# Differential calculus: linear and non-linear graphs

TLF ID L7826

Observe the linear and non-linear distance–time graphs of a rocket travelling at both constant and changing velocities. Calculate the average and instantaneous velocities of the rocket over different time intervals. Notice what happens to the average and instantaneous velocities as the time intervals become smaller. Work out the relationship between these velocities and the gradients of secants and tangents. This learning object is a combination of two objects in the same series.

#### Educational details

Key learning objectives
• Students interpret the gradient of a graph, and the gradient of the tangent to a curve, as being the rate of change of the function.
• Students identify that a linear function produces a straight line graph.
• Students identify that a non-linear function produces a curved graph.
• Students predict that the average and instantaneous velocity of a linear function will always be the same at any given point on the graph.
• Students apply the concept of a limit in the context of the rate of change of a function.
• Students apply the first principles method to differentiate basic polynomial functions.
Educational value
• Demonstrates the connection between average and instantaneous velocity and the rate of change of a linear and non-linear functions.
• Provides opportunities for students to calculate the average and instantaneous velocity of a rocket between different time intervals.
• Features graphs for students to interpret visual representations of average and instantaneous velocities.
• Summarises key statements related to linear and non-linear graphs, and average and instantaneous velocities.
• Allows students to manipulate where the secant intersects the curve in order to establish relationships between secant and gradient tangents and average and instantaneous velocities.
• Displays feedback for correct and incorrect answers.
Year level

11; 12

Learning area
• mathematics
Strand
• Mathematics/Algebra
Student activity
• Interactives;
• Experiment;
• Analysis;
• Modelling;
• Multiple choice questions;
• Data manipulation and interpretation

#### Other details

Contributors
• Script writer
• Educational validator
• Name: Professor Graham Jones
• Technical implementer
• Subject matter expert
• Name: Howard Reeves
• Address: Lindisfarne TAS 7015 Australia
• Educational validator
• Name: Doctor Paul White
• Address: Strathfield NSW 2135 Australia
• Publisher
• Date of contribution: 20 Sep 2013
• Organisation: Education Services Australia
• Address: Melbourne VIC 3000 Australia
• URL: http://www.esa.edu.au/
Access profile
• Device independence
• Hearing independence
Learning resource type
• Interactive Resource
Browsers
• Microsoft Internet Explorer - minimum version: 8.0 (MS Windows) - maximum version: 9.0 (MS-Windows)
• Firefox - minimum version: (MS Windows)
• Safari - minimum version: 5.1 (MacOS)
Operating systems
• MacOS - minimum version: 10.6
• MS-Windows - minimum version: XP - maximum version: 7
Platform requirements
• Initial load time and load time on some screens may exceed 10 seconds due to the use of rich media content.
Rights
• © Education Services Australia Ltd, 2013, except where indicated under Acknowledgements.