Download exponential function

HONR 297
Environmental Models
Chapter 3: Air Quality Modeling
3.4: Review of Exponential Functions
Functions
What is a function?
 Here is an informal definition:

◦ A function is a procedure for assigning a unique
output to any acceptable input.
◦ A function’s domain is the set of allowable
inputs.
◦ A function’s range is the set of outputs one gets
by putting in all domain values.

Functions can be described in many ways!
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Example 1 (Some Functions)

(a) Explicit algebraic formula
◦
◦
◦
◦
◦
f(x) = 4x-5
(linear function)
g(x) = x2
(quadratic function)
r(x) = (x2+5x+6)/(x+2) (rational function)
p(x) = ex
(exponential function)
Functions f and g given above are also called
polynomials.
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Example 1 (Some Functions)

(b) Graphical representation, such as the
following graph for the function y = x3+x.
y x3 x
10
5
0
-5
-10
-2
-1
0
1
2
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Example 1 (Some Functions)

(c) Description or procedure
◦ Assign to each house or business in a
neighborhood a street address.
◦ Assign to each house or business in a
neighborhood an air quality level.
◦ Record the number of NPL sites in each state.
◦ Measure the head level in each monitoring well at
a gas station with leaking storage tanks.
◦ Etc.
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Example 1 (Some Functions)

(d) Table of values
or data
◦ The table to the
right shows the
land area in
Australia colonized
by the American
marine toad (Bufo
marinis).
Year
Area(km^2)
1939
32800
1944
55800
1949
73600
1954
138000
1959
202000
1964
257000
1969
301000
1974
584000
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Example 1 (Some Functions)

Question: What are the domain and
range of each example given above?
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Functions with More Than One
Input
All of the examples of functions given
above are single variable functions, i.e. for
each single input, the function produces an
output.
 A function may also have more than a
single input value – we call this type of
function a multi-variable function.
 The ideas of domain and range can be
extended to multi-variable functions!

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Example 2 (Some Multi-Variable
Functions)
The function z = x2 + 3y2 is a multi-variable
function.
 For each choice of an x and y value, which we
can denote as an ordered pair, (x, y), this function
produces an output z.
 If (x, y) = (-1, 4), then

◦ z = (-1)2 + 3(4)2 = 1 + 3(16) = 49
We can graph this function – to do so we need
three coordinate axes!
 Other examples of multi-variable functions
include z = sin(x)*cos(y) and the contaminant
plume model we saw in our last lecture!

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Example 2 (Some Multi-Variable
Functions)
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Example 2 (Some Multi-Variable
Functions)
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Exponential Functions

An exponential function is a function of the form
◦ y = ax for a > 0.
We call a the base of the exponential function.
For exponential functions, the input is the power
to which the base a is raised!
 All exponential functions with base a ≠ 1 have the
following in common:


◦
◦
◦
◦
Domain: all real numbers x
Range: all real numbers y > 0
(0,1) is on the function’s graph
The graph is either always increasing or always
decreasing!
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Example 3 (Some Exponential
Functions)
f(x) = 2x
 g(x) = (1/2)x = 2-x by Laws of Exponents!
 h(x) = ex

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Exponential Functions in Models


It turns out that
exponential functions
are very useful as
models in many
settings.
For example,
exponential functions
can be used to
describe population
growth, radioactive
decay, investment
interest, cooling of
coffee, and air
pollution!
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The Base e
Any real number a > 0 can
be used as a base in an
exponential function.
 It turns out that a base
that is very convenient to
work with is Euler’s number,
e = 2.71828 …
 This number, which like pi
is irrational (i.e. it cannot be
written in terminating or
repeating decimal form).
 One reason e is chosen as
a base is that the graph of
y = ex has a slope of one at
the point (0,1).

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The Base e


Another reason for
choosing the base of e is
that the inverse of the
this function is y = ln x
and it is easier to work
with ln x than it is to
work with logarithms in
other bases.
Finally, any exponential
function with base a is
related to an exponential
function with base e via
ax = exlna.
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The Base e – Some Notes!

Notation and terminology:
◦ exp(x) = ex
◦ We call exp(x) the natural
exponential function.
◦ We call ln x the natural
logarithm function.
◦ The reason for this is that
the natural exponential
function appears in many
models of things found in
nature (such as those listed
above on slide 14).
◦ Euler probably chose e for
“exponential”, since e is
used as the base of an
exponential function …
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The Base e – Some ways to define
(and estimate) it!

e is the limit as x approaches infinity of
(1+1/x)^x.
◦ This means that as x gets larger and larger,
(1+1/x)^x gets closer and closer to the
number e.
e = 1 + 1/2! + 1/3! + 1/4! + 1/5! + …
 Here’s another way to define e in terms
of an area:

◦ http://www.youtube.com/watch?v=UwMhx8Jc
SJQ
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A Better Trendline for the Toads Data
Using Excel’s Trendline feature, we can
find a function that fits the data better
than a linear function!
 It turns out that an exponential function
does a much better job!

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Example 4

Using the exponential
trendline found by Excel,
along with the POWER
and EXP function, compare
the actual toad data to that
found with the exponential
trendline
◦ y = 9*10-62e0.0779x.

Note that Excel 2007 may
give
◦ y = 9*10-62e0.077x.
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Example 4
A way to fix the “missing digits” in the
trendline equation can be found here:
http://support.microsoft.com/kb/282135
 Unfortunately, this may introduce a new
problem!

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Scientific Notation
In the exponential function found by Excel
to model the toad data, the coefficient of the
exponential term was given as 9E-62.
 This number is to be interpreted as
9*10^(-62), which is in scientific notation.
 Recall that multiplying a number by an
integer power of 10 moves the decimal place
right if the power is positive and left if the
power is negative.
 Scientific notation is useful for working with
very large numbers or numbers very close
to zero.

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Example 5

Write each number using scientific notation
◦ - 45,000,000,000,000,000
= - 4.5 x 10^(16)
◦ 0.0000000000000124579
= 1.24579 x 10^(-14)

Write each number given in scientific
notation as a decimal
◦ - 2.345 x 10^(-7)
= - 0.0000002345
◦ 3.56 x 10^(72)
= 356000000000 … 000 (70 zeros!)
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Exercises
Using a graphing calculator or Excel, try
to sketch the functions given in problems
# 1 – 5 on page 79 of Hadlock.
 Repeat with Mathematica, Wolfram Alpha,
or some other online graphing package.

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Resources
Charles Hadlock, Mathematical Modeling
in the Environment – Chapter 3, Section 4
 James Stewart, Calculus – Early
Transcendentals (7th. ed.)
 St. Andrews History of Mathematics
Archive (Euler’s picture)

◦ http://www-groups.dcs.stand.ac.uk/~history/PictDisplay/Euler.html
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