Gus and Lyle's Precalculus Website

Formulas, Properties, etc.

Formula for Compound Interest

n compounding s per year.........A = P(1+^r⁄_n)^nt
continuous compound.........A = Pe^rt

(variables: A = amount, P = principal, r = interest rate (convert to decimal if it's a percent), n = number of times interest is compounded per unit t, t = time)

Example
A total of $12,000 is invested at an annual rate of 3%. Find the balance after 5 years if the interest is compounded a) quarterly and b) cotninuously.

a) A = P(1+^r⁄_n)^nt
= 12,000(1+^.03⁄₄)⁴⁽⁵⁾
= $13,094.21

b) A = Pe^rt
= 12,000e^.03(5)
= $13,942.01

Note: e is a number. To type it on your calculator, press "2nd" and then "÷".

To find out how long it will take your money to double, use this formula: T = ^𝓁n2⁄_r

Polar coordinates ⇄ Rectangular coordinates

P→R

x = rcos𝛳
y = rsin𝛳

Ex.
(2, ^𝜋⁄₃)
x = 2cos^𝜋⁄₃ = 1
y = 2sin^𝜋⁄₃ = √3
= (1, √3)

R→P

tan𝛳 = ^y⁄_x
r² = √(x² + y²)

Ex.
(-1, 1) (in QⅠ)
𝛳 = tan^-1(^y⁄_x)
tan^-1(¹⁄_-1) = ^3𝜋⁄₄
r = √(-1² + 1²) = √2
= (√2, ^3𝜋⁄₄)

Properties of Vector Addition and Scalar Multiplacation

u, v, and w are vectors. c and d are scalars.

1. u+v = v+u
2. (u+v)+w = u+(v+w)
3. u+0 = u
4. u+(-u) = 0
5. c(du) = (cu)d
6. (c+d)u = cu+du
7. c(u+v) = cu+cv
8. 1(u) = u and 0(u) = 0
9. ||cv|| = c||v||

Dot Product Properties

The dot product of u = ＜u₁, u₂＞ and v = ＜v₁, v₂＞ is given by u · v = u₁v₁+u₂v₂.

1. u · v = v · u
2. 0 · v = 0
3. u · (v+w) = u · v + u · w
4. v · v = ||v||²
5. c(u · v) = cu· v = u · cv