Lesson 13

Exponential Functions with Base $e$

13.1: $e$ on a Calculator (5 minutes)

Warm-up

In this lesson, students are expected to use calculators with an \(e\) button. This warm-up is meant to smooth out any technical difficulties with using their calculator to perform desired calculations.

Students may not have had much experience with entering an expression (rather than a single number) for the exponent and may not realize that parentheses may be needed to group the parts of the expression together. Expect calculation errors that stem from this issue.

Depending on the number of digits their calculator allows and the settings for displaying decimals, students may also notice that their calculation results have a different number of decimal places than those on the task statements.

Launch

Arrange students in groups of 1. Provide access to calculators that have buttons for \(e\) and exponent.

Student Facing

The other day, you learned that \(e\) is a mathematical constant whose value is approximately 2.718. When working on problems that involve \(e\), we often rely on calculators to estimate values.

  1. Find the \(e\) button on your calculator. Experiment with it to understand how it works. (For example, see how the value of \(2e\) or \(e^2\) can be calculated.)
  2. Evaluate each expression. Make sure your calculator gives the indicated value. If it doesn’t, check in with your partner to compare how you entered the expression.
    1. \(10\boldcdot e^{(1.1)}\) should give approximately 30.04166
    2. \(5 \boldcdot e^{(1.1)(7)}\) should give approximately 11,041.73996
    3. \(e^{\frac{9}{23}}+7\) should give approximately 8.47891

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share some issues they came across when evaluating the expressions with a calculator and how to resolve them. Make sure that students recognize that grouping symbols may be needed when entering an expression as an exponent, and that calculation results may vary because of the level of precision of the calculator.

13.2: Same Situation, Different Equations (20 minutes)

Activity

This activity exposes students to growth factors written as \(e^r\). An important takeaway for students is understanding that when we assume \(r\) is a continuous growth rate, that is, the rate \(r\) is happening at every moment and not just each unit of time \(t\), then \(e^r\) is how we can write the growth factor. If, however, we assume that the rate \(r\) is happening for each unit of time \(t\), then using a growth factor of \(1+r\) is appropriate.

This work prepares students to work with equations of the form \(f(t)= P \boldcdot e^{rt}\), which they will encounter in future activities throughout the rest of the unit. In the next activity, students will focus on interpreting the parameters of exponential equations written in this form. At this stage, students are not expected to understand \(e\) in depth. That work is reserved for future, more-advanced courses. The goal here is to connect this new form of expressing an exponential function to students’ prior work by centering student thinking around the idea of how a growth rate is being applied.

Launch

Read the introduction to the activity and display the equations for \(P\) and \(Q\). Ask students to think of at least one thing they notice and at least one thing they wonder about the functions. Give students 1 minute of quiet think time, and then invite students to share, recording and displaying their responses for all to see. Tell students that during this activity they will investigate pairs of functions that model the exponential growth of the same population, but make different assumptions about how the population is growing. 

Arrange students in groups of 2–4 and provide access to graphing technology and a spreadsheet tool, in case requested. Ask groups to complete the first question and then pause for a brief whole-class discussion.

Invite students to share their tables and observations. Highlight the observations that the predictions produced by the two models are fairly close, but not identical, and that the difference seems to increase for larger values of \(t\).

Next, tell students that they will now use graphs to compare the predictions of the two models for colonies that are growing at slower and faster rates. Suggest that they identify the two graphs for each colony with labels and different colors (if possible and simple to do). Consider asking students to split up the graphing work to optimize time.

Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. Begin by providing frames that encourage comparison of \(P(t)\) and \(Q(t)\), such as “How are _____ and _____ different?” and “What do _____ and _____ have in common?” Once students have agreed on some key comparisons, then display sentence frames that push their language toward more clearly articulating function behavior and understanding that \(P(t)\) assumes the growth rate of 0.02 is applied for each interval of time \(t\), while \(Q(t)\) assumes 0.02 is a continuous growth rate applied at every moment. Provide sentence frames such as, “What does this part of _____ mean?” and “Another way to look at it is. . .” to help students articulate the meaning of the functions beyond surface comparisons.
Supports accessibility for: Language; Organization

Student Facing

The population of a colony of insects is 9 thousand when it was first being studied. Here are two functions that could be used to model the growth of the colony \(t\) months after the study began.

\( P(t) = 9 \boldcdot (1.02)^t\)

\(Q(t) = 9 \boldcdot e^{(0.02t)}\)

  1. Use technology to find the population of the colony at different times after the beginning of the study and complete the table.
    \(t\) (time in months) \(P(t)\) (population in thousands) \(Q(t)\) (population in thousands)
    6    
    12    
    24    
    48    
    100    
  2. What do you notice about the populations in the two models?
  3. Here are pairs of equations representing the populations, in thousands, of four other insect colonies in a research lab. The initial population of each colony is 10 thousand and they are growing exponentially. \(t\) is time, in months, since the study began.

    \(\text{Colony 1}\\ f(t) = 10 \boldcdot (1.05)^t\\ g(t) = 10 \boldcdot e^{(0.05t)}\)

    \(\text{Colony 2}\\k(t) = 10 \boldcdot (1.03)^t\\ l(t) = 10 \boldcdot e^{(0.03t)}\)

    \(\text{Colony 3}\\p(t) = 10 \boldcdot (1.01)^t\\ q(t) = 10 \boldcdot e^{(0.01t)}\)

    \(\text{Colony 4}\\v(t) = 10 \boldcdot (1.005)^t\\ w(t) = 10 \boldcdot e^{(0.005t)}\)

    1. Graph each pair of functions on the same coordinate plane. Adjust the graphing window to the following boundaries to start: \(0 < x < 50\) and \(0 < y < 80\).
    2. What do you notice about the graph of the equation written using \(e\) and the counterpart written without \(e\)? Make a couple of observations.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Select groups of students to share their observations and graphs (or consider displaying the graphs in the Student Response for all to see). Here are some questions for discussion:

  • “Which pair of graphs seemed most alike?” (Colony 4's graphs were almost the same until you got to large values of \(t\). At the end of 2 year2, \(v(24)\) and \(w(24)\) were both about 11.3 thousand.)
  • “Which pair of graphs seemed least alike?” (Colony 1's graphs looked different after just a few months. After 2 years, \(f(24) \approx32.3\) while \(g(24)\approx33.2\).)
  • “For Colony 2, what is the growth factor for 1 month predicted by each model?”(\(k\) predicts a growth factor of 1.03 while \(l\) predicts a growth factor of \(e^{0.03}\), which is a bit more than 1.03.)

Tell students that exponential functions that involve small but ongoing growth (such as population growth or inflation) can be modeled in different ways. A factor written in the form of \(e^{0.03}\) highlights that the \(r\) percent growth rate (for example, 3%) is applied continuously. The \((1+r)^t\) model predicts a change by a factor of \(1 + r\) for each unit of time. While these sound similar, they are not equivalent.

Students will learn more about \(e\) in future courses. For now, it is sufficient simply to know that scientists often find it helpful to use \(e^r\) to model exponential growth and decay in cases where the growth or decay is assumed to happen continuously.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. To help students share their observations about the graphs of the equations written using \(e\) and their counterparts written without \(e\), provide sentence frames such as: “I notice that . . .”, “_____ and _____ are alike because. . .”, “_____ and _____ are different because. . . .” 
Design Principle(s): Optimize output (for comparison); Support sense-making

13.3: $e$ in Exponential Models (10 minutes)

Activity

This activity further familiarizes students with the way continuously compounded models are typically written. It makes explicit the format of an equation used to represent such a model, which students will encounter more of in future lessons. Students also interpret the parameters of such equations in context and use the given structure to complete partially built equations. In this course, though, students are not assessed on generating a model to represent a situation characterized by continuous growth.

Only examples of growth situations are included in the activity. Decay situations are intentionally excluded so students don’t have to make sense of a negative factor in the exponent of an expression while also deciphering a new form for writing exponential functions. In synthesizing the activity, however, the teacher could choose to mention that in decay situations the \(r\) takes on a negative value to represent a percent decrease.

Launch

Arrange students in groups of 2. Give students a few minutes of quiet work time, and then ask them to discuss their responses with their partner. Follow with a whole-class discussion.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, invite students to use the same colors in the diagram and color code corresponding elements of the questions in the activity.
Supports accessibility for: Visual-spatial processing

Student Facing

Exponential models that use \(e\) often use the format shown in this example:

A function, annotated.

Here are some situations in which a percent change is considered to be happening continuously. For each function, identify the missing information and the missing growth rate (expressed as a percentage).

  1. At time \(t=0\), measured in hours, a scientist puts 50 bacteria into a gel on a dish. The bacteria are growing and the population is expected to show exponential growth.
    • function: \(b(t) = 50 \boldcdot e^{(0.25t)}\)
    • continuous growth rate per hour:
  2. In 1964, the population of the United States was growing at a rate of 1.4% annually. That year, the population was approximately 192 million. The model predicts the population, in millions, \(t\) years after 1964.
    • function: \(p(t) = \underline{\hspace{0.5in}} \boldcdot e^{\underline{\hspace{0.25in}}t}\)
    • continuous growth rate per year: 1.4%
  3. In 1955, the world population was about 2.5 billion and growing. The model predicts the population, in billions, \(t\) years after 1955.
    • function: ​​\(q(t) = \underline{\hspace{0.5in}} \boldcdot e^{(0.0168t)}\)
    • continuous growth rate per year:

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may struggle relating the given exponential expression to the given situation. Draw their attention to what each part of the expression means. For example, 13 is the value of the expression when \(t = 0\). For the second question, the value when \(t = 0\) is the United States population, in millions, in 1964. What is this? (192). The coefficient of \(t\) is the continuous growth rate. What is the continuous growth rate predicted by the model? (It's 1.4% which corresponds to 0.014.)

Activity Synthesis

Select students to share how they completed and interpreted the missing information, where possible drawing attention to the structure of the exponential model given. Invite students to discuss how this form is like and unlike the equations students have seen prior to this point (\(f(x) = a \boldcdot b^x\)). In particular, highlight that the growth factor, \(e^r\), is expressed in terms of both \(e\) and \(r\). This is different than growth at a rate of \(r\) per unit time which is expressed only in terms of \(r\) (as \(1+r\)).

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “The growth rate is 25%” can be restated as a question such as, “Is the continuous growth rate per hour 25%?” This will help students listen and respond to each other as they explain how they use the structure of the exponential model to fill in the missing information.
Design Principle(s): Support sense-making

13.4: Graphing Exponential Functions with Base $e$ (15 minutes)

Optional activity

This activity is optional. Use this activity to allow students to practice analyzing graphs of exponential functions involving \(e\) and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.

Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base \(e\), the analytical work is fundamentally the same as that when the base is another number. Students will also have opportunities to graph exponential functions with base \(e\) later in the unit.

Launch

Provide access to devices that can run Desmos or other graphing technology.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses to the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How did you get . . . ?”, “Why did you . . . ?”, and “How do you know . . . ?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for the population of bacteria 10 hours after the scientist put the initial bacteria in the dish.
Design Principle(s): Optimize output (for justification); Cultivate conversation

Student Facing

  1. Use graphing technology to graph the function defined by \(f(t) = 50 \boldcdot e^{(0.25t)}\). Adjust the graphing window as needed to answer these questions:
    1. The function \(f\) models the population of bacteria in \(t\) hours after it was initially measured. About how many bacteria were in the dish 10 hours after the scientist put the initial 50 bacteria in the dish?
    2. About how many hours did it take for the number of bacteria in the dish to double? Explain or show your reasoning.
  2. Use graphing technology to graph the function defined by \(p(t)=192 \boldcdot e^{(0.014t)}\). Adjust the graphing window as needed to answer these questions:
    1. The equation models the population, in millions, in the U.S. \(t\) years after 1964. What does the model predict for the population of the U.S. in 1974?
    2. In which year does the model predict the population will reach 300 million?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Research what the population of the U.S. was in the year the model predicted 300 million people. How far off was the model? What factors do you think account for the actual population in that year being different from the prediction of the model? In what year did the U.S. actually reach 300 million people?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Focus the discussion on how students set the graphing window and how they used the graphs to answer the questions.

Depending on the graphing technology used, students may use the following strategies:

  • Visually estimate the value of the output given a certain input, or the input given an output.
  • Use a tracing tool to trace the graph and identify the coordinates when the input or output has a certain value.
  • To find an unknown input: graph a horizontal line at a given \(y\)-value (for example, \(y=100\), a doubled bacteria population), find the intersection of that line and the graph of the exponential function, and identify the \(x\)-value at that intersection.
  • To find an unknown output: use a table or type something like \(b(10)\) after \(b\) was defined.

Some students may also notice that they could calculate the output value given an exponential equation and an input value, but had to rely on the graph to find an unknown input value because they were unsure how to solve for \(x\) when the equation has \(e\) as a base. Tell students that in a future lesson they will learn how to use logarithms to solve this type of equation algebraically similar to how they have used logarithms previously.

Lesson Synthesis

Lesson Synthesis

In this lesson, we have seen that an exponential growth situation can be represented with models of different forms. Highlight the similarities and differences in the structure of the two forms students have seen so far. Consider using the equations representing Colony 4 from the first activity:  \(v(t) = 100 \boldcdot (1.005)^t\) and \( w(t) = 100 \boldcdot e^{(0.005t)}\).

Explain to students that while the first form can be used to represent both discrete and exponential functions, the second form (using base \(e\)) is used only to represent situations where change happens "continuously" or at every moment. The constant \(e\) is not used, for example, to describe how many layers of paper there are if we keep folding it in half \(n\) times.

13.5: Cool-down - Two Population Predictions (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Suppose 24 square feet of a pond is covered with algae and the area is growing at a rate of 8% each day.

We learned earlier that the area, in square feet, can be modeled with a function such as \(a(d) = 24 \boldcdot (1+0.08)^d\) or \(a(d) = 24 \boldcdot (1.08)^d\), where \(d\) is the number of days since the area was 24 square feet. This model assumes that the growth rate of 0.08 happens once each day.

In this lesson, we looked at a different type of exponential function, using the base \(e\). For the algae growth, this might look like \(A(d)=24 \boldcdot e^{(0.08d)}\). This model is different because the 8% growth is not just applied at the end of each day: it is successively divided up and applied at every moment. Because the growth is applied at every moment or "continuously," the functions \(a\) and \(A\) are not the same, but the smaller the growth rate the closer they are to each other. 

Many functions that express real-life exponential growth or decay are expressed in the form that uses \(e\). For the algae model \(A\), 0.08 is called the continuous growth rate while \(e^{0.08}\) is the growth factor for 1 day. In general, when we express an exponential function in the form \(P \boldcdot e^{rt}\), we are assuming the growth rate (or decay rate) \(r\) is being applied continuously and \(e^r\) is the growth (or decay) factor. When \(r\) is small, \(e^{rt}\) is close to \((1+r)^t\).