# Active Prelude to Calculus

## Section3.1Exponential Growth and Decay

Linear functions have constant average rate of change and model many important phenomena. In other settings, it is natural for a quantity to change at a rate that is proportional to the amount of the quantity present. For instance, whether you put $$$100$$ or$$$100000$$ or any other amount in a mutual fund, the investment’s value changes at a rate proportional the amount present. We often measure that rate in terms of the annual percentage rate of return.
Suppose that a certain mutual fund has a $$10$$% annual return. If we invest $$$100\text{,}$$ after $$1$$ year we still have the original$$$100\text{,}$$ plus we gain $$10$$% of $$$100\text{,}$$ so \begin{equation*} 100 \xrightarrow{\text{year } 1} 100 + 0.1(100) = 1.1(100)\text{.} \end{equation*} If we instead invested$$$100000\text{,}$$ after $$1$$ year we again have the original $$$100000\text{,}$$ but now we gain $$10$$% of$$$100000\text{,}$$ and thus
\begin{equation*} 100000 \xrightarrow{\text{year } 1} 100000 + 0.1(100000) = 1.1(100000)\text{.} \end{equation*}
We therefore see that regardless of the amount of money originally invested, say $$P\text{,}$$ the amount of money we have after $$1$$ year is $$1.1P\text{.}$$
If we repeat our computations for the second year, we observe that
\begin{equation*} 1.1(100) \xrightarrow{\text{year } 2} 1.1(100) + 0.1(1.1(100)) = 1.1(1.1(100)) = 1.1^2 (100)\text{.} \end{equation*}
The ideas are identical with the larger dollar value, so
\begin{equation*} 1.1(100000) \xrightarrow{\text{year } 2} 1.1(100000) + 0.1(1.1(100000)) = 1.1(1.1(100000)) = 1.1^2 (100000)\text{,} \end{equation*}
and we see that if we invest $$P$$ dollars, in $$2$$ years our investment will grow to $$1.1^2 P\text{.}$$
Of course, in $$3$$ years at $$10$$%, the original investment $$P$$ will have grown to $$1.1^3 P\text{.}$$ Here we see a new kind of pattern developing: annual growth of $$10$$% is leading to powers of the base $$1.1\text{,}$$ where the power to which we raise $$1.1$$ corresponds to the number of years the investment has grown. We often call this phenomenon exponential growth.

### Preview Activity3.1.1.

Suppose that at age $$20$$ you have $$$20000$$ and you can choose between one of two ways to use the money: you can invest it in a mutual fund that will, on average, earn $$8$$% interest annually, or you can purchase a new automobile that will, on average, depreciate $$12$$% annually. Let’s explore how the$$$20000$$ changes over time.
3. Suppose instead that the car’s value is modeled by a linear function $$L$$ and satisfies the values stated at the outset of this activity. Find a formula for $$L(t)$$ and determine both the purchase value of the car and when the car will be worth $$$1000\text{.}$$ 4. Which model do you think is more realistic? Why? ### Subsection3.1.4Summary • We say that a function is exponential whenever its algebraic form is $$f(t) = ab^t$$ for some positive constants $$a$$ and $$b$$ where $$b \ne 1\text{.}$$ (Technically, the formal definition of an exponential function is one of form $$f(t) = b^t\text{,}$$ but in our everyday usage of the term “exponential” we include vertical stretches of these functions and thus allow $$a$$ to be any positive constant, not just $$a = 1\text{.}$$) • To determine the formula for an exponential function of form $$f(t) = ab^t\text{,}$$ we need to know two pieces of information. Typically this information is presented in one of two ways. • If we know the amount, $$a\text{,}$$ of a quantity at time $$t = 0$$ and the rate, $$r\text{,}$$ at which the quantity grows or decays per unit time, then it follows $$f(t) = a(1+r)^t\text{.}$$ In this setting, $$r$$ is often given as a percentage that we convert to a decimal (e.g., if the quantity grows at a rate of $$7$$% per year, we set $$r = 0.07\text{,}$$ so $$b = 1.07$$). • If we know any two points on the exponential function’s graph, then we can set up a system of two equations in two unknowns and solve for both $$a$$ and $$b$$ exactly. In this situation, it is useful to consider the quotient of the two known outputs, as demonstrated in Example 3.1.6. • Exponential functions of the form $$f(t) = ab^t$$ (where $$a$$ and $$b$$ are both positive and $$b \ne 1$$) exhibit the following important characteristics: • The domain of any exponential function is the set of all real numbers and the range of any exponential function is the set of all positive real numbers. • The $$y$$-intercept of the exponential function $$f(t) = ab^t$$ is $$(0,a)$$ and the function has no $$x$$-intercepts. • If $$b \gt 1\text{,}$$ then the exponential function is always increasing and always increases at an increasing rate. If $$0 \lt b \lt 1\text{,}$$ then the exponential function is always decreasing and always decreases at an increasing rate. ### Exercises3.1.5Exercises #### 1. Suppose $$Q = 30.8(0.751)^t\text{.}$$ Give the starting value $$a\text{,}$$ the growth factor $$b\text{,}$$ and the growth rate $$r$$ if $$Q = a \cdot b^t = a(1+r)^t\text{.}$$ $$a =$$ $$b =$$ $$r =$$ % #### 2. Find a formula for $$P = f(t)\text{,}$$ the size of the population that begins in year $$t = 0$$ with $$2090$$ members and decreases at a $$3.7$$ % annual rate. Assume that time is measured in years. $$P = f(t) =$$ #### 3. (a) The annual inflation rate is $$3.8$$% per year. If a movie ticket costs$9.00 today, find a formula for $$p\text{,}$$ the price of a movie ticket $$t$$ years from today, assuming that movie tickets keep up with inflation.
$$P = f(t) =$$
(b) According to your formula, how much will a movie ticket cost in $$30$$ years?

#### 4.

In the year 2003, a total of 7.2 million passengers took a cruise vacation. The global cruise industry has been growing at 9% per year for the last decade. Assume that this growth rate continues.
(a) Write a formula for to approximate the number, $$N\text{,}$$ of cruise passengers (in millions) $$t$$ years after 2003.
$$N =$$
(b) How many cruise passengers (in millions) are predicted in the year 2011?
$$N =$$
(c) How many cruise passengers (in millions) were there in the year 2000?
$$N =$$

#### 5.

The populations, $$P\text{,}$$ of six towns with time $$t$$ in years are given by
 1 $$\displaystyle P = 800(0.78)^t$$ 2 $$\displaystyle P = 900(1.06)^t$$ 3 $$\displaystyle P = 1600(0.96)^t$$ 4 $$\displaystyle P = 1400(1.187)^t$$ 5 $$\displaystyle P = 500(1.14)^t$$ 6 $$\displaystyle P = 2800(0.8)^t$$
Answer the following questions regarding the populations of the six towns above. Whenever you need to enter several towns in one answer, enter your answer as a comma separated list of numbers. For example if town 1, town 2, town 3, and town 4, are all growing you could enter 1, 2, 3, 4; or 2, 4, 1, 3; or any other order of these four numerals separated by commas.
(a) Which of the towns are growing?
(b) Which of the towns are shrinking?
(c) Which town is growing the fastest?
What is the annual percentage growth RATE of that town? %
(d) Which town is shrinking the fastest?
What is the annual percentage decay RATE of that town? %
(e) Which town has the largest initial population?
(f) Which town has the smallest initial population?

#### 6.

(a) Determine whether function whose values are given in the table below could be linear or exponential.
• linear
• exponential
 $$x =$$ 0 1 2 3 4 $$h(x) =$$ 14 8 2 -4 -10
Find a possible formula for this function.
$$h(x) =$$
(b) Determine whether function whose values are given in the table below could be linear or exponential.
• linear
• exponential
 $$x =$$ 0 1 2 3 4 $$i(x) =$$ 14 12.6 11.34 10.206 9.1854
Find a possible formula for this function.
$$i(x) =$$

#### 7.

A population has size 8000 at time $$t = 0\text{,}$$ with $$t$$ in years.
(a) If the population decreases by 125 people per year, find a formula for the population, $$P\text{,}$$ at time $$t\text{.}$$
$$P(t) =$$
(b) If the population decreases by 6% per year, find a formula for the population, $$P\text{,}$$ at time $$t\text{.}$$
$$P(t) =$$

#### 8.

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. 2
1. Let $$G(t)$$ denote the area of Grinnell Glacier in acres in year $$t\text{,}$$ where $$t$$ is the number of years since 2007. Find a formula for $$G(t)$$ and define the function in Desmos.
2. How many acres of ice were in the glacier in 1997? In 2012? What does the model predict for 2022?
3. How many total acres of ice were lost from 2007 to 2012?
4. 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?
5. How would you you describe the overall behavior of $$G\text{,}$$ and thus what is happening to the Grinnell Glacier?

#### 9.

Consider the exponential function $$f$$ whose graph is given by Figure 3.1.12. Note that $$f$$ passes through the two noted points exactly.
1. Determine the values of $$a$$ and $$b$$ exactly.
2. Determine the average rate of change of $$f$$ on the intervals $$[2,7]$$ and $$[7,12]\text{.}$$ Which average rate is greater?
3. Find the equation of the linear function $$L$$ that passes through the points $$(2,20)$$ and $$(7,5)\text{.}$$
4. 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 $$[0,2]\text{?}$$

#### 10.

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 3.1.13.
1. Assume that the data in the table represents the overall trend of the behavior of $$F\text{.}$$ Is $$F$$ linear, exponential, or neither? Why?
2. Is it possible to determine an exact formula for $$F\text{?}$$ If yes, do so and justify your formula; if not, explain why not.
3. What is the average rate of change of $$F$$ on $$[4,6]\text{?}$$ Write a sentence that explains the practical meaning of this value in the context of the overall exercise.
4. 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?

#### 11.

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 $$A(6) = 15.2\text{.}$$
1. Find a formula for $$A$$ in the form $$A(t) = ab^t\text{,}$$ where you determine the values of $$a$$ and $$b$$ exactly.
2. What is the size of the initial dose the person was given?
3. How much of the drug remains in the person’s body $$8$$ hours after the dose was taken?
4. Estimate how long it will take until there is less than $$1$$ mg of the drug remaining in the body.
5. Compute the average rate of change of $$A$$ on the intervals $$[3,5]\text{,}$$ $$[5,7]\text{,}$$ and $$[7,9]\text{.}$$ 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.
6. 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).
It takes calculus to justify this claim fully and rigorously.
See Exercise 34 on p. 146 of Connally’s Functions Modeling Change.