Exponential Functions

1
Exponential Functions

• An exponential function is a function of the form
  the real constant
  a is called the base, and the independent
  variable x may assume any real value.
• The graph of y 2x is shown below. It is
  increasing as are all exponential functions with
  base gt 1.

2
More about Exponential Functions

• The graph of y is shown next. It is
  decreasing as are all exponential functions with
  0 lt base lt 1.
• Since exponential functions are increasing or
  decreasing, it follows that they are one-to-one.
  Why?
• By examining the graph we conclude that the range
  of an exponential function is the set of positive
  real numbers.

3
More about Exponential Functions

• The graph of always passes
  through the points (0, 1) and (1, a).
• The graph of is the reflection
  about the y-axis of the graph of
• The graph of has the horizontal
  asymptote y 0.
• If there are two exponential functions
  and a lt b, then

4
Solving Exponential Equations

• If ?
  ?
• Example. Solve 310 35x. By the previous
  bullet points,
• Example. Solve. 27 (x1)7. By the previous
  bullet points,

5
The Number e

• As the real number m gets larger and larger,
• The limiting value 2.71828... is an irrational
  number known as e.
• In order to simplify certain formulas,
  exponential functions are often written with base
  e.
• For x gt 0, 2x lt ex lt 3x.

6
Compound Interest

• When the money in an account receives compound
  interest, each interest payment includes interest
  on the previously accrued interest.
• Example. 100 compounded annually at 10
  interest for 3 years, and P dollars compounded
  annually at r interest for 3 years

Year Starting Amount Ending Amount StartingAmount Ending Amount
1 100 100(10.1) 110 P P(1r)
2 110 110(10.1) 121 P(1r) P(1r)2
3 121 121(10.1) 133.10 P(1r)2 P(1r)3
7
Compound Interest Formula

• If the interest on an account at r annually is
  compounded k times per year, the interest rate
  applied to each accounting period is r/k.
• When k 2, we say that interest is compounded
  semiannually, when k 4, we say that interest is
  compounded quarterly, and when k 12, we say
  that interest is compounded monthly.
• In general, if P dollars are invested at an
  annual interest rate r (expressed in decimal
  form) compounded k times annually, then the
  amount A available at the end of t years is

8
Compound Interest Example

• Suppose that 6000 is invested at an annual rate
  of 8. What will be the value of the investment
  after 3 years if (a) interest is compounded
  quarterly?
  (b) interest is compounded
  semiannually?
• In which case, (a) or (b), is the total amount of
  interest greater? Why?

9
Continuous Compounding

• Suppose we let the number of compounding periods
  k increase without bound. (imagine compounding
  every second, then every millisecond, etc.). The
  amount of the investment of P dollars after t
  years approaches a limit
• When the above situation pertains, we say that we
  are compounding continuously.
• In general, if P dollars are invested at an
  annual interest rate r (expressed in decimal
  form) compounded continuously, then the amount A
  available at the end of t years is

10
Compound Interest Examples

• Example. Suppose that 6000 is invested at an
  annual rate of 8. What will be the value of the
  investment after 3 years if interest is
  compounded continuously?
• Note that the amount of the investment after 3
  years is greater than it was when compounding was
  done semiannually or quarterly? Why?
• Example. Suppose that a principal P is to be
  invested at continuous compound interest of 8
  per year to yield 10,000 in 5 years. How much
  should be invested?

11
Exponential Growth Model--World Population

• A model which predicts the quantity Q, which is
  number or biomass, for a population at time t is
  the following exponential growth
  model
  Both q0 and k are constants specific to the
  particular population in question, and k is
  called the growth constant.
• For the world population, k 0.019 and q0 6
  billion when t 0 corresponds to the year
  2000. The model is
  In the year 2010, the model predicts a world
  population of

12
Exponential Decay Model

• A model which predicts the quantity Q, which is
  mass, for a particular radioactive element at
  time t is the following exponential decay
  model Both
  q0 and k are constants specific to the particular
  radioactive sample in question, and k is called
  the decay constant. We use the term half-life to
  describe the time it takes for half of the atoms
  of a radioactive element to break down.
• A radioactive substance has a decay rate of 5
  per hour. If 500 grams are present initially,
  how much remains after 4 hours?

13
Summary of Exponential Functions We discussed

• Definition of an exponential function and its
  base
• Fact that exponential functions are increasing or
  decreasing and therefore they are one-to-one
• Range of an exponential function
• Horizontal asymptote of an exponential function
• Solving exponential equations
• The number e
• The formula for compound interest
• The formula for continuous compounding
• Exponential growth and decay