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