Education Technology

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
  • Re-teach concepts as necessary