3.1 Exponential Growth and Decay (continued)

Trends in the behavior of exponential functions

Summary

A function $p$ that is always decreasing and decreases at a constant rate.

A function $q$ that is always increasing and increases at an increasing rate.

A function $r$ that is always increasing for $t \lt 2$, always decreasing for $t \gt 2$, and is always changing at a decreasing rate. (Recall that when a negative number decreases, it becomes more negative.)

A function $s$ that is always increasing and increases at a decreasing rate. (Hint: to find a formula, think about how you might use a transformation of a familiar function.)

A function $u$ that is always decreasing and decreases at a decreasing rate. (Recall that when a negative number decreases, it becomes more negative.)

Grinnell Glacier in Glacier National Park in Montana covered about $142$ acres in 2007 and was found to be shrinking at about $4.4$% per year.（See Exercise 34 on p. 146 of Connally's Functions Modeling Change.）

• Let $G(t)$ denote the area of Grinnell Glacier in acres in year $t$, where $t$ is the number of years since 2007. Find a formula for $G(t)$ and define the function in Desmos. - How many acres of ice were in the glacier in 1997? In 2012? What does the model predict for 2022? - How many total acres of ice were lost from 2007 to 2012? - What was the average rate of change of $G$ from 2007 to 2012? Write a sentence to explain the meaning of this number and include units on your answer. In addition, how does this compare to the average rate of change of $G$ from 2012 to 2017? - How would you you describe the overall behavior of $G$, and thus what is happening to the Grinnell Glacier?

Consider the exponential function $f$ whose graph is given by Figure. Note that $f$ passes through the two noted points exactly.

• Determine the values of $a$ and $b$ exactly. - Determine the average rate of change of $f$ on the intervals $[2,7]$ and $[7,12]$. Which average rate is greater? - Find the equation of the linear function $L$ that passes through the points $(2,20)$ and $(7,5)$. - Which average rate of change is greater? The average rate of change of $f$ on $[0,2]$ or the average rate of change of $L$ on $[0,2]$?

A cup of hot coffee is brought outside on a cold winter morning in Winnipeg, Manitoba, where the surrounding temperature is $0$ degrees Fahrenheit. A temperature probe records the coffee's temperature (in degrees Fahrenheit) every minute and generates the data shown in Table.

$t$ | $0$ | $2$ | $4$ | $6$ | $8$ | $10$

$F(t)$ | $175$ | $129.64$ | $96.04$ | $71.15$ | $52.71$ | $39.05$

• Assume that the data in the table represents the overall trend of the behavior of $F$. Is $F$ linear, exponential, or neither? Why? - Is it possible to determine an exact formula for $F$? If yes, do so and justify your formula; if not, explain why not. - What is the average rate of change of $F$ on $[4,6]$? Write a sentence that explains the practical meaning of this value in the context of the overall exercise. - How do you think the data would appear if instead of being in a regular coffee cup, the coffee was contained in an insulated mug?

The amount (in milligrams) of a drug in a person's body following one dose is given by an exponential decay function. Let $A(t)$ denote the amount of drug in the body at time $t$ in hours after the dose was taken. In addition, suppose you know that $A(3) = 22.7$ and $A(6) = 15.2$.

• Find a formula for $A$ in the form $A(t) = ab^t$, where you determine the values of $a$ and $b$ exactly. - What is the size of the initial dose the person was given? - How much of the drug remains in the person's body $8$ hours after the dose was taken? - Estimate how long it will take until there is less than $1$ mg of the drug remaining in the body. - Compute the average rate of change of $A$ on the intervals $[3,5]$, $[5,7]$, and $[7,9]$. Write at least one careful sentence to explain the meaning of the values you found, including appropriate units. Then write at least one additional sentence to explain any overall trend(s) you observe in the average rate of change. - Plot $A(t)$ on an appropriate interval and label important points and features of the graph to highlight graphical interpretations of your answers in (b), (c), (d), and (e).