20052006200720082009201020112012
\displaystyle C\left(y\right)C(y)2.312.622.843.302.412.843.583.68
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 the table above 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.
Average rate of change=\displaystyle \frac{\text{Change in output}}{\text{Change in input}}
Change in input
Change in output
=\displaystyle \frac{\Delta y}{\Delta x}
Δx
Δy
=\displaystyle \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}
x
2
−x
1
y
2
−y
1
=\displaystyle \frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}
x
2
−x
1
f(x
2
)−f(x
1
)
The Greek letter \displaystyle \DeltaΔ (delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in \displaystyle yy divided by the change in \displaystyle xx.” Occasionally we write \displaystyle \Delta fΔf instead of \displaystyle \Delta yΔy, 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
\displaystyle \frac{\Delta y}{\Delta x}=\frac{{1.37}}{\text{7 years}}\approx 0.196\text{ dollars per year}
Δx
Δy
=
7 years
1.37
≈0.196 dollars per year
On average, the price of gas increased by about 19.6¢ each year.
Other examples of rates of change include:
A population of rats increasing by 40 rats per week
A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)
The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
The amount of money in a college account decreasing by $4,000 per quarter
A GENERAL NOTE: RATE OF CHANGE
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.
\displaystyle \frac{\Delta y}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}
Δx
Δy
=
x
2
−x
1
f(x
2
)−f(x
1
)
HOW TO: GIVEN THE VALUE OF A FUNCTION AT DIFFERENT POINTS, CALCULATE THE AVERAGE RATE OF CHANGE OF A FUNCTION FOR THE INTERVAL BETWEEN TWO VALUES \DISPLAYSTYLE {X}_{1}X
1
AND \DISPLAYSTYLE {X}_{2}X
2
.
Calculate the difference \displaystyle {y}_{2}-{y}_{1}=\Delta yy
2
−y
1
=Δy.
Calculate the difference \displaystyle {x}_{2}-{x}_{1}=\Delta xx
2
−x
1
=Δx.
Find the ratio \displaystyle \frac{\Delta y}{\Delta x}
Δx
Δy
.
