Construction of Derivative Functions and Their Graphs

Objectives

12.4.2.1.c :

𝑓(𝑥) = 𝑐, 𝑓(𝑥) = 𝑥 𝑛 (𝑛 = 1, 2, 3, 4) fonksiyonlarının türevleri, türev tanımı kullanılarak hesaplatılır.

·         Students will be able to define the first and second derivatives of a given cubic function.

·         Students will be able to interpret the effect of changing the coefficients of the function on derivatives and on their graphs.

Pedagogical Explanation

   Function is one of the most important topic in Math. Sometimes students have a difficulty with it so we construct a tool for students to be able show them function and its' graph.

            The purpose of our activity is that to be able to show how coefficients affect the functions and its' first and second derivatives and also their graphs. Therefore, we used the Geometer's Sketchpad  to construct these functions and graphs. In our activity, teacher want students to find the derivatives of our function with hide of the answers and then, S/he can show the answer  and they can compare answers so basically our activity help students to do practice of derivatives. Also, in our activity, students can see the graph of function and derivatives so again, teacher can hide the answers and want students to estimate that  how the graph of functions and derivatives should be.

            Moreover, the main concept of our activity is that students can see the effects of coefficients on the function and its' derivatives and also, their graph so we construct sliders to change the coefficients. By using these sliders, students can see the change of both graph and functions and make an inference about them.

User Manual

This tool can be used basically to show relationships between the coefficients of a function and its first and second derivatives and also how the graph will be affected when the coefficients were changed. For example, with hide&show buttons you can provide students to calculate the first and second derivatives and to estimate how the graphs look like. For the calculations and estimations you can create a task like in below:

1) There is a cubic function such that; f(x) = ex³ + fx² + gx + h.

a)      Find the first and second derivative of this function.

b)      When the value of “e” increase, how will be the derivatives affected?

c)      When the value of “f” increase, how will be the derivatives affected?

d)     When the value of “g” increase, how will be the derivatives affected?

e)      When the value of “h” increase, how will be the derivatives affected?

2) Look at the first derivative and try to decide the degree of this equation. According to your decision, think how the graph of this equation will be.(hint: remember the parabolas)

3) Look at the second derivative and try to decide the degree of this equation. According to your decision, think how the graph of this equation will be.(hint: remember the linear equations)

3) Look at the graphs of y, y’ and y’’ at The Geometer Sketchpad. If there is any change, write + and if there is no change write -.

	y	y’	y’’
e
f
g
h

   Therefore, when teachers teach the functions and their graphs in their class, they can use our activity and create a task like that so for students, it can easy to understand the relationship among functions, their derivatives and graphs and also effects of coefficients on them. If you want to learn how to construct that tool, you can watch our videos from YouTube :)                                                                                                                                                  

  Link for video:    

https://www.youtube.com/watch?v=XncLCQkPlhk&feature=youtu.be

Link for Project:

https://drive.google.com/open?id=1GAhtHACJPL9GtMLhH9hhx73m5aw84RFo