Recovering a Function from its Derivative: A Graphical Approach

Published on
07/15/2005

Activity Overview

Students use the calculator to visualize a function using the equation describing its derivative and a single point on the function. They start with simple equations and later solve more complex and difficult differential equations.

Before the Activity

Set the calculator to the Differential Equation Graphing mode

See the attached PDF file for detailed instructions for this activity

Print pages 1 - 15 from the attached PDF file for your class

During the Activity

Distribute the pages to the class.
Follow the Activity procedures:
Visual Solution:

Enter the differential equation on the calculator

Set the initial conditions and the graphing axes

Graph the solution to the differential equation

Compare the analytic solution with the graphed solution

Changing Initial Values:

Change the initial values and graph the solution

Observe that the solution corresponding to each set of initial values moves the starting point of the graph

Entering Several Initial Values:

Enter several sets of initial values and graph the data

Observe a shift in the graph

Slope Fields:

Use the slope field function to get a general shape of the family of solutions

A line segment from any initial point that is tangent to the graph is one of the solutions

Normal Probability Density Functions:

Enter the differential equation with initial values of y = -1, and x = -3 and the Min and Max values set at -3 and +3

Set the calculator to connect points every tenth and graph the solution

Observe the graph is a scaled version of the normal probability density function

Interactive Specification of Initial Values:

Graph the slope field associated with the differential equation

Graph solutions corresponding to specific initial values

Graphing 2 Equations Simultaneously:

Enter two differential equations and different initial values

Select separate style for each graph

Graph the solutions together

After the Activity

Students solve problems on the exercise page.
Review student results:

As a class, discuss questions that appeared to be more challenging