Unit 4, Lesson 13
Exponential Functions with Base $e$
Integrated Math 3
13.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSF-LE.A.4
For exponential models, express as a logarithm the solution to where , , and are numbers and the base is 2, 10, or ; evaluate the logarithm using technology.
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity Narrative
In this lesson, students are expected to use calculators with a way to represent . 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 2. Calculators that have buttons for and exponents are needed for every student.
Activity
Noneis a mathematical constant whose value is approximately 2.718. When working on problems that involve , we often rely on calculators to estimate values.
- Find how to represent on your calculator. Experiment with it to understand how it works. (For example, see how the value of or can be calculated.)
- Evaluate each expression. Make sure that your calculator gives the indicated value. If it doesn’t, check in with your partner to compare how you entered the expression.
- should give approximately 30.04166
- should give approximately 11,041.73996
- should give approximately 8.47891
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
Activity
Materials
To Gather
Graphing technologyActivity Narrative
This activity exposes students to growth factors written as . An important takeaway for students is understanding that when we assume is a continuous growth rate, that is, the rate is happening at every moment and not just for each unit of time , then is how we can write the growth factor. If, however, we assume that the rate is happening for each unit of time , then using a growth factor of is appropriate.
This work prepares students to work with equations of the form , 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 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.
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
13.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
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 takes on a negative value to represent a percent decrease.
13.4
Activity
Optional
Materials
To Gather
Graphing technologyActivity Narrative
Use this activity to allow students to practice analyzing graphs of exponential functions involving and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.
Invite selected students to share their strategies for finding the output from a graph.
Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base , 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 later in the unit.
Lesson Synthesis
In this lesson, students 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: , and .
Explain to students that while the first form can be used to represent both discrete and continuous functions, the second form (using base ) is used only to represent situations where change happens "continuously," or at every moment. The constant is not used, for example, to describe how many layers of paper there are if we keep cutting a sheet of paper in half times.
Student Lesson Summary
Suppose 24 square feet of a pond is covered with algae, and the area is growing at a rate of 8% each day.
The area, in square feet, can be modeled with a function such as or , where 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 . For the algae growth, this might look like . This model is different because the 8% growth per day 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 and 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 . For the algae model , 0.08 per day is called the continuous growth rate while is the growth factor for 1 day. In general, when we express an exponential function in the form , we are assuming that the growth rate (or decay rate) is being applied continuously and that is the growth (or decay) factor. When is small, is close to .