文档介绍：Formulas from Algebra
r n − 1
1 + r + r 2 +···+r n−1 = log x = (log b)(log x)
r − 1 a a b
+ + +···+ = 1 ( + ) |x|−|y| |x ± y| |x|+|y|
1 2 3 n 2 n n 1
2 + 2 + 2 +···+ 2 = 1 ( + )( + )
1 2 3 n 6 n n 1 2n 1
Cauchy-Schwarz Inequality
n 2 n n
2 2
xi yi xi yi
i=1 i=1 i=1
Formulas from Geometry
Area of circle: A = πr 2 (r = radius) Circumference of circle: C = 2πr
= 1 ( + ) =
Area of trapezoid: A 2 h a b (h height; a and b are parallel bases)
= 1 = =
Area of triangle: A 2 bh (b base, h height)
Formulas from Trigonometry
sin2 x + cos2 x = 1 sin π− x = cos x
2
+ 2 = 2 π−=
1 tan x sec x cos 2 x sin x
sin x = 1/ csc x sin(x + y) = sin x cos y + cos x sin y
cos x = 1/ sec x cos(x + y) = cos x cos y − sin x sin y
tan x = 1/ cot x sin x + sin y = 2 sin 1 (x + y) cos 1 (x − y)
2 2
tan x = sin x/ cos x + = 1 ( + ) 1 ( −)
cos x cos y 2 cos 2 x y cos 2 x y
sin x =−sin(−x) = 1 ( x −−x )
sinh x 2 e e
cos x = cos(−x) = 1 ( x + −x )
cosh x 2 e e
Graphs
y
tan x y
os x
sin x cos x
1 arcsin x
– arctan x
2
x
– 3–– 2
2 2
Ϫ1 x
Ϫ1 0 1
Ϫ–
2
Formulas from Analytic Geometry
y2 − y1
Slope of line: m = (two points (x1, y1) and (x2, y2))
x2 − x1
Equation of line: y − y = m(x − x )
1 1
2 2
Distance formula: d = (x2 − x1) + (y2 − y1)
2 2 2
Circle: (x − x0) + (y − y0) = r (r = radius, (x0, y0) center)
(x − x )2 (y − y )2
Ellipse: 0 + 0 = 1(a and b semiaxes)
a2 b2
Definitions from Calculus
The limit statement lim f (x) = L means that for any ε>0, there is a δ>0 such that | f (x) − L| <ε
x→a
whenever 0 < |x − a| <δ.
A function f is continuous at x if lim f (x + h) = f (x).
h→0
1 d
If lim [ f (x + h) − f (x)] exists, it is denoted by f (x) or f (x) and is termed the derivative of f at x.
h→0 h dx
Formulas from Differential Calculus
d d −1
( f ± g) = f ± g log x = x −1 log e ot x =
dx a a dx 1 + x 2
d d 1