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}

__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}

r^{2} = √(x^{2} + y^{2})

**Ex.**

(-1, 1) (in QⅠ)

𝛳 = tan^{-1}(^{y}⁄_{x})

tan^{-1}(^{1}⁄_{-1}) = ^{3𝜋}⁄_{4}

r = √(-1^{2} + 1^{2}) = √2
**= (√2, ^{3𝜋}⁄_{4})**

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||

The dot product of u = ＜u_{1}, u_{2}＞ and v = ＜v_{1}, v_{2}＞ is given by u · v = u_{1}v_{1}+u_{2}v_{2}.

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