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Gasoline costs have experienced some wild fluctuations over the last several decades. [link] http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014. lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.
$y$ | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
$C\left(y\right)$ | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year . In this section, we will investigate changes such as these.
The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in [link] did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.
The Greek letter $\text{\Delta}\text{\hspace{0.17em}}$ (delta) signifies the change in a quantity; we read the ratio as “delta- y over delta- x ” or “the change in $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ divided by the change in $\text{\hspace{0.17em}}x.$ ” Occasionally we write $\text{\hspace{0.17em}}\text{\Delta}f\text{\hspace{0.17em}}$ instead of $\text{\hspace{0.17em}}\text{\Delta}y,\text{\hspace{0.17em}}$ which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.
In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was
On average, the price of gas increased by about 19.6¢ each year.
Other examples of rates of change include:
A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are “output units per input units.”
The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.
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