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Calculus and Algebra

Logarithm Rules

If logam = n    then    an = m, for all m > 0

logam + logan = logamn

logam - logan = loga m/n

nlogam = logamn

logam = logbm

             logba

Properties of Logarithms

 loga1 = 0                                            a0 = 1

 logaa = 1                                            a1 = a

 logaax = x                                           ax = ax

 

Exponent Rules

auav = au + v                                         (ab)x = axbx

au / av = au - v or   1/ av-u                       (a/b)x = ax/bx

(au)v = auv                                             a-x = 1/ax

ar/s = ( s√a   )r  =  s√ar                             apa/ ar  =  ap + q - r

 

Quadratic Formula

X = - b ± Ö b2 - 4ac

            2a

Growth and Decay

A = A0ekt           

[Growth]

A = A0e-kt          

[Decay]

[A0 = initial amount]

[k = growth constant]

[t = elapsed time]

[A = amount after some time t]

th = ln2

        k

[half-life formula]

 

Factoring

(x2 - y2) = (x + y)(x - y)

 

(x3 + y3) = (x + y)(x2 - xy +y2)

(x3 - y3) = (x - y)(x2 + xy + y2)

 

(x - y) = - (y -x)

 

Math Sequences

Additive Sequences (e.g. 2, 4, 6, 8, 10)

an = a1 + (n - 1)d

[to find the value of the term an]

[d = size of the increase between the terms]

Sn = (n/2)(a1 + an)

[to find the sum of the series]

Geometric (e.g. 2, 4, 8, 16, 32)

an = ran-1  or  an = a1r n-1

Sn = a1(1 - rn)

          1 - r

Infinite Geometric

S =  (a1)/(1-r)

Functions and Derivatives Review

Functions

(F ° g) = F(g(x)) 

[e.g. F(x) = 1/x2; g(x) = 3x2 - 1; (F ° g) = F(g(x)) = 1/(3x2 - 1)2; g(x) replaces the single x in the F(x) equation]

Derivatives

F(x) = a constant;  F¢ = 0

Constant

F(x) = a constant;  F¢ = 0

Power Rule

F(x) = cxn; F¢ = n(cx)n-1

Addition

F(x) = g(x) + h(x); F¢ = g'(x) + h'(x)

Subtraction

F(x) = g(x) - h(x); F¢ = g'(x) - h'(x)

Chain Rule

F(x) = g(h(x)); F¢ = g '(h(x)) · h'(x)

General Power Rule

F(x) = [g(x)]n;  F¢ = n[g(x)]n-1 ·g'(x)

[e.g. if F(x) = (x3 + 2)4 then F¢ = 4(x3 + 2)3(3x2 + 0)]

 

(Fg)¢ = F¢g + Fg¢

Multiplication

(g/F)¢ = Fg¢ - F¢g    or   (F/g)¢ = F¢g - Fg¢

                 F2                               g2

Division

y2 = (2yy')

Trigonometry Review

Converting Degrees and Radians

To go from degrees to radians, multiply by p/180

To go from radians to degrees, multiply by 180/p

Trigonomic Functions

sin q = y/r                     csc q = r/y

cos q = x/r                    sec q = r/x

tan q = y/x                    cot q = x/y

Trigonomic Identities

sin² u + cos² u = 1         1 + tan² u = sec² u         1 + cot² u = csc² u = csc² u

tan u = sin u                  cot u = cos u

           cos u                            sin u

sin(π/2 - u) = cos u        cos(π/2 - u) = sin u        tan(π/2 - u) = cot u

csc(π/2 - u) = sec u       sec(π/2 - u) = csc u       cot(π/2 - u) = tan u

sin(-α) = -sin α               cos(-α) = cos α             tan(-α) = -tan α

csc(-α) = -csc α             sec(-α) = sec α             cot(-α) = -cot α

 

sin(u +  v) = sin u cos v + cos u sin v

cos(u + v) = cosucosv + sin u sin v

tan(u + v) = tan u + tan v / (1 + tan u tan v)

 

sin(2u) = 2 sinucosu

cos(2u) = cos²u - sin²u

            = 2 cos² u - 1

            = 1 - 2sin² u

 

sin² u = 1 - cos(2u)

                   2

 

cos² u = 1 + cos(2µ)

                   2

 

tan² u = 1 - cos(2u)

            1 + cos(2u)

 

 

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