x coth x
dx
d
(sech x ) = − sech x tanh x
dx
d
(coth x ) = − csch 2 x
dx
Derivatives of Inverse Hyperbolic Functions
d
1
sinh −1 x =
dx
1 + x2
d
cosh −1 x =
dx
−1
d
csch −1 x =
dx
x 1 + x2
−1
d
sech −1 x =
, 0 < x <1
dx
x 1 − x2
(
)
(
(
)
)
(
1
2
x −1
)
The Derivative Rule for Inverse Functions
If a function f is differentiable on an interval ( a, b ) ,
and if f ′ ( x ) ≠ 0 for all x ∈( a, b ) , then f −1 both
exists and is differentiable on the image of the interval
( a, b) under f , denoted as f (( a, b)) in the formula
below. Further,
( )
if x ∈( a, b ) , then f −1 ′ ( f ( x )) =
d
1
tanh −1 x =
,
dx
1 − x2
x <1
d
1
coth −1 x =
,
dx
1 − x2
x >1
(
, x >1
)
(
)
The Mean Value Theorem
If f is continuous on the closed interval [ a, b ] and
differentiable on ( a, b ) , then there is at least one point
c ∈( a, b ) for which
f ′ (c ) =
1
,
f ′ ( x)
f (b ) − f ( a )
.
b−a
and
if x ∈ f
(( a, b)) , then ( f )′ ( x ) =
−1
1
(
f ′ f −1 ( x )
)
.
INTEGRATION
Properties of the Definite Integral
Given the integrable functions f and g on the interval
[ a, b] and any constant k, the following properties hold.
1. a
∫ f ( x ) dx = 0 a
3. ∫
5. ∫
b
a
b
a
k dx = k (b − a ) a
b
b
a
∫ f ( x ) dx = −∫ f ( x ) dx
2. 4. ∫
b
a
b
a
b
 f ( x ) ± g ( x ) dx = ∫ f ( x ) dx ± ∫ g ( x ) dx
a
a
b
b
c
a
each integral exists
7. If f ( x ) ≤ g ( x ) on [ a, b ] , then
b
b
a
a
∫ f ( x ) dx ≤ ∫ g ( x ) dx.
b
kf ( x ) dx = k ∫ f ( x ) dx
c
a
6. ∫ f ( x ) dx + ∫ f ( x ) dx = ∫ f ( x ) dx, assuming
8. If m = min f ( x ) and M = max f ( x ) , then
a ≤ x ≤b
a ≤ x ≤b
b
m (b − a ) ≤ ∫ f ( x ) dx ≤ M (b − a ).
a
The Fundamental Theorem of Calculus
Part I
Part II
Given a continuous function f on an interval I and
a fixed point a ∈ I, define the function F on I by
x
F ( x ) = ∫ f (t ) dt. Then F ′ ( x ) = f ( x ) for all x ∈ I.
a
The Substitution Rule
If u = g ( x ) is a differentiable function whose range is
the interval I, and if f is continuous on I, then
∫
f ( g ( x )) g ′ ( x ) dx = ∫ f (u ) du.
Hence, if F is an antiderivative of f on I,
∫ f ( g ( x )) g ′ ( x ) dx = F ( g ( x )) + C.
If f is a continuous function on the interval [ a, b ] and
if F is any antiderivative of f on [ a, b ] , then
b
∫ f ( x ) dx = F (b) − F ( a ) .
a
Integration by Parts
Given differentiable functions f and g,
∫ f ( x ) g ′ ( x ) dx = f ( x ) g ( x ) − ∫ g ( x ) f ′ ( x ) dx.
If we let u = f ( x ) and v = g ( x ) , then du = f ′ ( x ) dx
and dv = g ′ ( x ) dx and the equation takes on the more
easily remembered differential form
∫ u dv = uv − ∫ v du.
SEQUENCES AND SERIES
Summation Facts and Formulas
Constant Rule for Finite Sums:
n
∑ c = nc, for any constant c
i =1
Constant Multiple Rule for Finite Sums:
n
n
i =1
i =1
∑ cai = c∑ ai , for any constant c
Sum/Difference Rule for Finite Sums:
n
n
n
i =1
i =1
i =1
∑ (ai ± bi ) = ∑ ai ± ∑ bi
Sum of the First n Positive Integers:
n ( n + 1)
i=
∑
2
i =1
n
Geometric Series
For a geometric sequence {an } with common ratio r:
Partial Sum:
sn =
(
a 1− rn
1− r
Infinite Sum:
∞
∑ ar
n −1
n =1
∑i
2
=
i =1
For any real number m and -1 < x < 1,
∞
∑i
i =1
3
=
n ( n + 1)
4
m
(1 + x )m = ∑  n  x n
m ( m − 1) 2 m ( m − 1) ( m − 2) 3
x +
x +
2!
3!
m ( m − 1)( m − n + 1) n
x +
+
n!
= 1 + mx +
n ( n + 1) ( 2n + 1)
6
2
a
, if r < 1
1− r
n= 0
Sum of the First n Cubes:
n
=
Binomial Series
Sum of the First n Squares:
n
) , if r ≠ 0, 1
2
where
 m m ( m − 1)
 m
 m
,
 0  = 1,  1  = m,  2  =
2!
 m m ( m − 1) ( m − n + 1)
and   =
for n ≥ 3
 n
n!
Taylor Series and Maclaurin Series
Given a function f with derivatives of all orders throughout an open interval containing a, the power series
∞
∑
n=0
f (n) ( a )
f ′′ ( a )
f ′′′ ( a )
( x − a )n = f ( a ) + f ′ ( a ) ( x − a ) +
( x − a )2 +
( x − a )3 + n!
2!
3!
is called the Taylor series generated by f about a. The Taylor series generated by f about 0 is also known as the
Maclaurin series generated by f .