Chapter 5: Exponential and Logarithmic Functions

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Chapter 5 Exponential and Logarithmic Functions

• Section 5.3 Exponential Functions and Models

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Introduction to Exponential Functions

• Imagine the following scenario A local
  millionaire offers you a choice between receiving
  5 million today or .01 today, .02 tomorrow,
  .04 the next day, and so on for 30 days. Which
  will you choose?
• The mathematically inclined will always choose
  the latter scenario. At the end of 30 days, you
  will have accrued a total of 10,737,418.23 more
  than twice the original offer.
• The above scenario is simply one example of the
  use of a so-called exponential function. A
  definition is as follows
• Exponential FunctionA function f represented by
  f(x) Cax, a gt 0, a not equal to 1, and C gt
  0,is an exponential function with base a.
• The obvious primary difference between this type
  of function and the kind weve been studying
  until now is that the independent variable now
  appears in the exponent. Note that if a 1, we
  simply have the constant function f(x) C. Note
  also that when x 0, we have f(0) C. Thus, in
  applications, C often represents the initial
  value of our function (e.g., when x represents
  time).

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Examples and Characteristics

• Some examples of exponential functions f(x) 3x
  , g(x) 4(0.5)x, h(x) 2x2.
• The primary characteristic of an exponential
  function is its extremely rapid rate of growth as
  the independent variable x increases. In other
  words, exponential functions generally have very
  high average rates of change, even on small
  intervals. This fact is all the more obvious when
  viewing the graph of an exponential function.
• Example 1 Let f(x) 2x. Find the y-intercept
  of f (that is, f(0) ). What is f(5)? f(10)?
• Solution The y-intercept of f can be found by
  evaluating f when x 0 f(0) 20 1. On the
  other hand, f(5) 25 32, while by the time x
  reaches 10, we have f(10) 210 1024. Clearly,
  f grows very quickly.
• One of the prime examples of exponential growth
  is in applications involving population (e.g.,
  animal, bacterial, human) growth. For the moment,
  we examine a simpler example.

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A Simple Application

• Example 2 (text, p. 373, Example 2). A college
  graduate signs a contract for an annual salary of
  50,000 and can choose either Option A a fixed
  annual raise of 6000 each year, or Option B a
  10 raise each year. However, the graduate must
  stay with the same option. Which is the better
  option?
• Solution We can determine the better option by
  comparing salaries at various points along the
  career line. Let x represent the number of years
  worked and use only the integer values of x.
  Thus, during the first year of work, x 0 and
  the salary is 50,000. Our graph will contain the
  point (0,50000), and since the increase in
  salary per year is constant, we have a constant
  rate of change. That is, we have a linear
  function with slope m 6000 and passing through
  the point (0, 50000). The salary under option A,
  then, can be determined in year x by the function
  f(x) 6000x 50,000. Under option B, we
  determine that the salary can be given by g(x)
  50000(1.10)x, where x again represents the
  number of full years worked. By comparing
  salaries as in Table 5.14, it is clear that
  option B quickly becomes the more lucrative
  choice, a fact that is all the more clear when
  these functions are graphed together.

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Exponential Models

• Exponential functions are used to model an
  enormous number of phenomena, both in nature and
  in society. Examples of such phenomena are, as
  mentioned, population growth, radioactive decay,
  thermal diffusion, compounding interest. The
  list goes on and on. Here, we will consider
  several applications of compounding interest, and
  the student will be left to work on various other
  applications.
• Example 3 The parents of a college student
  place 10,000.00 in a 5-year CD earning 7
  interest, compounded monthly, to be offered upon
  graduation as a gift. To the nearest dollar, what
  amount will the student receive at the end of
  these five years?
• Solution Using the formula for compound
  interest (text, p. 379), we find that the
  student will receive, to the nearest dollar,
  14,176.00 at the end of the five-year period.
• Problem If the cost of college tuition is
  currently 5000 per year, inflating at 6 per
  year, how much will a current freshman save in
  tuition costs, to the nearest dollar, by
  finishing her degree in four years rather than in
  five?

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More Compound Interest

• Often, financial institutions attempt to attract
  clients with more frequent compounding rather
  than higher interest rates. In general, more
  frequent compounding at a similar rate will yield
  higher balances, though perhaps not as much as
  one might expect. Suppose, for example, that your
  bank offers daily compounding at 6, while one of
  its competitors offers hourly compounding at 5.
  Which is better? On the other hand, what if an
  institution were to offer compounding by the
  minute, or by the second? What if we just never
  stop compounding? This is done, and it is called
  continuous compounding. A formula for
  continuously compounded interest can be found on
  page 381 of the text.
• Problem The parents of a college student place
  10,000.00 in a 5-year CD earning 7 interest,
  compounded continuously, to be offered upon
  graduation as a gift. To the nearest dollar, what
  amount will the student receive at the end of
  these five years?

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The Natural Exponential Function

• The formula for continuously compounded interest
  given in the text is an example of a natural
  exponential functionthat is, an exponential
  function with base e. Natural exponential
  functions are found most often in applications to
  natural phenomena, and will be useful in later
  sections for solving exponential and logarithmic
  equations.
• Homework for this section