# Functions and Models

This chapter is mostly a review of pre-calculus., in
conjunction with part of the preview assignments .

In class we cover just the last two sections , on exponentials and logarithms ,
plus topics that

come up from questions on the preview assignment.

The exercises for this chapter start with four online
WebAssign diagnostic assignments. These

correspond to the diagnostic tests near the front of the textbook, but submit
your answers online

even if you work them all out on paper first; partly to learn how to use
WebAssign before we start

on the graded assignments.

A function like f (x) = 2^{x} is called exponential because
the argument x is the exponent in the

formula. Exponential functions are the most basic and common transcendental
functions, and are

probably the most important functions in mathematics and science after
polynomials . So this is

the early encounter mentioned in the sub-title of the text.

We will see how exponential functions can be defined to
have graphs that are continuous , unbroken

curves with well defined slopes, rather being only a collection of separate
points for integer values

of x.

**Non-negative integer powers of 2
**With basic algebra, exponential functions are defined first for positive
integer arguments, by formula

2^{n} = 2· 2 · 2 · · · 2, the product of n factors 2.

Then to satisfy the rule for the case m = 0 requires

2^{0} = 1

so all non-negative integers n are covered.

**Negative integer powers of 2
**For a negative integer n, |n| = −n is positive, and to satisfy the rule
we must have

and so dividing by

for n a negative integer.

** Rational powers of 2
**Next we can make sense of exponentials for rational exponents. To get the
exponential 2r for any

rational number r start with exponent 1/q, q a positive integer. To satisfy the rule

requires so taking the q-th root of both sides of this equation,

the q-th root of 2, for q a positive integer.

Finally, any rational number can be written as r = p/q
with p an integer, q a positive integer, and

the same rule requires

**Irrational powers of 2 (so all power of 2)
**The graph of 2

^{x}for all rational x looks like a dense collection of dots along a curve which increases

to the right. Can we fill in the gaps at irrational values of x and get an smooth, uninterrupted

curve? For example, can we make sense of an irrational power like ?

A number like
is approximated by a succession of decimal fractions 1, 1.7, 1.73,

1.732, 1.7320, 1.73205 and so on: it is the limit of this sequence of rational
numbers. Raising 2 to each

of these powers gives the following new sequence of numbers (everything rounded
to five decimal

places):

All of these should be less that
since the values are increasing as the exponent increases and

is greater than each of these exponents. On the other hand if we round up the
decimal approximations

of
,
the exponentials should all be greater than
:

So it appears that

so that rounded to four decimal places is 3.3220.

We could continue with either sequence to compute a value
for
to as many decimal places as

we wish. In this way, we can make sense of, and compute, any power of 2,
rational or irrational,

so we have made sense of the exponential function f(x) = 2^{x} for all real
arguments x.

**Irrational powers of any positive number
**There is nothing special about the base 2 used above except that it is
positive: we could do the same

thing with any positive real number a, to compute the exponential function f(x) = a

^{x}. The graphs

for the different functions vary mostly in that they are increasing for a > 1, and increase faster for

larger values of a, and are decreasing for 0 < a < 1, decreasing faster for smaller values of a. In the

borderline case of a = 1, the graph is a constant: 1

^{x}= 1.

**Rules for Exponential Functions
**The familiar rules for exponentials still hold just as with with rational
exponents: for a and b

positive and any real numbers x and y,

**Applications of Exponential Functions
**The textbook does an example of population growth which we will see again
later. For variety, let

us look at radioactive decay.

ADDED EXAMPLE A The half-life of strontium-90, is 25 years. This means that half of any

given quantity of will disintegrate in 25 years.

a. If a sample of
initially has a mass of 24mg, find an expression for the mass m(t) that

remains after t years.

b. Find the mass remaining after 40 years, correct to the nearest milligram.

c. Use a graphing device to graph m(t) and use the graph
to estimate the time required for the

mass to be reduced to 5 mg .

**The number e
**Of all possible choice of the base a of an exponential function a

^{x}, one is most convenient for mathematics

because it makes the slope of the graph simplest : the number called e whose value is

approximately e ≈ 2.71828.

The graphs of all exponential functions pass through the
point P(0, 1) on the y-axis, but the bigger

a is, the faster the function value grows as x increases, so the greater the
slope is at this point. The

slope is zero for a = 1, when the function is constant, and increases as a
increases. Experimenting

with a graphing calculator suggests that the slope is less than 1 for 2^{x}, but
greater than 1 for 3^{x}.

So it seems that by increasing a to somewhere between 2 and 3, the slope will be
1 at P(0, 1), with

the slope greater than 1 for greater values of a, less than 1 for lesser values.
That is, there is just one

special value for the base that gives slope 1: this is the value called e. We
have already seen that

e lies between 2 and 3, and with ever more careful computation of slopes we
could calculate the

more accurate value given above.

We will soon see that any other exponential function can
be written in terms of e^{x}, and this is very

convenient in calculus, making this particular exponential function so important
that it is often

called simply “the exponential function”.

Homework Exercises 1, 2, 7, 8, 17, 18, 25.

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