This is an interactive applet exploring the derivatives of exponential functions of the form f (x) = b^{x} where b is some positive number. This applet visually explores f (x) = b^{x} for various values of b, and shows how f ′(x) is related to f (x).

The applet allows the user to modify the value of b using a slider, and plots the graph of f (x) = b^{x} . It also shows the plot of f'(x) and the tangent line at x=0.

Type of Material:

Simulation

It is an interactive Graphing Calculator created using open-source software GeoGebra.

GeoGebra is a dynamic mathematics software that joins geometry, algebra and calculus.

Recommended Uses:

This material can be used in-class to graphically show the exponential function and to define e.

Assignments can be framed using the GeoGebra Graphing Calculator. For example: f ′(x) = m_{b} f (x). Confirm that when m_{b} = 0.5, then f ′(x) is half as big as f (x), when m_{b} = 1/3, then f ′(x) is a third as big as f (x), and so forth.

Technical Requirements:

Works on any browser.

Identify Major Learning Goals:

The major learning goals for learner/user of the applet are:

Ability to visualize the graph of f(x)=b^{x}, f'(x) and the tangent line at x=0 for various values of b from 0.25 to 4.

Recognize that f ′(x) is proportional to f (x). That is, f ′(x) = m_{b} f (x) for some number m_{b} that depends on b.

Understand the definition of e and visualize that if f (x) = e^{x}, then f ′(x) = e^{x}.

Recognize that every exponential function goes through the point (0,1).

Learn the proof of f ′(x) = m_{b} f (x)

Target Student Population:

Students taking differential calculus.

High School, College Lower Division.

Targeted user groups are: High School Students and Faculty (effective teaching tool)

Prerequisite Knowledge or Skills:

Basic knowledge of Algebra, Calculus and the graph of the exponential function.

Ability to learn and use the GeoGebra Graphing Calculator (Not a pre-requisite for users), however for faculty learning GeoGebra, it might help them to improve their teaching methodology and ability to present a complex topic in a simple way; for example, the definition of e.

Content Quality

Rating:

Strengths:

Thes is a great visualization tool that allows students to experience the relationship between an exponential function, its derivative and the tangent line at x = 0. Included is a clear explanation of the concepts followed by a proof that the derivative of an expontial function is proportional to the exponential function and that the derivative of e^{x }is e^{x}.

Highlights include:

Use of open-source GeoGebra Graphing Calculator

The clarity in understanding the concept of the exponential function. Every exponential function goes through the point (0,1), right? Why is this?

The definition of e

Visualize f(x), f'(x) and tangent at x=0 for various values of b using slider

The proof that the derivative of the exponential function is proportional to the exponential function.

Concerns:

It would be helpful if the user could enter e as the base in addition to using the slider that might get close if the student can remember what the value of e is.

Potential Effectiveness as a Teaching Tool

Rating:

Strengths:

Any student with the implied prerequisite knowledge will be able to learn about the relationship betwee the exponential and its derivative after using this app. It can be used either as an introduction to te concept or as a reinforcement of the exponential properties after the students are introduced to exponents and their derivatives.

Highlights are:

Calculus Applets using GeoGebra is an excellent teaching tool

Enables the student to explore the graph of f(x) and f'(x) w.r.t to b

Visualize the proportionality m_{b}= f'(x)/f(x)

Conceptual understanding of the subject

Definition of e

Concerns:

It would be helpful to include a reference link to material about the basics of exponential functions.

Teachers should have knowledge of GeoGebra Graphing Calculator. (Though not a pre-requisite for users)

Ease of Use for Both Students and Faculty

Rating:

Strengths:

The tool only requires the student to move a slider and click on the well labeled checkboxes. Students will have no difficulties using this app.

Highlights are:

Works on Desktop/notebook/tablet/smartphone chrome/Firefox/Microsoft edge

GeoGebra is an open-source software

Content & Questions in the material are pertinent and improves learning skill

Easy to navigate and use

Concerns:

The following message came up on one of the attempts to open the site:

"The information you're about to submit is not secure"

Because the site is using a connection that’s not completely secure, user information may be visible to others.

Creative Commons:

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