Explore Average Rates of Change in an Exponential Decay Context

5.10.3 Explore Average Rates of Change in an Exponential Decay Context

5.10.3 • Explore Average Rates of Change in an Exponential Decay Context

Activity

Here is a graph you saw in an earlier lesson. It represents the exponential function $p$ , which models the cost $p (t)$ , in dollars, of producing 1 watt of solar energy from 1977 to 1988, where $t$ is years since 1977.

Line graph showing a sharp decline in dollars per watt over 12 years since 1977, dropping from about $80 to below $10 per watt. The curve smooths as it approaches the lower values.

1. Clare said, “In the first five years, between 1977 and 1982, the cost fell by about $12 per year. But in the second five years, between 1983 and 1988, the cost fell only by about $2 a year.” Show that Clare is correct.

Compare your answer:

$\frac{p (5) - p (0)}{5 - 0} \approx - 12.2$

$\frac{p (11) - p (6)}{11 - 6} \approx - 2.17$

2. If the trend continues, will the average decrease in price be more or less than $2 per year between 1987 and 1992? Be prepared to show your reasoning.

Compare your answer:

From the first 5 years to the next 5 years, the rate of decrease slowed down, so if the trend continues, the decrease in price from 1987 to 1992 should be less than $2 per year.

Are you ready for more?

Extending Your Thinking

Suppose the cost of producing 1 watt of solar energy had instead decreased by $12.20 each year between 1977 and 1982. Compute what the costs would be each year and plot them on a graph. Compare your graph to the graph from the beginning of the 5.10.3 activity (top of page). How do these alternate costs compare to the actual costs shown?

Compare your answer:

A graph shows dollars per watt decreasing over 11 years since 1977. Data points form a downward curve; a dashed tangent line at t=2 decreases at a constant rate, illustrating the rate of price decline over time.

The alternate costs with the constant decrease would be the same in 1977 and 1982 and higher for each year in between.

Self Check

The graph below shows the bounce height of a basketball at each bounce. What is the approximate average rate of change from the first bounce to the third bounce?

-600
600
-300
300

Additional Resources

Trends in Average Rates of Change

A construction company purchased some equipment costing $300,000. The equipment depreciates in value over time.

A scatter plot showing equipment value depreciating over 15 years. The value drops steeply at first, from $300,000, then declines more gradually, reaching under $25,000 by year 15. Axes are labeled years and value in dollars.

Approximately how much did the value drop from years 0 to 5 compared to years 6 to 10?

$\frac{v (5) - v (0)}{5 - 0} \approx \frac{140, 000 - 300, 000}{5} = - 32, 000$

$\frac{v (10) - v (6)}{10 - 6} \approx \frac{70, 000 - 120, 000}{4} = - 12, 500$

Following this trend, we would expect the average rate of change of the value of the equipment from years 11 – 15 to be below –12,500 dollars per year.

Try it

Trends in Average Rates of Change

Graph showing the value of equipment in dollars, or v of t, as a function of the number of years after equipment was purchased, or t. plotted points representing exponential decay are shown, including the y-intercept 0 comma three hundred thousand and the points 5 comma one hundred forty thousand, 6 comma one hundred twenty thousand, and 10 comma seventy thousand.

Approximately how much did the value drop from year 11 to year 15?

Compare your answer:

Here is how to find the average value drop from year 11 to year 15:

(Note that approximate points are used, so your answer may vary a little, but the answer should still be below –10,000.)

$\frac{v (15) - v (11)}{15 - 11} \approx \frac{30, 000 - 60, 000}{4} = - 7, 500$