3.1 (part 2) Compound Interest & e Functions

I.. Compound Interest: A = P ( 1 + r/n)nt

A = Account balance after time has passed.

P = Principal: $ you put in the bank.

r = interest rate (written as a decimal).

n = number of times a year the interest is compounded. (annual = 1, semi-annual = 2, quarterly = 4, monthly = 12, etc.)

t = time (in years) the money is in the bank.

A) To determine the Account balance after time has passed, plug all the #s into the formula and simplify.

3.1 (part 2) Compound Interest & e Functions

Example: 1) If you deposit $4000 in an account that pays 2.92% interest semi-annually, what is the balance after 5 years? How much did the account earn in interest?

A = P ( 1 + r/n )nt A = 4000 ( 1 + .0292/2 )2•5

A = 4000 ( 1 + .0146 )10

A = 4000 (1.0146)10

A = $ 4623.90

So the account gained $623.90 dollars in the 5 years.

3.1 (part 2) Compound Interest & e Functions

Example: 2) If you deposit $12,500 in an account that pays 4.5% interest quarterly, what is the balance after 8 years? How much did the account earn in interest?

A = P ( 1 + r/n )nt A = 12500 ( 1 + .045/4 )4•8

A = 12500 ( 1 + .01125 )32

A = 12500 (1.01125)32

A = $ 17,880.64

So the account gained $5380.64 dollars in the 8 years.

3.1 (part 2) Compound Interest & e Functions

II.. The Natural Base e.

A) e ≈ 2.72

B) “e” occurs in nature and in math/science formulas.

C) definition: e = as “n” approaches + ∞.

D) y = a•ebx+c + d is the natural base exponential function.

E) y = ebx is the parent function: critical pt at (0 , 1)

1) + b = Growth graph 2) – b = Decay graph

3) Horizontal asymptote: y = 0.

n

n

11

3.1 (part 2) Compound Interest & e Functions

III.. Continuously Compounded Interest: A = Pert

A) Convert the interest rate to a decimal.

1) Move the decimal two places to the left

Example: 3) Find the account balance after you invest $5,000 in a 3.5% continuously compounding account for 8 years.

A = 5000 e^(.035•8)

A = $ 6615.649062

Round money off to the nearest penny (2 decimal places).

A = $ 6615.65

3.1 (part 2) Compound Interest & e Functions

IV. Solving Exponential Equations (that have common bases).

A) Break down both sides of the = sign into the same base #.

1) Break all bases down into a common base.

a) Remember that

2) Exponent property: (34)x+2 = 3^(4x + 8)

3) Set “ exponent = exponent” by crossing off the bases.

a) base expo = base expo (gives expo = expo)

4) Solve for the variable.

Homework page 227 # 33 – 40 all, 45 – 54 all

nnx

x

1