HONR 297Environmental ModelsChapter 3: Air Quality Modeling3.4: Review of Exponential Functions

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FunctionsWhat 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.

-2 -1 0 1 2-10

-5

0

5

10yx3x

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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 InputAll 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 FunctionsAn 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 eAnother 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=Uw

Mhx8JcSJQ

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A Better Trendline for the Toads DataUsing 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!

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 4A 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 NotationIn 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 5Write 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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ExercisesUsing 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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ResourcesCharles 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.st-and.ac.uk/~

history/PictDisplay/Euler.html