c Roberto Barrera, Fall 2015
Math 142 1
1.3 Exponential Models
Mathematical models of Interest
If the principal is invested for a period of time at an interest rate of i, where i is given
as a decimal, then the amount at the end of the first period is:
i is called the interest per period.
Compound interest:
Example: The annual interest for a bank account is 6% a year and is compounded
monthly. Find the interest per period.
Computing compound interest after n periods:
c Roberto Barrera, Fall 2015
Math 142 2
Suppose the principal P earns interest at the annual rate of r, expressed as a decimal,
and interest is compounded m times a year. The amount F after t years in terms of m,
t, and r is
Example: (p.53 of text) Suppose $1000 is deposited into an account that yields 9%
annually. Find the ammont at the end of the fifth year if the compounding is quarterly.
Present value:
Example: (p.57, text) How much money must grandparents set aside at the birth of
their grandchild if they wish to have $20,000 when the grandchild reaches his or her
eighteenth birthday? They can earn 9% compounded quarterly.
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c Roberto Barrera, Fall 2015
Math 142 Effective Yield
Effective annual yield:
Nominal rate:
Example: If $1000 is invested at an annual rate of 9% compounded monthly, then at
the end of the year there is
F=
in the account.
Effective annual yield:
Nominal rate:
Calculating effective yield:
c Roberto Barrera, Fall 2015
Math 142 4
Compounding Continuously
Increasing the number of compoundings per year increases the amount of money in the
account. This amount has an upper bound. When the effective yield is set at this upper
bound, we say that the interest is compounding continuously.
Example: Suppose $1000 is deposited into an account that yields 6% annually. Find
the principal at the end of the first year if it is compounded
(a) annually.
(b) monthly.
(c) weekly.
(d) daily.
(e) hourly.
(f) by the minute.
Compounding continuously when effective yield is
c Roberto Barrera, Fall 2015
Math 142 5
If a principal P earns interest at the annual rate of r, expressed as a decimal, and interest
is compounded continuously, then the amount F after t years is
Example: Suppose $1000 is invested at an annual rate of 9% compounded continuously.
(a) How much is in the account after 3 years?
(b) When will the amount become $1500?
As before, P is calld the present value.
The effective yield, re f f is given by re f f = er − 1.
c Roberto Barrera, Fall 2015
Math 142 1.5 Logarithms
Logarithm base a:
Natural logarithm:
Common logarithm:
Basic properties and rules
1. aloga x = x if x > 0.
2. loga ax = x for all x.
3. loga xy = loga x + loga y
4. loga xy = loga x − loga y
5. loga xc = c loga x
Example: Solve
(a) 2 log(x + 7) + 3 = 0
(b) 3 · 102−5x = 4
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c Roberto Barrera, Fall 2015
Math 142 7
Change of bases
We can write exponential functions in terms of e:
We can write the logarithm to one base interms of the logarithm to another base:
Example: An accont with an initial amount of $1000 is growing at 9% a year, compounding continuously. Find the time it takes for this account to double.