# Differential calculus: cubic and quartic functions

TLF ID L7828

Observe the non-linear time graph of a rocket travelling at a changing velocity. The distance, s, travelled by the rocket after t seconds is determined by the formulas: s(t) = t³ – 2 and s(t) = t⁴ + t². Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and look for a pattern. Use your knowledge of limits to derive a formula for finding the instantaneous velocity at a given point. This learning object is a combination of two objects in the same series.

#### Educational details

Key learning objectives
• Students apply the concept of a limit in the context of the rate of change of a non-linear function.
• Students interpret the gradient of the tangent to a curve as being the rate of change of the function.
• Students apply first principles methods to differentiate the functions: s(t) = t³ – 2 and s(t) = t⁴ + t².
• Students interpret the concept of a derivative of a function. They identify that for s(t) = t³ – 2 the derivative is f'(t) = 3t², and that for s(t) = t⁴ + t² the derivative is f'(t) = 4t³ + 2t.
Educational value
• Introduces students to the language, notation and methods of differential calculus.
• Provides opportunities for students to engage in first principles differentiation of a basic polynomial function.
• Tabulates results to help students find patterns in the gradients at different points on the curve.
• Summarises key statements related to derivatives 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

#### 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
• Colour independence
• Device independence
• Hearing independence
• Generic
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
Rights
• © Education Services Australia Ltd, 2013, except where indicated under Acknowledgements.

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