Lesson 4: The exponential function and natural logarithms (Pt. 1)

Learning Goals

At the end of the lesson, we will be able to:

• Identify Euler's number as being a unique value

• Recall that when differentiating e^x, this term remains unchanged.

• Recall and correctly apply differentiation laws when differentiating functions involving the term e^x.

ICT

• Website (Interactive) - Desmos

• Video (YouTube) - Eddie Woo

• Video (YouTube) - The number e is everywhere

LESSON STRUCTURE

1. Introduction (5 minutes) - Calculus refresher

2. Body Activity 1 (20 minutes) - The special case of differentiation: d(e^x)/dx = e^x (AFL)

3. Body Activity 2 (20 minutes) - What is this number "e"? (LIT) (NUM)

4. Conclusion (5 minutes) - Where can we see "e"?

Duration: 5 minutes

Resource:

N/A for this activity

Activity Description

The purpose of this activity is to recall and refresh students' knowledge of the rules of differentiation from the previous Calculus unit to lead them into Body Activity 1.

Teacher Action

• The teacher

  • begins the introduction activity by writing the following three functions on the board for students to differentiate

    • 1. y = 3x^2 + 5

      2. y = x^3*x^7

      3. y = 5x/(3x + 2x^4)

  • allocates a couple of minutes for students to individually differentiate the above three questions

  • asks the class to share their answers with the teacher and write these on the board along with the steps students had taken to reach their answers so that the whole class is on the same page.

    • Answers

      1. dy/dx = 6x

      2. dy/dx = (17/2)*x^7*sqrt(x)

      3. dy/dx = -(30x^2)/(2x^3+3)^2

Student Action

• Students

  • work to solve the 3 questions posed by the teacher, showing their working out, in their books.

  • share their working out with the teacher

Duration: 20 minutes

Resource:

N/A for this activity

Activity Description

The purpose of this activity is to introduce the formula for differentiation using e, d(e^x)/dx = e^x. This activity implements a predict, observe, practice style of learning as students use their prior knowledge on how differentiation works (refreshed from the Introduction activity) to predict what will happen when differentiating e^x, observe the actual result "d(e^x)/dx = e^x" as explained by the teacher and lastly, practice this new concept with a couple of questions to solidify this. The questions are the end of this activity are scaffolded to increase in difficulty as the questions go on. This activity is also a pivotal learning point for students as they learn further calculus rules and expand their knowledge of exponentials. Therefore, as the teacher receives answers from the students about the questions below, they should take note to ensure students understand these concepts and whether or not further explanation is needed after the activity or at the start of the next lesson.

Teacher Action

• The teacher

  • asks students to keep in mind how they differentiated the function from the introduction activity as they write "y=e^x" on the board

  • asks students where they have seen "e" of this type of function before.

    • Possible student responses

      1. "In previous lessons of this unit involving log functions where "e" was one of the base values used."

      2. "In a previous lesson of the unit where we saw y=a^x was somehow related to log functions."

  • gives students a couple of minutes to think with the person next to them and write down how they would differentiate this function

  • asks for some responses from the class

    • Responses are expected to be incorrect at this stage. Here the teacher is showing the class what not to do to highlight potential misconceptions in front of the class so that they do not make these later on.

    • Possible responses from the class could include:

      1. d(e^x)/dx = x*e^(x-1)

         • Misconception: Students are simply using the chain rule here and are treating 'x' as a value and 'e' as the function.

      2. d(e^x)/dx = x*e

         • Misconception: Students are also just using the chain rule to try and differentiate the function however are also differentiating the exponent 'x' as d(x)/dx = 1.

      3. d(e^x)/dx = e

         • Misconception: Students are simply differentiating the exponent 'x' as d(x)/dx = 1.

  • introduces the formal rule: d(e^x)/dx= e^x

  • draws attention to the function of form f(x) = k*e^(a*x), where k and a are constants. This is what the basic structure of a simple exponential function appears as.

  • writes up three example questions for students to solve and check answers from the students at the end of this activity

Student Action

• Students

  1. predict - think with the person next to them about how they may use what they know to suggest possible methods/answers to how to differentiate e^x

  2. observe - take note of the new differential law d(e^x)/dx = e^x

  3. practice - solve the questions on the board by practising this new law

Differentiated Version

• Easier version

• If students are struggling, students may only be able to complete questions 1-4.

• The teacher can give them the following hints to help students progress:

  1. Hint: Try taking out the pro-numeral to make differentiation easier

  2. Hint: We know that (3^1)* (3^2) = 3^(1+2) = 3^3, how can we use this to simplify questions such as question 4?

     • This hint is trying to get students to identify concepts they are more familiar with to what is seemingly much different with e^x until this connection is made.

• Harder version

• If students have time, they can extend their learning by completing questions 9 and 10 above which involve the use of other functions in addition to functions involving e^x

Duration: 20 minutes

Resource:

1. Desmos activity - https://teacher.desmos.com/activitybuilder/custom/5d66db7521bdbe6585439de4?collections=5cba1ada3d8baa0c055fd5ba

2. Teacher's Guide (see Worksheet 4.1)

3. Eddie Woo explanation - https://www.youtube.com/watch?v=pg827uDPFqA&ab_channel=EddieWoo

Activity Description

The purpose of this activity is for students to discover the value of e through an investigation using exponential functions and the gradient function. It is an investigative task that enables students to work out why e is a unique number where e ≈ 2.71828182845. This activity involves the use of Desmos to step through the different stages of thinking for this investigation so that students arrive at the same conclusion about e. This program also allows the teacher to see where students are up to and their responses throughout the activity. There is also a Teacher's Guide below (see Worksheet 4.1) that dissects the activity and what the teacher is looking for in student responses. The activity is to be done in pairs. It also exercises the student's graphical literacy and numeracy skills as they work to understand the numerical connection between different exponential graphs to eventually estimate the value of "e".

Teacher Action

• Before the activity, read through the Teacher's Guide (see Worksheet 4.1) to

• Start the activity with a little session of teacher dialogue

  • "So in the previous activity, you were working with the fact that d(e^x)/dx = e^x and completed some differentiation of functions using this rule. However, does anyone have any idea why the derivative of this function involving e^x is itself? What could the value of e be so that this is true?"

• The teacher then

  • introduces the activity as an investigation where students are going to explore these questions to find what e could be.

  • assigns students to work with the person next to them for this activity

• During the activity the teacher

  • monitors the Desmos activity from the teacher's perspective, looking at responses and student progress to gauge whether students are moving along smoothly with the task or need help

  • walks around the classroom to check in with what students are thinking about as they work to determine e

  • refers to the Teacher's Guide (see Worksheet 4.1) for how students should be responding to the Desmos activity

• After the activity, the teacher should confirm the value of "e" with students and state that the best approximation of "e" to 11 d.p. is e ≈ 2.71828182845

Student Action

• In pairs, students

  • work together to explore the investigative Desmos task

  • write down this final unique approximation of e in their notes

Differentiated version

• Easier version

  • Students can watch Eddie Woo's explanation of the value of "e" (Resource 3) as another explanation of how "e" becomes the approximated value of 2.71828182845. He explains it very logically and foreshadows how it can be written as a sum function.

  • Students should then return to the Desmos activity or try to compare the graphs of 2.71828182845^x and its derivative to see that it is the same.

• Harder version

  • Pages 10 and 11 on the Desmos activity can be worked through by students who wish to explore alternative ways to evaluate "e". This is just for students to see another representation of e.

Worksheet 4.1 - Teacher's Guide

Duration: 5 minutes

Resource

Video - https://www.youtube.com/watch?v=b-MZumdfbt8&ab_channel=RandellHeyman

Activity Description

The purpose of this activity is to ground the concept of the value of "e" in other real-world and abstract examples through a short video.

Teacher Action

• The teacher

  • places up the video (see Resource) for the whole class to watch together

  • asks students to write down three examples where "e" is used in the real world to explain everyday situations while the video is playing

  • ensures students can access the video in case they wish to watch it later and pause and play it at their convenience to explore these real-world examples further

Student Action

• Students

  • actively listen to and watch the video

  • note down three examples of "e" in the real world