# Gus and Lyle's Precalculus Website

## Formulas, Properties, etc.

### Formula for Compound Interest

n compounding s per year.........A = P(1+rn)nt
continuous compound.........A = Pert

(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+rn)nt
= 12,000(1+.034)4(5)
= \$13,094.21

b) A = Pert
= 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 = 𝓁n2r

### Polar coordinates ⇄ Rectangular coordinates

P→R

x = rcos𝛳
y = rsin𝛳

Ex.
(2, 𝜋3)
x = 2cos𝜋3 = 1
y = 2sin𝜋3 = √3
= (1, √3)

R→P

tan𝛳 = yx
r2 = √(x2 + y2)

Ex.
(-1, 1) (in QⅠ)
𝛳 = tan-1(yx)
tan-1(1-1) = 3𝜋4
r = √(-12 + 12) = √2
= (√2, 3𝜋4)

### 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

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